FNCE102 FMI
In this post, I’ll be focusing on the module Financial Markets & Investments (FMI). This post will consist a comprehensive interactive study guide I built to consolidate my understanding of the module.
After completing my Year 2 Semester 2, I've gained a lot of new and well-rounded knowledge. There's no better time to consolidate what I've learned than now. This post covers all 12 lectures: from capital allocation and portfolio theory, to derivatives, bonds, and performance evaluation. It is filled with concept explanations, worked examples, formula references, practice questions, and interactive calculators. Hope this is useful for whoever is trying to grasp an understanding of this module.
Disclaimer: This post reflects my personal experience with the module during the term from January to May 2026, as taught by this specific professor.
FNCE102
Financial Markets
& Instruments
A comprehensive self-study hub. Theory, solving guide, formulas, and exam strategy in one place.
Utility & y*
Markowitz
Single-Index
CAPM
APT / FF3
Bonds
Futures & Hedging
Performance
⚡ Crash Course
The Big Picture — How Everything Connects
Financial Markets & Instruments asks one central question: how should a rational investor price and choose assets? Each lecture adds one layer to the answer.
1. What markets exist?→2. Measure utility→3. How much risk? (CAL → y*)→4. Which assets? (Markowitz)
5. Simplify with index model→6. Price assets: CAPM & APT→7. Are markets efficient?→8. Biases explain anomalies
9. Manage risk with derivatives→10. Invest via funds & bonds→11. Evaluate performance
Lecture 1
Financial Markets
The stage where all investing happens
The financial system channels funds from savers to borrowers. This guide approaches markets from the investor’s perspective: how to price, choose, and manage assets.
Roles of Financial Markets
Informational Role
Stock prices reflect firm prospects. Capital flows to companies with best outlooks. When the market is optimistic, stock price rises — prices aggregate dispersed information.
Consumption Timing
Move purchasing power across time. Use securities to store wealth in high-earning periods; sell in low-earning periods. Separates earning from consumption.
Allocation of Risk
Wide variety of securities lets investors select those matching their risk appetite. This benefits issuers who can then issue at best possible prices.
Separation of Ownership & Management
Enables large-scale enterprise. Agency problems arise when managers pursue own interests. Mitigated by: compensation plans, board monitoring, analyst scrutiny, takeover threats.
Four Market Types
| Market | What Trades | Examples |
|---|---|---|
| Fixed Income | Debt instruments | T-bills, T-bonds, corporate bonds |
| Equity | Ownership stakes | Common & preferred stock |
| Forex | Currencies | Exchange rates, forwards |
| Derivatives | Derived value | Futures & forwards (covered here); Options (separate topic) |
Fixed Income: Money Market vs Capital Market
Money Market (≤1 year)
- Treasury Bills: Government-issued, discount basis. Highly liquid. No coupons — investor earns from discount to par.
T-Bill Price FormulaPrice = 100 − (n × r) / 360
n = days to maturity, r = rate (decimal)
E.g. n=109, r=0.00298: Price = 100 − (109×0.00298)/360 = 99.91
n = days to maturity, r = rate (decimal)
E.g. n=109, r=0.00298: Price = 100 − (109×0.00298)/360 = 99.91
Capital Market (>1 year)
- T-Notes: maturity up to 10yr; T-Bonds: 10–30yr; semiannual coupons; par = $1,000
- Corporate Bonds: higher default risk; secured (collateral) vs debentures (no collateral); callable / convertible
- Mortgage-Backed Securities: proportional ownership of a mortgage pool; securitised
Singapore Bond Market
| Instrument | Tenor | Interest | Min. Inv. | SGX-Tradable? |
|---|---|---|---|---|
| T-Bills | 6 months or 1 year | Discount basis | S$1,000 | Yes |
| SGS Bonds | 2–50 years | Fixed semiannual | S$1,000 | Yes |
| SSBs | Up to 10 years | Fixed step-up semiannual | S$500 | No (redeem via MAS) |
Corporate Bond Types
- Retail Bonds: all investors, min S$1,000
- Wholesale Bonds: institutional & accredited investors only, larger amounts
- Seasoning Bonds: wholesale bonds that meet eligibility criteria, offered to retail after 6 months SGX listing, min S$1,000
Primary Dealers
Approved financial institutions (banks) that facilitate MAS auctions for Singapore Government Securities. They submit competitive bids and distribute bonds to investors, ensuring liquidity in the government bond market.
Equity Securities
Common vs Preferred Stock
| Common | Preferred | |
|---|---|---|
| Ownership & voting | Yes — residual claim | No voting rights |
| Dividends | Variable, discretionary | Fixed (like perpetuity) |
| Priority in liquidation | Last | Before common |
| Liability | Limited | Limited |
Dividend Yield = Annual Dividends / Purchase Price
P/E Ratio = Price per Share / Earnings per Share
Stock Indices
| Index | Method | Notes |
|---|---|---|
| DJIA | Price-weighted | 30 blue-chips; since 1896 |
| S&P 500 | Market-value-weighted | 500 firms; broader |
| STI | Market-cap-weighted | Singapore benchmark |
Forex: Law of One Price
Exchange rates fluctuate due to differences in price levels (inflation) and interest rates across countries.
Law of One Price: If US$1 = JPY100, then a US$2 burger = JPY200 burger. If exchange rate were US$1 = JPY200 instead, the Japanese burger costs US$1 — cheaper than the US burger. Demand shifts to Japanese goods, supply of US goods rises, and exchange rate converges back to US$1 = JPY100.
If Japanese prices then rise to JPY250: New equilibrium = US$1 = JPY125. Japanese inflation → JPY depreciates, US$ appreciates.
If Japanese prices then rise to JPY250: New equilibrium = US$1 = JPY125. Japanese inflation → JPY depreciates, US$ appreciates.
ERSG — Ethics, Social Responsibility, Sustainability & Governance
ERSG trends are reshaping financial markets and investment decisions. Knowing them is increasingly essential for investors.
Components
- Ethics: Moral principles guiding unbiased, non-prejudiced decisions
- Social Responsibility: Entities responsible for societal benefit, not just individual gain
- Sustainability — Environmental: Preserve environment for future generations; anti-pollution, climate risk
- Sustainability — Business: Resilience in face of disruptions (Covid-19, geopolitical risks, tech changes)
Corporate Governance Principles
- Transparency
- Responsibility
- Accountability
- Fairness
- Risk Management
Concern: conflict between shareholders’ and managers’ interests. Corporate governance rules help ensure markets are efficient avenues for capital allocation.
Concept Check
A Singapore investor wants liquidity before maturity. They hold an SGS bond (5-year) and a Singapore Savings Bond. Which can they sell on SGX?
AOnly the SGS bond
BOnly the SSB
CBoth are SGX-tradable
DNeither can be traded before maturity
Lecture 2
Capital Allocation to Risky Assets
How much of your wealth into risky vs safe assets?
The central question: given risky portfolio P and risk-free asset F, what fraction y to invest in P? The answer depends on your risk aversion coefficient A.
Risk Aversion & Utility
Utility Score (KEY)U = E(r) − (1/2) × A × sigma²
A > 0 → Risk-averse (penalises risk) ← normal
A = 0 → Risk-neutral (ignores risk)
A < 0 → Risk-lover (enjoys risk)
Risk-free: U = rf (sigma = 0, no penalty)
A > 0 → Risk-averse (penalises risk) ← normal
A = 0 → Risk-neutral (ignores risk)
A < 0 → Risk-lover (enjoys risk)
Risk-free: U = rf (sigma = 0, no penalty)
Higher expected returns ↑ utility. Higher risk ↓ utility. The more risk-averse (higher A), the greater the penalty for taking on risk. Investors choose the portfolio with the highest utility score.
Speculation vs Gambling
| Speculation | Gambling | |
|---|---|---|
| Risk premium | Positive | Zero (fair game) |
| Expected profit | Positive | Zero |
| Risk-averse accept? | Yes | No |
A coin flip = gambling (p=0.5 each side, expected profit = 0). But if you believe p > 0.5 (heterogeneous expectations), it looks like speculation to you.
sigma² trap: if sigma = 20%, then sigma² = (0.20)² = 0.04. Never use 20 or 0.20 directly in the formula.
Mean-Variance Criterion
Portfolio A dominates Portfolio B if E(rA) ≥ E(rB) and sigmaA ≤ sigmaB, with at least one strict inequality.
- Quadrant IV portfolios (lower E(r), higher sigma than P): dominated by P — always avoid.
- Quadrant I portfolios (higher E(r), lower sigma than P): dominate P — always prefer.
- Quadrants II & III: neither clearly dominates. Use indifference curves to decide. Portfolio must lie northwest of P to be superior.
Measuring Risk Aversion (KYC)
Know-Your-Client (KYC)
Financial advisers gauge risk aversion by:
- Questionnaires — score various risk scenarios
- Observing decisions when confronted with risk
- Observing how much people are willing to pay to avoid risk
Capital Allocation Line (CAL)
Mix risky portfolio P (proportion y) with risk-free F (proportion 1−y).
Complete PortfolioE(rC) = rf + y × [E(rP) − rf]
sigmaC = y × sigmaP
E.g. rf=7%, E(rP)=15%, sigmaP=22%:
y=0.5: E(rC)=11%, sigmaC=11%
y=1.4 (leverage): E(rC)=18.2%, sigmaC=30.8%
sigmaC = y × sigmaP
E.g. rf=7%, E(rP)=15%, sigmaP=22%:
y=0.5: E(rC)=11%, sigmaC=11%
y=1.4 (leverage): E(rC)=18.2%, sigmaC=30.8%
CAL Slope = Sharpe RatioSharpe = [E(rP) − rf] / sigmaP
= reward per unit of total risk
ALL points on the CAL share the same Sharpe ratio — leverage does not improve Sharpe, only scales it.
CML = special CAL using the market portfolio.
= reward per unit of total risk
ALL points on the CAL share the same Sharpe ratio — leverage does not improve Sharpe, only scales it.
CML = special CAL using the market portfolio.
Example: Portfolio Z (Asset Allocation)
Total portfolio Z = $300,000. Currently: 30% in risk-free F ($90,000), 70% in risky P ($210,000). Within P: 54% stocks ($113,400), 46% bonds ($96,600).
Key rule: when shifting between F and P, the internal weights of P remain unchanged. Only the overall split between F and P changes. E.g. shifting from y=70% to y=56%: each stock/bond holding scales down proportionally within P.
Optimal Allocation y*
Optimal Weight in Risky Asset (KEY)y* = [E(rP) − rf] / [A × sigmaP²]
y* ↑ when: risk premium ↑, A ↓, sigmaP ↓
y* > 1 → Leverage (borrow to invest >100% in P)
y* < 0 → Short the risky portfolio
E.g. rf=7%, E(rP)=15%, sigmaP=22%, A=4:
y* = (15%−7%) / [4 × (0.22)²] = 0.08/0.1936 = 0.413
y* ↑ when: risk premium ↑, A ↓, sigmaP ↓
y* > 1 → Leverage (borrow to invest >100% in P)
y* < 0 → Short the risky portfolio
E.g. rf=7%, E(rP)=15%, sigmaP=22%, A=4:
y* = (15%−7%) / [4 × (0.22)²] = 0.08/0.1936 = 0.413
Indifference Curves
An indifference curve connects all (E(r), sigma) pairs giving the same utility score. To build one:
- Start with risk-free: U = rf (sigma = 0)
- Increase sigma. Solve for E(r) that maintains same U: E(r) = U + ½Aσ²
- Repeat for all sigma values → plot pairs
Steeper indifference curve = more risk averse (investor needs more return compensation per unit of added risk). Optimal complete portfolio = tangency point between highest reachable indifference curve and the CAL.
If borrowing rate rb > lending rate rf (kinked CAL): use rb when y > 1. E.g. new Sharpe = [E(rP)−rb] / sigmaP < lending Sharpe. Two separate line segments with different slopes.
Concept Check
A=4, rf=3%, E(rP)=11%, sigmaP=20%. What is optimal y*?
Ay* = 1.00
By* = 0.75
Cy* = 0.50
Dy* = 0.25
Lecture 3
Optimal Risky Portfolio
Markowitz diversification — build the best risky portfolio
We go inside portfolio P and ask: how to combine assets D (bonds) and E (equities) to form the best possible P? The goal: efficient diversification — lowest risk for any given expected return.
Two Types of Risk
Systematic (Market) Risk
Marketwide risk sources. Remains even after diversification. Also called: non-diversifiable. Eg. GDP, interest rates, global financial crises. Investors are compensated with the market risk premium.
Firm-Specific (Idiosyncratic) Risk
Risk unique to a firm or industry. Eliminated by holding many uncorrelated securities. Also called: diversifiable or nonsystematic. No compensation — investors can diversify it away for free.
Two-Asset Portfolio Formulas
Expected ReturnE(rP) = wD × E(rD) + wE × E(rE)
Unaffected by correlation — always weighted avg.
Unaffected by correlation — always weighted avg.
Portfolio VariancesigmaP² = wD²sigmaD² + wE²sigmaE²
+ 2×wD×wE×Cov(rD,rE)
Cov(rD,rE) = rho × sigmaD × sigmaE
+ 2×wD×wE×Cov(rD,rE)
Cov(rD,rE) = rho × sigmaD × sigmaE
Effect of Correlation
| Correlation rho | Portfolio SD | Benefit |
|---|---|---|
| +1 | = weighted avg of SDs | Zero — no diversification |
| 0 to +1 | < weighted avg | Partial benefit |
| 0 | Significantly lower | Good benefit |
| −1 | Can reach zero! | Maximum benefit (perfect hedge) |
Perfect Hedge (rho=−1)Set sigmaP = 0: wD = sigmaE/(sigmaD+sigmaE), wE = sigmaD/(sigmaD+sigmaE)
E.g. sigmaD=12%, sigmaE=20%: wD=20/32=62.5%, wE=12/32=37.5% → sigmaP=0
E.g. sigmaD=12%, sigmaE=20%: wD=20/32=62.5%, wE=12/32=37.5% → sigmaP=0
Good to add assets with LOW or NEGATIVE correlation to an existing portfolio. This reduces total risk without necessarily reducing E(r).
Minimum Variance & Tangency Portfolio
Minimum Variance WeightwD(min) = [sigmaE² − Cov] / [sigmaD² + sigmaE² − 2Cov]
wE = 1 − wD
Minimises sigma. Does NOT maximise Sharpe ratio.
wE = 1 − wD
Minimises sigma. Does NOT maximise Sharpe ratio.
Tangency Portfolio (Max Sharpe)Use EXCESS returns: E(RD) = E(rD) − rf
wD = [E(RD)×sigmaE² − E(RE)×Cov] / [E(RD)×sigmaE² + E(RE)×sigmaD² − (E(RD)+E(RE))×Cov]
wD = [E(RD)×sigmaE² − E(RE)×Cov] / [E(RD)×sigmaE² + E(RE)×sigmaD² − (E(RD)+E(RE))×Cov]
Tangency portfolio formula uses EXCESS returns E(RD) = E(rD) − rf. Using total returns gives wrong weights.
Separation Property
The optimal risky portfolio P is the same for ALL investors regardless of risk aversion. Only y* differs. One mutual fund can serve many clients — everyone holds P in different proportions combined with the risk-free asset.
Concept Check
sigmaD=15%, sigmaE=20%, rho=0. What is minimum variance weight wD?
AwD = 0.36
BwD = 0.47
CwD = 0.64
DwD = 0.75
Lecture 4
Single-Index Model
One market factor drives all covariances
Markowitz needs n² inputs. For 50 stocks: 1,325 estimates. For 200 stocks: 20,099 estimates! The SIM reduces this drastically by using the market index as the single common factor.
Core Equations
Single-Factor Market ModelRi(t) = alpha_i + beta_i × RM(t) + ei(t)
Ri = excess return of security i (= ri − rf)
RM = excess return of market index
alpha_i = nonmarket expected excess return (stock-specific premium)
beta_i = systematic sensitivity to market; uncorrelated with ei
ei = firm-specific residual (mean=0, uncorrelated with market)
Ri = excess return of security i (= ri − rf)
RM = excess return of market index
alpha_i = nonmarket expected excess return (stock-specific premium)
beta_i = systematic sensitivity to market; uncorrelated with ei
ei = firm-specific residual (mean=0, uncorrelated with market)
Variance Decomposition
Total Variance of Stock isigma_i² = beta_i² × sigmaM² + sigma²(ei)
Systematic: beta_i² × sigmaM² ← non-diversifiable
Firm-specific: sigma²(ei) ← diversifiable
R² = beta_i²×sigmaM² / sigma_i² (systematic proportion)
Systematic: beta_i² × sigmaM² ← non-diversifiable
Firm-specific: sigma²(ei) ← diversifiable
R² = beta_i²×sigmaM² / sigma_i² (systematic proportion)
Covariance Between Two StocksCov(ri, rj) = beta_i × beta_j × sigmaM²
ALL covariance comes ONLY from the common market factor. Firm-specific residuals are mutually uncorrelated — this is the key simplification of the SIM.
ALL covariance comes ONLY from the common market factor. Firm-specific residuals are mutually uncorrelated — this is the key simplification of the SIM.
Diversification Under SIM
For an equally-weighted portfolio of n stocks: portfolio’s excess return RP = alphaP + betaP × RM + eP
Portfolio VariancesigmaP² = betaP² × sigmaM² + sigma²(eP)
Systematic (betaP²×sigmaM²): persists regardless of n
→ Non-diversifiable
Systematic (betaP²×sigmaM²): persists regardless of n
→ Non-diversifiable
Firm-Specific Variancesigma²(eP) = sigma²(e_avg) / n
As n → ∞: sigma²(eP) → 0
→ Diversifiable — cancels out!
As n → ∞: sigma²(eP) → 0
→ Diversifiable — cancels out!
Beta, Adjusted Beta & Alpha
| Beta | Meaning | Examples |
|---|---|---|
| beta = 1 | Moves with market | Index fund |
| beta > 1 | Amplifies market (cyclical) | Ford ~1.33 |
| beta < 1 | Defensive, less sensitive | Utilities, staples |
| beta = 0 | No systematic risk | T-bills |
Adjusted Beta (Blume) — Forward-LookingAdj beta = (2/3) × Historical beta + (1/3) × 1
Betas revert toward 1 over time.
E.g. Historical beta = 1.9:
Adj beta = 2/3×1.9 + 1/3×1.0 = 1.6
Use E(r) = rf + 1.6×[E(rM)−rf]
Betas revert toward 1 over time.
E.g. Historical beta = 1.9:
Adj beta = 2/3×1.9 + 1/3×1.0 = 1.6
Use E(r) = rf + 1.6×[E(rM)−rf]
Always use Adjusted Beta when asked for "expected future beta" or "forward-looking beta".
Expected Return-Beta Relationship
Taking expected values of Ri = alpha_i + beta_i × RM + ei:
E(Ri) = alpha_i + beta_i × E(RM)Security risk premium has 2 parts:
1. alpha_i = nonmarket premium (stock-specific; >0 = underpriced, <0 = overpriced)
2. beta_i × E(RM) = systematic risk premium = beta × market risk premium
1. alpha_i = nonmarket premium (stock-specific; >0 = underpriced, <0 = overpriced)
2. beta_i × E(RM) = systematic risk premium = beta × market risk premium
alpha > 0: superior manager/mispriced stock → BUY. alpha < 0: underperformer → SELL. CAPM assumes alpha = 0 in equilibrium.
Concept Check
Stock A: beta=1.2, sigmaM=25%, sigma(eA)=18%. Total variance sigma_A²?
A0.0900
B0.1224
C0.0624
D0.0300
Lecture 5
CAPM & Security Market Line
The centrepiece of modern finance
CAPM adds the assumption of homogeneous expectations: all investors agree on return distributions. They all choose the same optimal risky portfolio — the market portfolio M. So CAL = CML for all investors.
CAPM / SML Equation (KEY)E(ri) = rf + beta_i × [E(rM) − rf]
rf = risk-free rate, beta_i = systematic risk
[E(rM)−rf] = market risk premium
Applies to individual assets AND portfolios.
Portfolio beta = betaP = ∑(wi × beta_i)
rf = risk-free rate, beta_i = systematic risk
[E(rM)−rf] = market risk premium
Applies to individual assets AND portfolios.
Portfolio beta = betaP = ∑(wi × beta_i)
SML vs CML
| SML | CML | |
|---|---|---|
| x-axis | Beta (systematic risk only) | Sigma (total risk) |
| Applies to | ANY asset or portfolio | Efficient portfolios only |
| Use for | Pricing: is it fair value? | Optimal portfolio construction |
| Slope | Market risk premium [E(rM)−rf] | Sharpe ratio of market |
SML x-axis = beta. CML x-axis = sigma. SML applies to ANY asset (diversified or not). CML only applies to efficient (fully diversified) portfolios. Never confuse them.
Alpha, Mispricing & Practical Use of CAPM
Jensen’s Alphaalpha = Actual return − CAPM predicted
= ri − [rf + beta_i(rM−rf)]
alpha > 0 → above SML → underpriced → BUY
alpha < 0 → below SML → overpriced → SELL
alpha = 0 → on SML → fairly priced
= ri − [rf + beta_i(rM−rf)]
alpha > 0 → above SML → underpriced → BUY
alpha < 0 → below SML → overpriced → SELL
alpha = 0 → on SML → fairly priced
CAPM Worked Example
Worked Example:
Market return = 14%, Stock A beta = 1.2, T-bill rate = 6%
CAPM: 6% + 1.2(14%−6%) = 15.6%
If actual return = 17%: alpha = 17%−15.6% = +1.4% → underpriced → BUY
Practical use:
Firm beta = 0.6, mkt premium = 8%, rf = 6%
Fair return = 6% + 0.6(8%) = 10.8%
Firm must price products to earn $10.8M per $100M invested.
Portfolio Beta Worked Example
25% Toyota (beta=1.10) + 75% Ford (beta=1.25):
betaP = 0.25×1.10 + 0.75×1.25 = 0.275 + 0.9375 = 1.2125
Portfolio risk premium = 1.2125 × 8% = 9.7%
Concept Check
rf=3%, E(rM)=10%, beta=1.4. Actual expected return=13.2%. Alpha and verdict?
Aalpha=+1.0%, overpriced
Balpha=+0.4%, underpriced
Calpha=−0.4%, overpriced
Dalpha=0, fairly priced
Lecture 6
APT & Multifactor Models
Multiple sources of systematic risk
CAPM uses one factor (market). But systematic risks come from many sources. APT uses only no-arbitrage (not full equilibrium) to derive expected returns.
Arbitrage Pricing Theory
Law of One Price & No Arbitrage
Arbitrage = riskless profit with zero net investment. Example: same stock at NYSE $165, NASDAQ $163 → buy at $163, sell at $165, riskless $2 profit. Arbitrageurs eliminate such gaps.
In well-diversified portfolios, eP → 0, so: RP = E(RP) + betaP × F. Any two well-diversified portfolios with the same beta must have the same expected return — otherwise arbitrage exists.
Arbitrage ExamplePortfolio A: beta=1.0, E(r)=10%. Portfolio B: beta=1.0, E(r)=8%.
→ Long $1M in A, Short $1M in B:
Profit = (10%+1F)×$1M − (8%+1F)×$1M = $20,000 riskless!
→ Arbitrage pressure forces prices to converge.
→ Long $1M in A, Short $1M in B:
Profit = (10%+1F)×$1M − (8%+1F)×$1M = $20,000 riskless!
→ Arbitrage pressure forces prices to converge.
Two-Factor Model
Two common macro risk factors: (1) unanticipated GDP growth, (2) unanticipated interest rate changes.
Two-Factor ModelRi = E(Ri) + beta_GDP × GDP_surprise + beta_IR × IR_surprise + ei
Power Utility Company
Low beta_GDP (stable cashflows, not GDP-sensitive). High beta_IR (cashflows like bonds, sensitive to rates). News of rising GDP & rates = BAD (rate rise dominates).
Airline Company
High beta_GDP (more passengers when economy grows). Low beta_IR. News of rising GDP & rates = GOOD (GDP rise dominates).
Multifactor APT Numerical Example
Multi-Factor APT Expected ReturnE(ri) = rf + beta_1[E(r1)−rf] + beta_2[E(r2)−rf] + ...
Factor Portfolio 1: E(r)=10%, rf=4%, premium=6%, betaA1=0.5 → contribution=3%
Factor Portfolio 2: E(r)=12%, rf=4%, premium=8%, betaA2=0.75 → contribution=6%
Overall: E(rA) = 4% + 3% + 6% = 13%
If Portfolio B has same betas but E(rB)=12%: Arbitrage! Long A, Short B → riskless $0.01 per dollar.
Factor Portfolio 1: E(r)=10%, rf=4%, premium=6%, betaA1=0.5 → contribution=3%
Factor Portfolio 2: E(r)=12%, rf=4%, premium=8%, betaA2=0.75 → contribution=6%
Overall: E(rA) = 4% + 3% + 6% = 13%
If Portfolio B has same betas but E(rB)=12%: Arbitrage! Long A, Short B → riskless $0.01 per dollar.
Fama-French Three-Factor Model
FF3 ModelE(ri)−rf = a_i + b_i[E(rM)−rf] + s_i×E(RSMB) + h_i×E(RHML)
SMB = Small Minus Big (size factor): Long small-cap, Short big-cap, zero net investment
HML = High Minus Low B/M (value factor): Long high B/M (value), Short low B/M (growth)
If model correct: a_i = 0 (intercept = zero)
SMB = Small Minus Big (size factor): Long small-cap, Short big-cap, zero net investment
HML = High Minus Low B/M (value factor): Long high B/M (value), Short low B/M (growth)
If model correct: a_i = 0 (intercept = zero)
How FF3 Factors Are Constructed
SMB Construction
- Use NYSE median size to split ALL US stocks (NYSE, NASDAQ, AMEX) into BIG and SMALL
- Create 2 value-weighted portfolios
- RSMB = Return(small portfolio) − Return(big portfolio)
- s_i > 0 → small-cap tilt. s_i < 0 → large-cap tilt.
HML Construction
- Sort all US stocks into 3 B/M groups: bottom 30% (low), middle 40%, top 30% (high)
- RHML = Return(high B/M) − Return(low B/M)
- h_i > 0 → value tilt (high B/M). h_i < 0 → growth tilt (low B/M).
FF3 interprets size and value premiums as extra risk factors, NOT alpha. CAPM wrongly labels them as alpha. If FF3 is correctly specified, a_i = 0. A positive a_i in FF3 = genuine skill or mispricing BEYOND the 3 factors.
CAPM vs FF3
FF3 shows much better fit than CAPM for actual cross-sectional returns. When 25 size/B⁄M-sorted US portfolios are plotted: FF3 predictions cluster along the 45° line (predicted = actual), while CAPM shows large deviations. Smaller firms and high-B/M firms earn risk premiums that CAPM calls “alpha” but FF3 correctly attributes to size and value risk factors.
Lecture 7
EMH & Asset Pricing Anomalies
Markets are mostly efficient — except when they’re not
Maurice Kendall (1953): stock prices show no predictable pattern. Prices follow a random walk because any forecast of future price rises leads to immediate price rises, eliminating the opportunity.
Efficient Market Hypothesis
| Form | Prices Reflect | Implication |
|---|---|---|
| Weak | All past prices & trading data | Technical analysis cannot generate consistent alpha |
| Semi-Strong | All public information | Fundamental analysis cannot generate consistent alpha |
| Strong | All info including insider | Even insiders cannot consistently earn alpha |
Strong-form implies semi-strong which implies weak form. Evidence generally supports weak and semi-strong; strong-form is violated by insider trading regulations.
Event Studies & Cumulative Abnormal Returns (CAR)
Market Model (Benchmark Return)ri(t) = alpha_i + beta_i × rM(t) + ei(t)
ei(t) = ri(t) − [alpha_i + beta_i×rM(t)] = unexpected return due to event
E.g. alpha=0.05%, beta=0.8, market rises 1%:
Predicted: 0.05% + (0.8×1%) = 0.85%
Actual = 2%: Abnormal return = 2% − 0.85% = +1.15%
ei(t) = ri(t) − [alpha_i + beta_i×rM(t)] = unexpected return due to event
E.g. alpha=0.05%, beta=0.8, market rises 1%:
Predicted: 0.05% + (0.8×1%) = 0.85%
Actual = 2%: Abnormal return = 2% − 0.85% = +1.15%
CAR = Cumulative Abnormal Return: sum of all abnormal returns over the event window. Captures total price reaction including pre-announcement leakage. Example: takeover announcement — prices drift up 30 days before (information leakage), then jump sharply on Day 0, then plateau. Rapid post-announcement plateau = semistrong efficiency.
Technical vs Fundamental Analysis
Technical Analysis
- Believes historical prices predict future prices
- Identifies trends using past data
- Implies: No weak-form efficiency
- Moving Average: 200-day or 10-day MA. Bullish signal: price breaks above MA from below. Bearish signal: price falls below MA.
- Breadth: spread between advancing and declining stocks. Positive breadth → bullish; negative → bearish.
- Resistance levels: prices historically bounce off old highs (sellers rush to break even).
Technical strategies that become widely known may become self-destructing — traders exploit them until the opportunity is gone.
Fundamental Analysis
- Uses economic and accounting information to predict stock prices
- Investigates financial statements, management quality, competitive position
- Calculates fair values and compares to actual prices
- Implies: No semi-strong efficiency
- Only useful if analyst can find mispricing before it is reflected in prices
Active vs Passive Management
Active Management
Attempts to identify mispriced securities. Only worthwhile if the market is not fully efficient or if the manager has superior information/skill. Requires beating the market net of fees.
Passive Management
Accept market prices as fair. Hold index funds. Lower cost. Consistent with EMH. Evidence shows most active managers underperform passive strategies after fees.
Key Anomalies
Fundamental Anomalies
- P/E effect: Low P/E stocks outperform high P/E stocks
- Book-to-market: High B/M (value) stocks outperform
- Small-firm effect: Small caps earn higher risk-adjusted returns
- Neglected-firm effect: Less-followed firms earn more (less analyst coverage)
Technical Anomalies
- January effect: Small stocks outperform in January (tax-loss selling in Dec)
- Momentum: Past winners continue short-term (3–12 months)
- Reversal: Past winners underperform long-term (3–5 years)
- Post-earnings-announcement drift (PEAD): Prices continue drifting after surprise
Anomalies persist because: (1) genuine risk factors (FF3 explanation), (2) transaction costs and short-selling constraints prevent arbitrage, or (3) behavioural biases delay correction.
Lecture 8
Behavioural Finance
Why rational models don’t fully explain reality
Two-part critique: (1) Investors have information processing errors and behavioural biases. (2) Even if prices are wrong, limits to arbitrage prevent rational investors from fully correcting them.
Part 1A: Information Processing Errors
Forecasting Errors
- Anchoring: Too much weight on the first piece of information. Creates an “anchor” that is hard to move away from.
- Overconfidence: Overestimate precision of own forecasts. Ranked ownself better than others.
- Confirmation bias: Only notice information that ties with existing beliefs.
- Conservatism: Slow to update beliefs on new info → underreaction → momentum in stock returns.
- Representativeness / Sample Size Neglect: Infer patterns from small samples; extrapolate too far. When corrected → price reversals.
Behavioural Biases
- Framing: Decisions depend on how choices are presented. “Risky gain” vs “risky loss” framing — same outcome, different psychological reaction.
- Mental Accounting: Treat funds differently based on origin/purpose. House money effect: take more risks with gains than principal.
- Regret Avoidance: Refuse to accept a bad decision. Hold losers hoping for recovery. Attribute losses to bad luck, not bad decisions.
- Affect: Feeling of good/bad about an asset. Socially responsible or popular products get bid up.
Prospect Theory
Conventional View (CAPM-style)
Utility depends on level of wealth. Higher wealth = higher utility at diminishing rate (concave curve). Implies consistent risk aversion.
Behavioural View (Prospect Theory)
Utility depends on changes in wealth from a reference point. Losses pane: convex (very sensitive to even small losses). Gains pane: concave (less sensitive to gains).
Loss aversion: A $1,000 gain increases utility LESS than a $1,000 loss decreases it. This gives rise to risk aversion around the reference point.
Prospect Theory (Kahneman & Tversky, 1979): investors don’t evaluate outcomes in absolute terms but relative to a reference point. This explains why investors hold losers too long (hoping to recover to their purchase price) and sell winners too early.
Part 2: Limits to Arbitrage
1. Fundamental Risk
“Markets can remain irrational longer than you can remain solvent.” (Keynes). Intrinsic value and market value may take too long to converge. E.g. short an index at $1,000 thinking it is overpriced vs normal $500; it continues to $2,000 before reverting — you lose before being right.
2. Implementation Costs
High transaction costs and restrictions on short-selling limit arbitrage activity. Net profit after costs may be zero or negative.
3. Model Risk
What if your pricing model is wrong and the market price is actually correct? You’ve made a wrong estimate of fair value — no mispricing exists, yet you’ve taken a position.
Bubbles & Behavioural Biases
Bubbles: easier to spot after they end. Prices feed on themselves:
- Investors become overconfident → rapid purchases based on expected continuing price rises
- Representativeness bias: extrapolate short-term observations too far into the future
- Even a small change in growth rate assumptions causes massive valuation changes
Bubble Valuation Example (Slides)S&P 500 dividends = $154.6M, k = 9.2%
If g = 8.0%: PV = $154.6M / (9.2%−8.0%) = $12,883M
If g = 7.4%: PV = $154.6M / (9.2%−7.4%) = $8,589M
→ Small 0.6% change in g assumption = 33% difference in valuation!
If g = 8.0%: PV = $154.6M / (9.2%−8.0%) = $12,883M
If g = 7.4%: PV = $154.6M / (9.2%−7.4%) = $8,589M
→ Small 0.6% change in g assumption = 33% difference in valuation!
Famous bubbles: Tulip Mania, Internet Dot-com Bubble, Housing Bubble (2008 GFC).
Lecture 9
Derivatives & Risk Management
Risk types, futures mechanics, FX & interest rate hedging
Risk management process: (1) Risk assessment, (2) Decide whether to hedge, (3) Set risk appetite/policy, (4) Monitor ongoing exposures.
Types of Risk
Financial Risks
- Market risk (systematic): Equity price, interest rate, FX rate, commodity price
- Credit risk: Risk of non-payment. Assessed by credit rating agencies. Higher credit risk → higher yield.
- Liquidity risk: Cannot sell without significant price discount
Non-Financial Risks
- Operational: System failures, human error, disasters, concealed trading
- Model risk: Wrong valuation models or fair value calculations
- Settlement/Counterparty: Delay or failure; cash flow problems
- Regulatory: Unexpected changes in laws/regulations
- Accounting/Tax: Changes in accounting rules or tax policies
- Sovereign/Political: Country risk, political instability
Futures Mechanics
Futures Contract BasicsAgreement to buy/sell an asset at a set price on a future date. Standardised: quantity, quality, delivery.
Profit to Long = PT − F0
Profit to Short = F0 − PT
Zero-sum game. At maturity: FT = PT (convergence). Most traders close before delivery.
Profit to Long = PT − F0
Profit to Short = F0 − PT
Zero-sum game. At maturity: FT = PT (convergence). Most traders close before delivery.
Mark-to-Market & MarginGains/losses realised daily via margin account.
IMR (Initial Margin): deposit to open position. E.g. T-Bond: $2,530
MMR (Maintenance Margin): minimum to keep open. E.g. $2,300
If balance falls below MMR → MARGIN CALL → must top up to IMR (not just MMR).
Broker can close position if margin not maintained.
IMR (Initial Margin): deposit to open position. E.g. T-Bond: $2,530
MMR (Maintenance Margin): minimum to keep open. E.g. $2,300
If balance falls below MMR → MARGIN CALL → must top up to IMR (not just MMR).
Broker can close position if margin not maintained.
Clearinghouse & Convergence Property
Clearinghouse
Acts as third party to all futures contracts — buyer to every seller, seller to every buyer. Guarantees all trades. Removes counterparty risk from individual traders.
Convergence Property
At maturity, futures price must equal spot price (FT = PT), otherwise arbitrage arises. As maturity approaches, (1+r+c−d−v) → 1, so futures converges to spot.
Futures Pricing
Futures Parity FormulaF = S × (1 + r + c − d − v)^t
r = risk-free rate (opportunity cost of money)
c = storage/carrying cost (for physical commodities)
d = dividend/convenience yield (benefit of owning underlying directly)
v = convenience yield (benefit of physical ownership)
Contango: F > S (usual for commodities with storage costs)
Backwardation: F < S (when convenience yield dominates)
r = risk-free rate (opportunity cost of money)
c = storage/carrying cost (for physical commodities)
d = dividend/convenience yield (benefit of owning underlying directly)
v = convenience yield (benefit of physical ownership)
Contango: F > S (usual for commodities with storage costs)
Backwardation: F < S (when convenience yield dominates)
Basis Risk
Basis = Futures price − Spot price. At maturity: basis = 0 (convergence). If hedge is unwound BEFORE maturity, basis may have changed unexpectedly → basis risk.
Example: Investor holds stock worth $50, needs to sell in 6 months. Hedges by shorting futures at F0=$50. If unwinds early when spot=$45 and futures=$47 (not yet converged): Net = $45 + ($50−$47) = $48 (not the expected $50). The $2 gap is basis risk.
Forwards vs Futures
| Futures | Forwards | |
|---|---|---|
| Traded | Exchange (standardised) | OTC (customised) |
| Mark-to-market | Daily | Not |
| Counterparty risk | Low (clearinghouse) | Higher (bilateral) |
| Flexibility | Less (standard contracts) | More (any size/date) |
| Most common type | T-bonds, currencies, indices | Foreign exchange forwards |
Long vs Short — The Core Hedging Rule
Identifying Your PositionLONG: You are happy if the price of "something" goes UP.
SHORT: You are happy if the price goes DOWN.
To hedge: take the OPPOSITE futures position to your spot position.
Own bonds (long) → fear rate rise → SHORT bond futures
Will receive JPY (long JPY) → fear JPY fall → SHORT JPY futures
Need to buy oil (short oil) → fear price rise → LONG oil futures
Car owner → LONG oil (fear price rise reduces driving value) → LONG oil futures?
Farmer who grows wheat → LONG wheat → SHORT wheat futures
SHORT: You are happy if the price goes DOWN.
To hedge: take the OPPOSITE futures position to your spot position.
Own bonds (long) → fear rate rise → SHORT bond futures
Will receive JPY (long JPY) → fear JPY fall → SHORT JPY futures
Need to buy oil (short oil) → fear price rise → LONG oil futures
Car owner → LONG oil (fear price rise reduces driving value) → LONG oil futures?
Farmer who grows wheat → LONG wheat → SHORT wheat futures
FX Futures Hedging
FX Hedge ExampleSingapore exporter receives JPY. Current futures: S$0.012/JPY. JPY depreciates 5% to S$0.0114.
Profits from short futures = $0.012−$0.0114 = $0.0006/JPY.
If business loss = S$600 per S$0.0006 depreciation:
Amount needed = S$600/S$0.0006 = JPY 1,000,000
Contract size = JPY 100,000
Contracts = JPY 1,000,000 / JPY 100,000 = 10 SHORT contracts
Profits from short futures = $0.012−$0.0114 = $0.0006/JPY.
If business loss = S$600 per S$0.0006 depreciation:
Amount needed = S$600/S$0.0006 = JPY 1,000,000
Contract size = JPY 100,000
Contracts = JPY 1,000,000 / JPY 100,000 = 10 SHORT contracts
Interest Rate ParityF0 = E0 × [(1+rA) / (1+rB)]^T
E.g. E0=$2/£, rUS=4%, rUK=5%:
F0 = $2 × (1.04/1.05) = $1.981/£
Even though UK rate is higher, US$ appreciates in forward market — the “forward premium” offsets the higher UK rate. No free lunch.
E.g. E0=$2/£, rUS=4%, rUK=5%:
F0 = $2 × (1.04/1.05) = $1.981/£
Even though UK rate is higher, US$ appreciates in forward market — the “forward premium” offsets the higher UK rate. No free lunch.
Interest Rate Futures Hedging
PVBP of PortfolioPVBP = Portfolio Value × D* × 0.0001
E.g. Portfolio = $10M, D* = 9yr:
If rates rise 10bp: loss = $10M × 9 × 0.10% = $90,000
PVBP = $90,000 / 10bp = $9,000 per bp
E.g. Portfolio = $10M, D* = 9yr:
If rates rise 10bp: loss = $10M × 9 × 0.10% = $90,000
PVBP = $90,000 / 10bp = $9,000 per bp
Number of Futures ContractsT-Bond futures: par=$100,000, price=$90/$100, D*=10yr
Contract value = $90,000
Loss per contract per 10bp = $90,000×10×0.10% = $900
PVBP per contract = $900/10 = $90/bp
Contracts = PVBP_portfolio / PVBP_futures
= $9,000 / $90 = 100 contracts SHORT
Contract value = $90,000
Loss per contract per 10bp = $90,000×10×0.10% = $900
PVBP per contract = $900/10 = $90/bp
Contracts = PVBP_portfolio / PVBP_futures
= $9,000 / $90 = 100 contracts SHORT
Costs of Hedging
- Fees charged by financial institutions for structuring hedging strategies
- Opportunity cost: forgoing other attractive investments
- Initial margin deposit at outset of futures contract
- Benefit: removes downside risks, provides stable returns
- Key trade-off: hedging removes downside but also removes upside
Concept Check
Singapore exporter will receive USD 500,000 in 3 months. Each USD futures contract = USD 100,000. To hedge FX risk, the exporter should:
ABuy 5 USD futures (long)
BSell 5 USD futures (short)
CBuy 10 USD futures (long)
DNo action needed
Lecture 10
Mutual Funds & Hedge Funds
Pooled investment vehicles and their strategies
Investment Companies — Why They Exist
- Record keeping & administration: Track dividends, capital gains, redemptions
- Diversification & divisibility: Small investors access wide variety of securities
- Professional management: Full-time staff monitor portfolios
- Lower transaction costs: Large-block trading reduces brokerage fees
NAV & Fund Types
NAV FormulaNAV = (Market value of assets − Liabilities) / Shares outstanding
E.g. Portfolio = $120M, Liabilities = $5M, Shares = 5M:
NAV = ($120M − $5M) / 5M = $23 per share
E.g. Portfolio = $120M, Liabilities = $5M, Shares = 5M:
NAV = ($120M − $5M) / 5M = $23 per share
Open-End Fund
- Issues shares when investors buy; redeems at NAV when they sell
- Priced at NAV at end of day
- Does NOT trade on exchanges
- Always possible to buy/sell at fair value
Closed-End Fund
- Fixed number of shares after initial offering
- Trades on exchanges like stocks — price set by supply/demand
- Can trade at premium or discount to NAV
- Must sell to another investor (not redeem from fund)
Mutual Fund Investment Policies
| Type | What It Invests In |
|---|---|
| Money Market | Short-maturity instruments: commercial paper, repo, CDs |
| Equity | Primarily stocks |
| Sector | Equity concentrated in one industry |
| Bond | Fixed-income securities |
| Index | Matches performance of a market index; holds exact index proportions |
Fee Structure & ETFs
Mutual Fund Costs
- Operating expenses: 0.2%–2% of AUM annually
- Front-end load: up to 6% paid when buying
- Back-end load: up to 6% paid when selling
- No-load, no-fee funds are cheapest but may offer less advice
ETFs (Exchange Traded Funds)
- Examples: SPDR (S&P 500), DIA (DJIA), QQQ (NASDAQ 100)
- Listed on exchanges; trade continuously throughout the day
- Lower costs than buying all stocks individually
- Disadvantage: price may deviate from NAV (like closed-end fund)
- Must buy from broker (unlike no-load mutual funds)
Mutual Fund vs Hedge Fund
| Feature | Mutual Fund | Hedge Fund |
|---|---|---|
| Investors | Anyone — no minimum | Accredited/sophisticated only (high net worth & income) |
| Transparency | Full public disclosure of strategy & portfolio | Minimal disclosure |
| Fees | 0.5%–1.25% of AUM | “2 and 20”: 2% mgmt + 20% profits |
| Strategies | Predictable; limited short/leverage | Very flexible: short, leverage, derivatives |
| Liquidity | Redeem anytime at NAV | Lock-up periods; advance notice required |
Hedge Fund Strategies
Directional
Bets that one sector outperforms. Takes broad market exposure. Profits depend on market direction.
Non-Directional (Market Neutral)
Exploits temporary misalignments in relative valuation. Long-short positions cancel market exposure. Profits regardless of market direction. E.g. buy 29.5yr bond, short 30yr bond (convergence arbitrage).
Statistical Arbitrage
Quantitative systems exploit many small misalignments. Pairs trading: buy undervalued, sell overvalued from correlated pairs. Short holding periods.
Portable Alpha & Hedge Fund Fee Structure
Portable Alpha (Pure Play)Goal: isolate stock-specific alpha, remove market (beta) exposure.
Strategy: Long stock + Short index futures to drive beta to zero.
Contracts = (Portfolio value / (Index × Multiplier)) × beta
E.g. $2.1M, beta=1.2, S&P=2,016, mult=$50:
= ($2,100,000 / (2,016×$50)) × 1.2 = 25 short contracts
Result: beta≈0, monthly return = rf + alpha + residual
Alpha=2%, rf=1%: earn 3% per month regardless of market
Strategy: Long stock + Short index futures to drive beta to zero.
Contracts = (Portfolio value / (Index × Multiplier)) × beta
E.g. $2.1M, beta=1.2, S&P=2,016, mult=$50:
= ($2,100,000 / (2,016×$50)) × 1.2 = 25 short contracts
Result: beta≈0, monthly return = rf + alpha + residual
Alpha=2%, rf=1%: earn 3% per month regardless of market
High Water Mark
Incentive fee only paid when fund NAV exceeds its previous peak. If fund loses 20% then recovers 25%, no incentive fee is earned during recovery until the old high is surpassed. Prevents manager from collecting fees on “recovered” losses.
Hurdle Rate
Minimum rate of return the fund manager must achieve before incentive fees are collected. E.g. if hurdle = 5%, manager only collects 20% of profits ABOVE 5%.
Hedge Fund Performance Biases
| Bias | What It Is | Effect on Measured Performance |
|---|---|---|
| Survivorship bias | Failed funds drop from database | Overstates average performance |
| Backfill bias | Funds only report when performing well | Overstates historical returns |
| Liquidity bias | Illiquid assets earn liquidity premium mistaken for alpha | Overstates true risk-adjusted alpha |
| Changing factor loadings | Strategies change frequently, hard to measure risk | Understates true risk taken |
| Tail events | Strategies profit most of the time but expose to rare extreme losses | Understates downside risk |
Lecture 11
Managing Bond Portfolios
Valuation, yields, duration, and interest rate sensitivity
Bond Structure
- Face/Par value: principal repaid at maturity
- Coupon rate: determines periodic interest payments
- Indenture: legal contract specifying all terms and conditions between issuer and bondholder
- Bond Trustee: official (usually a bank) representing bondholders; ensures indenture terms are met
Bond Valuation
Bond Price FormulaP = CPN × PVIFA(r,n) + Par × PVIF(r,n)
Annual: CPN = Par × coupon rate
Semi-annual: CPN = Par×coupon/2, n=years×2, r=YTM/2
E.g. 30yr 8% coupon $1,000 par, YTM=10% semiannual:
n=60, CPN=$40, r=5% → Price = $810.71
Annual: CPN = Par × coupon rate
Semi-annual: CPN = Par×coupon/2, n=years×2, r=YTM/2
E.g. 30yr 8% coupon $1,000 par, YTM=10% semiannual:
n=60, CPN=$40, r=5% → Price = $810.71
Inverse relationship: rates ↑ → prices ↓. Price-yield curve is convex: bond price rises more on rate decreases than it falls on equal rate increases.
Semi-annual bond: HALVE coupon and rate, DOUBLE periods. Forgetting any one gives a completely wrong answer.
Three Yield Measures
| Measure | Definition | Decision Rule |
|---|---|---|
| Required Return (Rrr) | Risk-adjusted fair rate given default risk, liquidity risk, etc. Used to compute Fair Present Value. | If FPV > market price → BUY (undervalued) |
| Expected Return / YTM | Rate equating PV of all promised cashflows to current market price. Also = Bond Equivalent Yield. | If YTM > Rrr → BUY |
| Realized Return | Actual ex-post return. Uses actual cashflows received and actual sale price. Ex-post measure. | Performance evaluation |
Worked ExampleBond: 4 annual coupons of $100, principal $960, mkt price $925, Rrr=11.25%
FPV = $935.31 > $925 → undervalued → BUY
ERR: solve 925 = 100/(1+r) + ... + 1060/(1+r)&sup4; → ERR = 11.607% > 11.25% → BUY
Realized return (bought at $890 two years ago): 890 = 100/(1+r) + 1025/(1+r)² → 13.08%
FPV = $935.31 > $925 → undervalued → BUY
ERR: solve 925 = 100/(1+r) + ... + 1060/(1+r)&sup4; → ERR = 11.607% > 11.25% → BUY
Realized return (bought at $890 two years ago): 890 = 100/(1+r) + 1025/(1+r)² → 13.08%
Yield Relationships & Convergence
| Situation | Ordering | Price vs Par |
|---|---|---|
| Coupon rate > YTM | Coupon rate > CY > YTM | PREMIUM (price > par) |
| Coupon rate < YTM | Coupon rate < CY < YTM | DISCOUNT (price < par) |
| Coupon rate = YTM | All three equal | AT PAR (price = par) |
As maturity approaches: premium bond price falls toward par. Discount bond price rises toward par. Both converge to par ($1,000) at maturity.
YTM, BEY, EAR, and Yield to Call
YTM, BEY & EARYTM = Bond Equivalent Yield (annualised) = semiannual rate × 2
E.g. 8% coupon, 30yr, $1,000 par, price=$1,276.76:
Solve: r = 3% per half year
BEY = 3% × 2 = 6% pa
EAR = (1.03)² − 1 = 6.09%
Current Yield = Annual coupon / Price = $80/$810.71 = 9.87%
E.g. 8% coupon, 30yr, $1,000 par, price=$1,276.76:
Solve: r = 3% per half year
BEY = 3% × 2 = 6% pa
EAR = (1.03)² − 1 = 6.09%
Current Yield = Annual coupon / Price = $80/$810.71 = 9.87%
Yield to CallCallable bond: issuer may retire bond early at call price.
Use call price instead of par, and call date instead of maturity.
E.g. 8% coupon, 30yr, price=$1,150, call price=$1,100, callable in 10yr:
→ YTC = 6.64% pa (BEY)
Low rates: callable bond price FLAT (high call risk, price capped)
High rates: callable bond behaves like normal bond (call unlikely)
Use call price instead of par, and call date instead of maturity.
E.g. 8% coupon, 30yr, price=$1,150, call price=$1,100, callable in 10yr:
→ YTC = 6.64% pa (BEY)
Low rates: callable bond price FLAT (high call risk, price capped)
High rates: callable bond behaves like normal bond (call unlikely)
Holding-Period Return & Price Convergence
If YTM remains unchanged, the Holding-Period Return = YTM. Capital gain or loss exactly offsets the coupon income difference.
Holding-Period Return ExampleBond: 2yr, 6% coupon, YTM=7%, price=$981.92
1 year later (1yr left): price = $990.65
Capital gain = $990.65 − $981.92 = $8.73
Total gain = $8.73 + $60 coupon = $68.73
HPR = $68.73 / $981.92 = 7% = YTM ✓
1 year later (1yr left): price = $990.65
Capital gain = $990.65 − $981.92 = $8.73
Total gain = $8.73 + $60 coupon = $68.73
HPR = $68.73 / $981.92 = 7% = YTM ✓
Price Sensitivity: Key Factors
| Factor | Effect on Price Sensitivity (Duration) |
|---|---|
| Longer maturity | Higher sensitivity (non-linear, increasing at decreasing rate) |
| Lower coupon | Higher sensitivity (zero-coupon = maximum sensitivity) |
| Lower initial YTM | Higher sensitivity (more room for price to move) |
Duration — Calculation & Meaning
Macaulay DurationD = ∑[t × PV(CFt)] / ∑[PV(CFt)]
= weighted average time to cashflows
Concept 1: Higher coupon → shorter D
Concept 2: Higher YTM → shorter D
Concept 3: Shorter maturity → shorter D
Zero-coupon bond: D = Maturity always (one cashflow)
= weighted average time to cashflows
Concept 1: Higher coupon → shorter D
Concept 2: Higher YTM → shorter D
Concept 3: Shorter maturity → shorter D
Zero-coupon bond: D = Maturity always (one cashflow)
Duration Worked Example2yr bond, 10% coupon, 8% YTM ($1,035.67):
t=1: CF=$100, PV=$92.59, t×PV=$92.59
t=2: CF=$1,100, PV=$943.07, t×PV=$1,886.14
Sum(t×PV) = $1,978.73
D = 1,978.73 / 1,035.67 = 1.911 years
Same bond, 6% coupon: D = 1.942 yrs (lower coupon = longer D ✓)
t=1: CF=$100, PV=$92.59, t×PV=$92.59
t=2: CF=$1,100, PV=$943.07, t×PV=$1,886.14
Sum(t×PV) = $1,978.73
D = 1,978.73 / 1,035.67 = 1.911 years
Same bond, 6% coupon: D = 1.942 yrs (lower coupon = longer D ✓)
Modified Duration & Price Sensitivity
Modified Duration (KEY)D* = D / (1 + R) R = yield per period
ΔP/P ≈ −D* × ΔR
If D* given directly: use ΔP/P = −D*×ΔR (do NOT divide by 1+R again!)
PVBP = D* × Portfolio Value × 0.0001
Note: Duration is a LINEAR approximation. For large rate changes (>1%), actual price change deviates from estimate due to convexity (price-yield curve bends away from the tangent line).
ΔP/P ≈ −D* × ΔR
If D* given directly: use ΔP/P = −D*×ΔR (do NOT divide by 1+R again!)
PVBP = D* × Portfolio Value × 0.0001
Note: Duration is a LINEAR approximation. For large rate changes (>1%), actual price change deviates from estimate due to convexity (price-yield curve bends away from the tangent line).
If D* (modified duration) is given directly, use ΔP/P = −D*×ΔR. Do NOT apply D/(1+R) conversion again — that conversion is already done.
Default Risk & Protective Covenants
Default Risk & Credit Ratings
- Default premium = Corporate YTM − Equivalent govt bond YTM
- Stated YTM > Expected YTM due to default probability
- Rating agencies: Moody’s, S&P, Fitch
- Investment grade: BBB/Baa and above
- Speculative/junk: below BBB or Baa
- Bad economic times → spreads widen (perceived default risk rises)
Protective Covenants & Indenture
- Sinking funds: systematic retirement of bond issue each year → safer, lower yield
- Subordination: restricts future borrowing that would rank above existing bonds
- Dividend restrictions: retain assets in firm rather than paying out to shareholders
- Collateral: specific asset that bondholders receive upon default
- Indenture: legal document specifying rights of both bondholders and issuer
- Trustee: usually a bank; represents bondholders; ensures indenture compliance
Concept Check
Modified duration D*=7.2, portfolio value $5,000,000. Yields rise 25bp. Approximate portfolio value change?
A+$90,000
B−$36,000
C−$90,000
D+$36,000
Lecture 12
Portfolio Performance Evaluation
Returns alone don’t tell the whole story — adjust for risk
A fund with 35% return but enormous volatility may be worse than a 10% return fund with low risk. Always adjust for the risk taken. Simplest approach: compare to a benchmark universe with similar risk characteristics.
Universe Comparison
Rank relative performance of each fund within a comparison group (funds with similar risk characteristics). Display as percentile rankings (5th to 95th percentile). Limitation: sub-groups within a universe may not be truly comparable (e.g. high-beta stocks within an equity universe).
Five Performance Measures
1. Sharpe RatioS = (rP − rf) / sigmaP
Denominator = TOTAL sigma (all risk)
Use when: P = investor’s ENTIRE holding
Denominator = TOTAL sigma (all risk)
Use when: P = investor’s ENTIRE holding
2. Treynor MeasureT = (rP − rf) / betaP
Denominator = BETA only (systematic)
Use when: P = one component of many (nonsystematic diversified away)
Denominator = BETA only (systematic)
Use when: P = one component of many (nonsystematic diversified away)
3. Jensen’s Alphaalpha_P = rP − [rf + betaP(rM−rf)]
alpha_P > 0 → outperformed CAPM prediction
alpha_P > 0 → outperformed CAPM prediction
4. M² Measure (Modigliani)Create P* = (sigmaM/sigmaP) in P + rest in T-bills
P* has same SD as market. Compare returns:
M² = rP* − rM
Shortcut: M² = sigmaM × (Sharpe_P − Sharpe_M)
M² > 0 → outperformed market on same risk basis
P* has same SD as market. Compare returns:
M² = rP* − rM
Shortcut: M² = sigmaM × (Sharpe_P − Sharpe_M)
M² > 0 → outperformed market on same risk basis
5. Information RatioIR = alpha_P / sigma(eP) sigma(eP) = tracking error (nonsystematic SD)
Use when: selecting an active fund to mix with an index fund. Measures alpha earned per unit of avoidable (diversifiable) risk.
Use when: selecting an active fund to mix with an index fund. Measures alpha earned per unit of avoidable (diversifiable) risk.
Worked Example
rf=6%, Portfolio P: rP=35%, betaP=1.20, sigmaP=42%, tracking error=18%. Market: rM=28%, sigmaM=30%.
All Five Measures ComputedSharpe P = (35%−6%)/42% = 0.690 | Sharpe M = (28%−6%)/30% = 0.733
Treynor P = (35%−6%)/1.2 = 24.2% | Treynor M = (28%−6%)/1.0 = 22%
Jensen alpha = 35% − [6% + 1.2×(28%−6%)] = 35% − 32.4% = +2.6%
M²: P* = 30/42=71.4% in P + 28.6% T-bills → rP* = 0.714×35% + 0.286×6% = 26.7%
M² = 26.7% − 28% = −1.3% (underperformed market on same-risk basis)
IR = 2.6%/18% = 14.4%
Treynor P = (35%−6%)/1.2 = 24.2% | Treynor M = (28%−6%)/1.0 = 22%
Jensen alpha = 35% − [6% + 1.2×(28%−6%)] = 35% − 32.4% = +2.6%
M²: P* = 30/42=71.4% in P + 28.6% T-bills → rP* = 0.714×35% + 0.286×6% = 26.7%
M² = 26.7% − 28% = −1.3% (underperformed market on same-risk basis)
IR = 2.6%/18% = 14.4%
Which Measure is Appropriate?
| Situation | Measure | Why |
|---|---|---|
| P = investor’s entire portfolio | Sharpe / M² | Total risk (both systematic + nonsystematic) matters |
| P = one sub-portfolio of many | Treynor | Nonsystematic diversified away; only beta matters |
| Selecting active to mix with index | Information Ratio | Alpha per unit of avoidable (diversifiable) risk |
| Absolute skill vs CAPM | Jensen’s Alpha | Return above CAPM for given beta level |
Interpretation of Multiple Measures
Comparing Portfolios P, Q, and Market
- If P/Q represents the entire investment: choose by Sharpe/M²
- If P/Q are competing sub-portfolios: use Treynor
- If seeking active fund to mix with index: use Information Ratio
- A fund can have positive Jensen’s alpha but negative M² simultaneously — high alpha plus even higher nonsystematic risk = above SML but below market Sharpe. Always check which measure the question asks for.
Treynor of P > Treynor of Q does NOT mean P is better overall. Valid only when P is ONE component of a broader diversified portfolio where nonsystematic risk has been diversified away.
Concept Check
Fund P: rP=35%, sigmaP=42%, betaP=1.2. Market: rM=28%, sigmaM=30%, rf=6%. Which is correct?
ASharpe P > Sharpe Market
BTreynor P < Treynor Market
CSharpe P < Sharpe Market, but Treynor P > Treynor Market
DBoth measures agree P outperformed
Crash Course
The Solving
Playbook
Step-by-step strategy for every topic. Know the traps before you sit down to solve.
Topic 01
Utility, Speculation & Gambling
1
Read A carefully. A>0=risk-averse (normal). A=0=risk-neutral. A<0=risk-lover.
2
Plug in sigma² (variance), NOT sigma. If sigma=20%, use (0.20)²=0.04. Never use 20 or 0.20 directly.
3
Compare U scores. Investor picks highest U. Risk-free: U=rf (zero variance penalty).
Indifference curves show same utility for ONE investor. Cannot compare curves across investors with different A values.
Speculation = positive risk premium + commensurate risk → risk-averse accept. Gambling = zero expected profit → risk-averse reject.
Topic 02
Capital Allocation — Finding y*
1
Identify inputs: E(rP), sigmaP, rf, A. Consistent units throughout (all decimals or all %).
2
Compute y*:
y* = [E(rP) − rf] / [A × sigmaP²]
3
Complete portfolio:
E(rC) = rf + y*×[E(rP)−rf] sigmaC = y*×sigmaP
4
Verify Sharpe: Sharpe of complete portfolio = Sharpe of P (same CAL). Both divide by sigmaP.
If y>1 (leverage): use borrowing rate rb (not rf) to compute E(rC) for the borrowed portion. This creates a kinked CAL.
Topic 03
Optimal Risky Portfolio
1
Compute Cov if given rho:
Cov(rD,rE) = rho × sigmaD × sigmaE (use decimal form)
2
Minimum variance portfolio:
wD(min) = [sigmaE²−Cov] / [sigmaD²+sigmaE²−2Cov] ← minimises sigma only
3
Tangency portfolio (max Sharpe):
Use EXCESS returns: E(RD) = E(rD)−rf, E(RE) = E(rE)−rf
wD = [E(RD)×sigmaE²−E(RE)×Cov] / [E(RD)×sigmaE²+E(RE)×sigmaD²−(E(RD)+E(RE))×Cov]
wD = [E(RD)×sigmaE²−E(RE)×Cov] / [E(RD)×sigmaE²+E(RE)×sigmaD²−(E(RD)+E(RE))×Cov]
4
Compute E(rP) and sigmaP:
E(rP) = wD×E(rD)+wE×E(rE) sigmaP² = wD²sigmaD²+wE²sigmaE²+2wDwECov
Cross-term is 2×wD×wE×Cov, NOT 2×wD×wE×rho. Compute Cov first, or use the rho version consistently. Don't mix them.
Topics 04–05
Single-Index Model & CAPM
1
SIM variance decomposition:
sigma_i² = beta_i²×sigmaM²+sigma²(ei) Cov(ri,rj) = beta_i×beta_j×sigmaM²
R² = systematic / total variance
R² = systematic / total variance
2
Adjusted beta: Adj beta = (2/3)×hist beta + (1/3)×1
3
CAPM:
E(ri) = rf + beta_i×[E(rM)−rf]
alpha = actual − CAPM predicted alpha>0 → BUY, alpha<0 → SELL
alpha = actual − CAPM predicted alpha>0 → BUY, alpha<0 → SELL
SML x-axis = beta. CML x-axis = sigma. SML applies to ANY asset. CML only to efficient portfolios.
CAPM assumes alpha=0 in equilibrium. If given alpha from SIM and asked to apply CAPM, use CAPM formula only.
Topic 09
Bond Valuation & Yields
Before You Calculate — Checklist
- Annual or semi-annual coupon? (semi: halve coupon & rate, double periods)
- Which yield? Rrr, YTM (Expected), or Realized Return?
- Is the bond callable? Need YTC instead of YTM?
- Par value? ($1,000 default — verify)
1
Bond price: PV = ∑[CPN/(1+r)^t] + Par/(1+r)^n. Semi: r=YTM/2, n=years×2, CPN=annual/2.
2
YTM (calculator): PV=−price, FV=par, PMT=coupon, N=periods → CPT I/Y. Multiply by 2 for annual BEY if semi.
3
YTC: Replace N with periods to call date, replace FV with call price.
4
EAR from BEY:
EAR = (1 + BEY/2)² − 1
Topic 10
Duration — Calculation & Hedging
The single most trap-filled topic
Calculating Macaulay DurationD = ∑[t × PV(CFt)] / ∑[PV(CFt)]
Steps:
1. PV each cashflow: CFt / (1+r)^t
2. Multiply each PV by time t
3. Sum all (PV × t)
4. Divide by total PV (= bond price)
Steps:
1. PV each cashflow: CFt / (1+r)^t
2. Multiply each PV by time t
3. Sum all (PV × t)
4. Divide by total PV (= bond price)
Using Duration (Price Sensitivity)Given Macaulay D:
D* = D / (1+r) ← convert to modified
ΔP/P ≈ −D* × ΔR
Given D* directly:
ΔP/P ≈ −D* × ΔR ← no conversion needed!
D* = D / (1+r) ← convert to modified
ΔP/P ≈ −D* × ΔR
Given D* directly:
ΔP/P ≈ −D* × ΔR ← no conversion needed!
1
PVBP of portfolio:
PVBP = Portfolio value × D* × 0.0001
2
Futures contract value:
F_value = (futures price / 100) × par value
3
PVBP of one futures contract:
PVBP_futures = F_value × D*_futures × 0.0001
4
Number of contracts:
Full hedge (D*=0): N = PVBP_portfolio / PVBP_futures → SHORT
Partial (target D*): N = [VP × (D*_current − D*_target)] / [F_value × D*_futures]
Positive N → SHORT (reducing duration). Negative N → LONG (increasing).
Partial (target D*): N = [VP × (D*_current − D*_target)] / [F_value × D*_futures]
Positive N → SHORT (reducing duration). Negative N → LONG (increasing).
If targeting D*=4 from D*=7.5: reduction=3.5yr → SHORT. If targeting D*=9 from D*=6: increase=3yr → LONG. Sign of (current−target) tells you direction.
Duration is a LINEAR approximation. For large rate changes (>1%), actual price change differs from estimate due to convexity.
Topic 13
Performance Evaluation
Pick the right measure for the context
Decision Matrix
| Situation | Measure | Formula |
|---|---|---|
| P = investor's ENTIRE portfolio | Sharpe | S = (rP−rf) / sigmaP |
| P = ONE component of many | Treynor | T = (rP−rf) / betaP |
| Select active to mix with index | Info Ratio | IR = alpha / sigma(eP) |
| Absolute skill vs CAPM | Jensen's Alpha | alpha = rP−[rf+betaP(rM−rf)] |
| Compare P vs market, equal risk | M² | M² = sigmaM×(SP−SM) |
Treynor of P>Treynor of Q does NOT mean P is better overall. Valid only when P is ONE component (nonsystematic risk diversified away in the broader portfolio).
Jensen's alpha positive + M² negative is possible. High alpha + even higher nonsystematic risk = above SML but below market Sharpe.
Reference
Master
Formula Sheet
All key formulas in exam-style notation, organised by topic.
Portfolio Theory & Capital Allocation
Utility
U = E(r) − (1/2) × A × sigma²
Optimal Weight y*
y* = [E(rP) − rf] / [A × sigmaP²]
Complete Portfolio
E(rC) = rf + y×[E(rP)−rf]
sigmaC = y × sigmaP
sigmaC = y × sigmaP
Sharpe Ratio (CAL Slope)
Sharpe = [E(rP) − rf] / sigmaP
Two-Asset Portfolio Return
E(rP) = wD×E(rD) + wE×E(rE)
Two-Asset Portfolio Variance
sigmaP² = wD²sigmaD² + wE²sigmaE²
+ 2×wD×wE×Cov(rD,rE)
Cov = rho × sigmaD × sigmaE
+ 2×wD×wE×Cov(rD,rE)
Cov = rho × sigmaD × sigmaE
Minimum Variance Weight
wD(min) = [sigmaE²−Cov] / [sigmaD²+sigmaE²−2Cov]
Perfect Hedge (rho=−1)
wD = sigmaE / (sigmaD+sigmaE)
wE = sigmaD / (sigmaD+sigmaE) → sigmaP=0
wE = sigmaD / (sigmaD+sigmaE) → sigmaP=0
Single-Index Model & Asset Pricing
Single-Index Model
Ri = alpha_i + beta_i×RM + ei
sigma_i² = beta_i²×sigmaM² + sigma²(ei)
Cov(ri,rj) = beta_i×beta_j×sigmaM²
sigma_i² = beta_i²×sigmaM² + sigma²(ei)
Cov(ri,rj) = beta_i×beta_j×sigmaM²
CAPM / SML
E(ri) = rf + beta_i × [E(rM)−rf]
alpha = ri − [rf + beta_i×(rM−rf)]
alpha = ri − [rf + beta_i×(rM−rf)]
R² & Adjusted Beta
R² = beta_i²×sigmaM² / sigma_i²
Adj beta = (2/3)×hist beta + (1/3)×1
Adj beta = (2/3)×hist beta + (1/3)×1
Fama-French Three-Factor
E(ri)−rf = a_i + b_i[E(rM)−rf]
+ s_i×E(RSMB) + h_i×E(RHML)
+ s_i×E(RSMB) + h_i×E(RHML)
Multi-Factor APT
E(ri) = rf + beta_1[E(r1)−rf]
+ beta_2[E(r2)−rf] + ...
+ beta_2[E(r2)−rf] + ...
Portfolio Beta
betaP = ∑(wi × beta_i)
Derivatives
Futures P&L
Profit (Long) = PT − F0
Profit (Short) = F0 − PT
Profit (Short) = F0 − PT
Futures Parity
F = S × (1 + r + c − d − v)^t
Interest Rate Parity (FX)
F0 = E0 × [(1+rA) / (1+rB)]^T
E0 = spot rate (A per B)
E0 = spot rate (A per B)
FX Hedge
Contracts = Exposure / Contract size
SHORT when long the foreign currency
SHORT when long the foreign currency
PVBP
PVBP = D* × Portfolio Value × 0.0001
(dollar change per 1 basis point)
(dollar change per 1 basis point)
IR Hedge (# Contracts)
N = PVBP_portfolio / PVBP_futures
SHORT to hedge rate rise on long bond portfolio
SHORT to hedge rate rise on long bond portfolio
Portable Alpha
N = (Portfolio value / (Index×Mult)) × beta
SHORT index futures to zero beta
SHORT index futures to zero beta
Duration Hedge (Partial)
N = [VP × (D*_current − D*_target)] / [F_value × D*_futures]
Positive N → SHORT, Negative N → LONG
Positive N → SHORT, Negative N → LONG
Bonds & Duration
Bond Valuation
P = CPN×PVIFA(r,n) + Par×PVIF(r,n)
Semi: halve CPN & r, double n
Semi: halve CPN & r, double n
EAR from BEY
EAR = (1 + BEY/2)² − 1
Macaulay Duration
D = ∑[t×PV(CFt)] / ∑[PV(CFt)]
Zero-coupon: D = Maturity (always)
Zero-coupon: D = Maturity (always)
Modified Duration & Sensitivity
D* = D / (1+r) r = yield per period
ΔP/P ≈ −D* × ΔR
ΔP/P ≈ −D* × ΔR
NAV
NAV = (Assets−Liabilities) / Shares outstanding
T-Bill Price
Price = 100 − (n × r) / 360
n = days to maturity, r = rate (decimal)
n = days to maturity, r = rate (decimal)
Performance Evaluation
Sharpe Ratio
S = (rP − rf) / sigmaP
Use when: P = entire portfolio
Use when: P = entire portfolio
Treynor Measure
T = (rP − rf) / betaP
Use when: P = one sub-portfolio
Use when: P = one sub-portfolio
Jensen's Alpha
alpha_P = rP − [rf + betaP×(rM−rf)]
Information Ratio
IR = alpha_P / sigma(eP)
sigma(eP) = tracking error (nonsystematic SD)
sigma(eP) = tracking error (nonsystematic SD)
M² Measure
M² = rP* − rM
Shortcut: M² = sigmaM × (SP − SM)
Shortcut: M² = sigmaM × (SP − SM)
Exam Strategy
Exam
Strategy
Time is limited and marks are tight. Here's how to maximise your score.
High-Probability Topics
Almost Certain to Appear
- Utility calculation & y* optimisation
- CAPM / SML: compute E(r), find alpha, classify under/overpriced
- Single-index model: variance decomposition, R², covariance between stocks
- Bond pricing (semi-annual), YTM, duration & price sensitivity
- Performance measure selection (Sharpe vs Treynor vs IR)
Likely to Appear
- Two-asset portfolio: variance, min-variance weights, tangency weights
- Futures hedging: FX or interest-rate, number of contracts
- Fama-French FF3: interpret factor loadings (SMB/HML direction)
- Adjusted beta (Blume formula)
- M² measure calculation
- Portable alpha / hedge fund strategy questions
Universal SAQ Checklist
- State all inputs explicitly before substituting. "Given: E(rP)=12%, rf=3%, sigmaP=18%, A=4"
- Write the formula first, then substitute. Never substitute without showing the formula.
- Show intermediate working steps. Marks are awarded at each step, not just the final answer.
- For "Explain why" questions: Name the mechanism → link to formula or concept → state direction of effect.
- For comparison questions: Compute both values → state which is higher → give the economic interpretation.
- Units matter: State whether returns are % or decimal, whether duration is in years.
- Check semi-annual conversion every time you see a bond question. Halve, double, halve — then proceed.
The 8 Most Dangerous Traps
Trap 01
sigma vs sigma² in utility
Always use sigma² in U=E(r)−½Aσ². If sigma=20%, then sigma²=0.04, not 20 or 0.20.
Trap 02
Semi-annual bond: halve and double
N=years×2, CPN=annual coupon/2, r=YTM/2. Forgetting any one gives a completely wrong answer.
Trap 03
D* given directly — don't double-convert
If modified duration D* is given directly, use ΔP/P=−D*×ΔR. Do NOT divide by (1+r) again.
Trap 04
Tangency portfolio uses excess returns
The tangency portfolio weight formula needs E(RD)=E(rD)−rf (excess return), not total return E(rD).
Trap 05
Sharpe vs Treynor context
Sharpe when fund = entire portfolio. Treynor when fund = one component. Confusing these is a guaranteed mark loss.
Trap 06
Hedging direction
Long underlying → SHORT futures. Short underlying → LONG futures. Always state direction before computing hedge ratio.
Trap 07
Alpha ≠ "good" in all contexts
Positive Jensen's alpha is consistent with underperforming on Sharpe ratio if nonsystematic risk is very high. M²<0 is possible even when alpha>0.
Trap 08
Leverage: use borrowing rate when y>1
When computing E(rC) for a leveraged portfolio (y>1), use borrowing rate rb (not rf) for the negative risk-free component.
Key Relationships to Memorise
Bond Price vs Yield
- Coupon rate > YTM → Premium bond (price > par)
- Coupon rate < YTM → Discount bond (price < par)
- Longer maturity → more interest rate sensitive
- Lower coupon → more sensitive
- Zero-coupon → most sensitive; duration = maturity
Duration Rules
- Higher coupon → shorter duration
- Higher YTM → shorter duration
- Longer maturity → longer duration
- Zero-coupon bond: D = maturity (always)
- Duration approximation less accurate for large rate changes
CAPM / SML
- Beta of market portfolio = 1 (always)
- Beta of T-bills = 0 (risk-free)
- Above SML → underpriced → alpha > 0 → BUY
- Below SML → overpriced → alpha < 0 → SELL
Diversification
- Lower correlation → more diversification benefit
- Only systematic risk is compensated
- Firm-specific risk → diversified away for free
- As n→∞: firm-specific risk → 0, systematic risk remains
Practice Mode
Test Your
Knowledge
25 MCQs · 3 SAQs with model answers · Calculation drills
25 Questions
Multiple Choice Questions
All topics — instant feedback — hard difficulty
Score
0 / 25
0%
Q1
An investor (A=3) holds risky portfolio P: E(rP)=14%, sigmaP=20%, rf=4%. She discovers sigmaP is actually 25%. What happens to y* and Sharpe ratio of P?
Ay* increases; Sharpe increases
By* increases; Sharpe decreases
Cy* decreases; Sharpe decreases
Dy* decreases; Sharpe increases
C is correct.
y* = [E(rP)−rf] / [A×sigmaP²]: (14%−4%) / [3×(0.25)²] = 0.10/0.1875 = 0.533 (was 0.833) → DECREASES
Sharpe = [E(rP)−rf]/sigmaP = 10%/25% = 0.40 (was 0.50) → DECREASES
Higher sigma in the denominator of both formulas reduces both. Same reward, higher risk = worse risk-adjusted performance.Sharpe = [E(rP)−rf]/sigmaP = 10%/25% = 0.40 (was 0.50) → DECREASES
Q2
Which CORRECTLY distinguishes the minimum-variance portfolio from the tangency (optimal risky) portfolio?
AThe minimum-variance portfolio always has the highest Sharpe ratio
BThe tangency portfolio maximises Sharpe ratio; the minimum-variance portfolio only minimises sigma and generally has a lower Sharpe ratio
CThey are the same portfolio when rho = 0
DThe optimal complete portfolio is always 100% in the tangency portfolio
B is correct. Min-variance minimises sigma but does NOT maximise Sharpe. Tangency portfolio = where CAL from rf is tangent to the efficient frontier = maximum Sharpe. D is wrong: the optimal complete portfolio mixes the tangency portfolio with the risk-free asset based on investor's A.
Q3
Historical beta=1.80. After Blume adjustment, adjusted beta=? If rf=2%, E(rM)=9%, actual return=14%, alpha=?
AAdj beta=1.87; alpha=+1.27%
BAdj beta=1.53; alpha=−1.27%
CAdj beta=1.60; alpha=+2.00%
DAdj beta=1.533; alpha=+1.27%
D is correct.
Blume: (2/3)×1.80 + (1/3)×1 = 1.200 + 0.333 = 1.533
CAPM: 2% + 1.533×(9%−2%) = 2% + 10.731% = 12.731%
Alpha = 14% − 12.731% = +1.27% → above SML → underpriced → BUY
CAPM: 2% + 1.533×(9%−2%) = 2% + 10.731% = 12.731%
Alpha = 14% − 12.731% = +1.27% → above SML → underpriced → BUY
Q4
Portfolio X: b=0.9 (market), s=0.6 (SMB), h=−0.3 (HML). Risk premia: mkt=7%, SMB=3%, HML=4%, rf=2%. FF3 return & what does h=−0.3 imply?
A10.9%; negative HML = value stock tilt
B8.9%; negative HML = large-cap tilt
C10.1%; negative HML = growth stock tilt
D8.9%; negative HML = growth stock tilt (low book-to-market)
D is correct.
FF3 = rf + b×mrp + s×SMB + h×HML
= 2% + 0.9×7% + 0.6×3% + (−0.3)×4%
= 2% + 6.3% + 1.8% − 1.2% = 8.9%
HML is long high B/M (value), short low B/M (growth). Negative h loading → portfolio behaves like growth stocks.= 2% + 0.9×7% + 0.6×3% + (−0.3)×4%
= 2% + 6.3% + 1.8% − 1.2% = 8.9%
Q5
A 10-year 6% coupon bond and a 10-year zero-coupon bond have the same YTM. Which statement about durations is correct?
ABoth have duration = 10 years
BCoupon bond has LONGER duration than zero-coupon
CZero-coupon D=10yr; coupon bond D<10yr. Zero-coupon is MORE sensitive to rate changes.
DCoupon bond D=10yr; zero-coupon D<10yr
C is correct. Zero-coupon: D=Maturity=10yr always (single cashflow at maturity). Coupon bond: coupons arrive before maturity, pulling weighted avg time forward → D<10yr. Higher D → more rate sensitive → zero-coupon is more sensitive.
Q6
Stock Z beats consensus by 15%. Price rises 8% on announcement then drifts +3%/month for 3 months before plateauing. Which anomaly and which EMH form does it violate?
AReversal effect; weak-form EMH
BPost-earnings-announcement drift (PEAD); semistrong-form EMH
CMomentum effect; weak-form EMH only
DSmall-firm effect; semistrong-form EMH
B is correct. PEAD: after positive earnings surprise, price drifts upward for months instead of fully adjusting immediately. Earnings = public information → violates semistrong-form EMH. Behavioural driver: conservatism bias (investors update beliefs too slowly).
Q7
$50M bond portfolio, D*=7.5yr. Reduce to D*=4yr using T-bond futures: par=$100K, futures price=$95/$100, D*_futures=8yr. How many contracts SHORT?
AShort 230 contracts
BShort 493 contracts
CShort 375 contracts
DShort 247 contracts
A is correct.
N = [VP × (D*_current − D*_target)] / [F_contract × D*_futures]
F_contract = ($95/$100) × $100,000 = $95,000
N = [$50M × (7.5−4.0)] / [$95,000 × 8]
= $175,000,000 / $760,000 = 230.3 → 230 contracts SHORT
Positive N → SHORT (reducing duration). PVBP check: gap = $17,500; PVBP_futures = $76; N = 17,500/76 = 230 ✓F_contract = ($95/$100) × $100,000 = $95,000
N = [$50M × (7.5−4.0)] / [$95,000 × 8]
= $175,000,000 / $760,000 = 230.3 → 230 contracts SHORT
Q8
Fund P: rP=28%, sigmaP=35%, betaP=1.3. Fund Q: rQ=22%, sigmaQ=20%, betaQ=0.85. Mkt: rM=18%, sigmaM=20%, rf=4%. Investor uses Fund P as ENTIRE portfolio. Most appropriate measure and winner?
ASharpe is most appropriate; Fund Q wins (Sharpe Q=0.90 > Sharpe P=0.686)
BTreynor; Fund P wins
CJensen's alpha; Fund P wins
DInformation Ratio; Fund Q wins
A is correct. P = entire portfolio → use Sharpe (total risk matters).
Sharpe P = (28%−4%)/35% = 0.686 | Sharpe Q = (22%−4%)/20% = 0.900
Q wins. Despite lower return, Q is far more efficient per unit of total risk.
Q wins. Despite lower return, Q is far more efficient per unit of total risk.
Q9
Three well-diversified portfolios: A: beta=0.8, E(r)=9%; B: beta=1.2, E(r)=12%; C: beta=1.6, E(r)=14%. rf=3%. Which creates an arbitrage?
ANo arbitrage; all on APT line
BPortfolio A is mispriced; short A, long B
CPortfolio C is overpriced; APT implies 15% for beta=1.6. Short C, long replicating portfolio of A+B.
DPortfolio B is mispriced
C is correct.
APT slope from A and B: (12%−9%)/(1.2−0.8) = 7.5% per unit beta
Intercept: 9% − 0.8×7.5% = 3% = rf ✓
APT for C (beta=1.6): 3% + 1.6×7.5% = 15%
Actual C = 14% < 15% → C is OVERPRICED (below APT line)
Strategy: Short C, go long replicating portfolio at beta=1.6 earning 15%. Net riskless profit = 1% per dollar.Intercept: 9% − 0.8×7.5% = 3% = rf ✓
APT for C (beta=1.6): 3% + 1.6×7.5% = 15%
Actual C = 14% < 15% → C is OVERPRICED (below APT line)
Q10
Stock index futures: F₀=4,100, S₀=4,000, rf=6%/yr, dividend yield=2%/yr, maturity=6 months. Is the futures fairly priced?
AFairly priced; theoretical F = 4,100
BOverpriced; theoretical F ≈ 4,079. Buy index, short futures (cash-and-carry).
CUnderpriced; theoretical F = 4,160. Buy futures, short index.
DOverpriced; theoretical F = 4,082.
B is correct.
F = S × (1+r−d)^t = 4,000 × (1.04)^0.5 = 4,000 × 1.0198 = 4,079.2
Actual F=4,100 > Fair F=4,079 → OVERPRICED
Cash-and-carry: borrow, buy index at 4,000, short futures at 4,100. Deliver at maturity. Riskless profit ≈ $20.8 per unit.Actual F=4,100 > Fair F=4,079 → OVERPRICED
Q11
Investor holds Stock X (bought $100, now $60, refuses to sell). Sold Stock Y for quick 10% gain. Which TWO biases are simultaneously at work?
AAnchoring and Framing
BOverconfidence and Conservatism
CRegret Avoidance (holding X) & Disposition Effect / Prospect Theory (selling Y early)
DMental Accounting and Representativeness
C is correct. Regret Avoidance (X): Selling crystallises the loss and forces confronting a bad decision → holds the loser instead. Disposition Effect (Y): Prospect Theory → investors sell winners early to "lock in" gains and hold losers hoping for recovery. Selling Y early while holding X = textbook disposition effect.
Q12
Equally-weighted 100-stock portfolio: avg beta=1.1, avg sigma²(ei)=0.04, sigmaM²=0.0225. Total portfolio variance?
AsigmaP² ≈ 0.02763 (systematic: 0.027225 + firm-specific: 0.0004)
BsigmaP² = 0.0625 (no diversification)
CsigmaP² = 0.027225 (firm-specific = zero)
DsigmaP² = 0.0225
A is correct.
Systematic: (1.1)² × 0.0225 = 1.21 × 0.0225 = 0.027225
Firm-specific: sigma²(eP) = 0.04/100 = 0.0004
Total: 0.027225 + 0.0004 = 0.027625
C is wrong: firm-specific risk only →0 as n→∞, not with finite n=100.Firm-specific: sigma²(eP) = 0.04/100 = 0.0004
Total: 0.027225 + 0.0004 = 0.027625
Q13
Hedge fund: $200M start. Yr1: +30%=$260M. Yr2: −20%=$208M. HWM=$260M. Yr3: +25%=$260M. Management=2%, incentive=20%. Total Yr3 fees?
AManagement=$5.2M; Incentive=$10.4M; Total=$15.6M
BManagement=$4.16M; Incentive=$0; Total=$4.16M
CManagement=$5.2M; Incentive=$13M; Total=$18.2M
DManagement=$5.2M; Incentive=$0; Total=$5.2M
D is correct.
Yr3 end NAV = $208M × 1.25 = $260M
Management fee = 2% × $260M = $5.2M
Profit above HWM = $260M − $260M = $0 → Incentive fee = $0
High water mark: incentive fee only when NAV EXCEEDS previous peak. Year 3 only recovered to HWM → no incentive fee earned.Management fee = 2% × $260M = $5.2M
Profit above HWM = $260M − $260M = $0 → Incentive fee = $0
Q14
$80M bonds, D*=6yr. Expects rates to FALL, wants D*=9yr using T-bond futures (D*=10yr, price=$100/$100, par=$100K). How many contracts LONG?
ALong 192 contracts
BLong 240 contracts
CShort 240 contracts
DLong 480 contracts
B is correct.
N = [VP × (D*_target − D*_current)] / [F_contract × D*_futures]
F_contract = $100,000
N = [$80M × (9−6)] / [$100,000 × 10]
= $240,000,000 / $1,000,000 = 240 LONG
Positive result → LONG (increasing duration). Rates expected to fall → longer duration = greater capital gains.F_contract = $100,000
N = [$80M × (9−6)] / [$100,000 × 10]
= $240,000,000 / $1,000,000 = 240 LONG
Q15
Investor Alpha (A=2) and Investor Beta (A=5) face the same opportunity set. Under Separation Property, what is TRUE?
AAlpha holds more in P; Beta is on a higher indifference curve so Beta is better off
BBoth hold the same proportion in P
CAlpha (lower A) invests larger y* in P; Beta holds more in risk-free. Both on same CAL with same Sharpe. Cannot say who is "better off" — indifference curves incomparable across different utility functions.
DAlpha's portfolio has a lower Sharpe ratio because Alpha takes more risk
C is correct. y*=1/(A×sigmaP²) denominator: lower A → higher y*. Both lie on the SAME CAL → Sharpe identical. Comparing indifference curves across investors with different A values is meaningless — they have different utility scales.
Q16
rUS=3%, rUK=5%, E₀=$1.25/£, T=1yr. IRP forward rate? If actual F=$1.20/£, what is the arbitrage?
AIRP: F=$1.2262/£. Actual F=$1.20<fair → £ cheap in forward → go long £ forward, borrow USD, invest in UK. Riskless profit ≈$0.026/£.
BIRP: F=$1.30/£. Actual F=$1.20 → sell £ forward.
CIRP: F=$1.2262/£. Actual F=$1.20 > fair → sell £ forward.
DNo arbitrage.
A is correct.
IRP: F = E × [(1+rUS)/(1+rUK)]^T = $1.25 × (1.03/1.05) = $1.2262/£
Actual F=$1.20 < Fair F=$1.2262 → £ underpriced forward. Buy £ forward at $1.20, profit from the gap when settled.Q17
Merger event study: target earns CAR +22% in 5 days around announcement, BUT CAR begins drifting 45 days BEFORE. Best interpretation?
AMarket is strongly efficient — pre-announcement drift proves market knew
BLarge CAR on day 0 is impossible under any EMH form
C45-day drift means semistrong efficiency is violated
D45-day drift suggests information leakage (possible strong-form violation). Large day-0 jump means most public investors were still surprised → broadly consistent with semistrong efficiency.
D is correct. Pre-drift = insiders/advisors trading on non-public info → possible strong-form violation (NOT semistrong, since info was not yet public). Day-0 jump = most of the market didn't know. Post-announcement plateau = semistrong efficiency holds after the announcement.
Q18
5-year 10% annual coupon bond (par=$1,000) callable in 3yr at $1,050. Trades at $1,080. YTM=8.00%. Which is correct about YTC?
AYTC < YTM always for premium callable bonds
BYTC ≈ 8.4–8.5% > YTM=8.00%. Since YTC > YTM, issuer is UNLIKELY to call — they'd pay $1,050 vs $1,000 at maturity.
CYTC = YTM since premium bonds are always called
DYTC cannot be computed without the future interest rate
B is correct.
YTC calc: PV=−1080, FV=1050, PMT=100, N=3 → I/Y ≈ 8.4–8.5%
YTC > YTM because call price ($1,050) > par ($1,000). The premium call price pushes YTC above YTM. Issuer has NO incentive to call — would pay $1,050 when they could let it run to maturity and pay $1,000.Q19
Portfolio P: rP=40%, sigmaP=50%. Market M: rM=20%, sigmaM=25%, rf=5%. Compute M².
A50% in P, 50% T-bills; M²=+2%
B25% in P, 75% T-bills; M²=−3%
C50% in P, 50% T-bills; P* return=22.5%; M²=+2.5% → P outperformed
D50% in P, 50% T-bills; P* return=20%; M²=0%
C is correct.
y = sigmaM/sigmaP = 25%/50% = 0.5 → 50% in P, 50% T-bills
rP* = 0.5×40% + 0.5×5% = 20% + 2.5% = 22.5%
M² = rP* − rM = 22.5% − 20% = +2.5%
Confirm: Sharpe P=(40−5)/50=0.70 > Sharpe M=(20−5)/25=0.60 ✓
rP* = 0.5×40% + 0.5×5% = 20% + 2.5% = 22.5%
M² = rP* − rM = 22.5% − 20% = +2.5%
Confirm: Sharpe P=(40−5)/50=0.70 > Sharpe M=(20−5)/25=0.60 ✓
Q20
T-bill: 91 days to maturity, ask discount rate=4.80%. Ask price per $100 par and Bond Equivalent Yield (BEY)?
AAsk price=$98.787; BEY≈4.92%
BAsk price=$98.787; BEY=4.80%
CAsk price=$98.80; BEY=4.80%
DAsk price=$99.22; BEY=4.92%
A is correct.
Price = 100 − (91×0.048)/360 = 100 − 1.2133 = $98.787
BEY = (Discount/Price) × (365/n) = (1.2133/98.787) × (365/91) = 0.01228 × 4.011 = 4.926%
BEY > bank discount yield because: (1) divides by price (smaller denominator) → larger %; (2) uses 365-day year not 360 → larger annualisation factor.BEY = (Discount/Price) × (365/n) = (1.2133/98.787) × (365/91) = 0.01228 × 4.011 = 4.926%
Q21
$3M portfolio in Stock Z: beta=1.5, alpha=3%/month, rf=0.5%/month. S&P=3,000, multiplier=$50. Zero-beta pure play: contracts to short and expected monthly return?
A20 contracts short; return=3%
B30 contracts short; return=3%
C20 contracts short; return=rf+3%
D30 contracts short; expected monthly return = rf + alpha = 0.5% + 3% = 3.5%
D is correct.
N = (Portfolio value / (Index × Multiplier)) × beta
= ($3M / (3,000 × $50)) × 1.5 = 20 × 1.5 = 30 contracts SHORT
After hedging: r = rf + alpha = 0.5% + 3% = 3.5%/month
B is wrong (says return=3%, forgetting rf). C has wrong contract count.= ($3M / (3,000 × $50)) × 1.5 = 20 × 1.5 = 30 contracts SHORT
After hedging: r = rf + alpha = 0.5% + 3% = 3.5%/month
Q22
Closed-end fund: NAV=$25, trades at $22. Open-end fund: NAV=$25. Which statement is correct?
AClosed-end fund is overpriced and should be sold immediately
BClosed-end trades at 12% discount to NAV. Open-end MUST trade at NAV. Discounts persist in closed-end funds because no arbitrage mechanism forces price=NAV.
COpen-end fund can trade above NAV
DBoth should trade at the same price in an efficient market
B is correct. Discount = (25−22)/25 = 12%. Open-end funds: investors can redeem at NAV daily → arbitrage forces price=NAV. Closed-end funds: no redemption mechanism after IPO → prices can deviate from NAV indefinitely. The persistent discount is called the "closed-end fund puzzle."
Q23
Stock A: E(r)=12%, beta=1.2. Stock B: E(r)=8%, beta=0.7. rf=3%, E(rM)=10%. 60% A + 40% B. Which is CORRECT?
ABeta=1.0; exactly on SML; alpha=0
BE(rP)=10.4%; alpha=0
CBeta=1.0; E(rP)=10.4%; CAPM predicts 10% for beta=1.0; alpha=+0.4% (slightly underpriced)
DBeta=0.98
C is correct.
betaP = 0.6×1.2 + 0.4×0.7 = 0.72 + 0.28 = 1.00
E(rP) = 0.6×12% + 0.4×8% = 7.2% + 3.2% = 10.4%
CAPM: 3% + 1.0×(10%−3%) = 10.0%
Alpha = 10.4% − 10.0% = +0.4% → above SML
E(rP) = 0.6×12% + 0.4×8% = 7.2% + 3.2% = 10.4%
CAPM: 3% + 1.0×(10%−3%) = 10.0%
Alpha = 10.4% − 10.0% = +0.4% → above SML
Q24
Adding Asset X (rho=−0.8 with portfolio) slightly reduces E(r) but dramatically reduces sigma. Investor declines: "I don't want lower returns." Correct rebuttal?
AAdding X shifts POS northwest. Investor can re-lever the new portfolio to achieve the SAME return but LESS risk — or higher return at same risk. The sigma reduction creates opportunities along a superior CAL.
BInvestor is correct — lower E(r) is always bad
CBeneficial only if it increases Sharpe ratio of the market portfolio
DWith rho=−0.8, adding X automatically increases E(r)
A is correct. The investor evaluates E(r) in isolation — a mean-variance error. If X improves the Sharpe ratio of the new portfolio P', the investor can increase y* (leverage) to regain the original expected return while achieving LOWER total risk. Diversification reduces sigma, not E(r). D is wrong: correlation affects risk, not expected return.
Q25
Which statement about CAPM, APT, and EMH is most accurate?
ACAPM and APT always produce identical expected return predictions
BEMH implies CAPM must hold exactly in all markets
CAPT requires stronger assumptions than CAPM
DCAPM uses full equilibrium (homogeneous expectations, all investors optimise). APT uses only no-arbitrage with fewer assumptions. EMH is consistent with both but requires neither — markets can be efficient even if CAPM is mis-specified.
D is correct. CAPM: strong assumptions — homogeneous expectations, mean-variance optimisers, single period. APT: only needs factor structure + no arbitrage in well-diversified portfolios — weaker assumptions. EMH: prices reflect info, independent of which asset pricing model is correct. Anomalies may reflect omitted risk factors, not inefficiency.
3 Questions
Short Answer Questions
Write your answer — then reveal the model answer to compare
SAQ 1
[22 marks]
NovaStar Capital — Full Portfolio Construction
STOCK D: alpha_D=0.5%, beta_D=0.6, sigma(eD)=15%, E(rM)=11%, sigmaM=18%, rf=3%
STOCK E: alpha_E=1.2%, beta_E=1.4, sigma(eE)=22%
CLIENT Rajan: A=3.5
STOCK E: alpha_E=1.2%, beta_E=1.4, sigma(eE)=22%
CLIENT Rajan: A=3.5
Part (a) — SIM: E(ri), sigma_i², Cov[4 marks]
Using SIM, compute E(ri), sigma_i², and Cov(rD,rE) for both stocks. Show all formula workings.
Model Answer — Part (a)
E(rD) = 3% + 0.6×(11%−3%) + 0.5% = 3% + 4.8% + 0.5% = 8.3%
E(rE) = 3% + 1.4×8% + 1.2% = 3% + 11.2% + 1.2% = 15.4%
E(rE) = 3% + 1.4×8% + 1.2% = 3% + 11.2% + 1.2% = 15.4%
sigmaM² = (0.18)² = 0.0324
sigmaD² = (0.6)²×0.0324 + (0.15)² = 0.011664 + 0.0225 = 0.034164 → sigmaD = 18.49%
sigmaE² = (1.4)²×0.0324 + (0.22)² = 0.063504 + 0.0484 = 0.111904 → sigmaE = 33.45%
sigmaD² = (0.6)²×0.0324 + (0.15)² = 0.011664 + 0.0225 = 0.034164 → sigmaD = 18.49%
sigmaE² = (1.4)²×0.0324 + (0.22)² = 0.063504 + 0.0484 = 0.111904 → sigmaE = 33.45%
Cov(rD,rE) = 0.6 × 1.4 × 0.0324 = 0.027216
Part (b) — Tangency Portfolio[5 marks]
Find optimal weights wD, wE (max Sharpe), then compute E(rP) and sigmaP. Use excess returns.
Model Answer — Part (b)
E(RD)=5.3%, E(RE)=12.4%
Numerator = 0.053×0.111904 − 0.124×0.027216 = 0.005931 − 0.003375 = 0.002556
Denominator = 0.005931 + 0.124×0.034164 − 0.177×0.027216 = 0.010167 − 0.004817 = 0.005350
wD = 0.002556/0.005350 = 47.78% | wE = 52.22%
Numerator = 0.053×0.111904 − 0.124×0.027216 = 0.005931 − 0.003375 = 0.002556
Denominator = 0.005931 + 0.124×0.034164 − 0.177×0.027216 = 0.010167 − 0.004817 = 0.005350
wD = 0.002556/0.005350 = 47.78% | wE = 52.22%
E(rP) = 0.4778×8.3% + 0.5222×15.4% = 12.01%
sigmaP² = 0.22830×0.034164 + 0.27269×0.111904 + 0.49935×0.027216 = 0.051901
sigmaP = 22.78% | Sharpe P = 9.01%/22.78% = 0.396
sigmaP² = 0.22830×0.034164 + 0.27269×0.111904 + 0.49935×0.027216 = 0.051901
sigmaP = 22.78% | Sharpe P = 9.01%/22.78% = 0.396
Part (c) — Rajan's Complete Portfolio[4 marks]
Compute optimal y* (A=3.5), E(rC) and sigmaC. Verify Sharpe equals P's Sharpe.
Model Answer — Part (c)
y* = (12.01%−3%) / [3.5×(0.2278)²] = 0.0901 / [3.5×0.051901] = 0.0901/0.181654 = 0.4960
E(rC) = 3% + 0.4960×9.01% = 3% + 4.469% = 7.47%
sigmaC = 0.4960×22.78% = 11.30%
Sharpe C = (7.47%−3%)/11.30% = 4.47%/11.30% = 0.395 ≈ 0.396 ✓
E(rC) = 3% + 0.4960×9.01% = 3% + 4.469% = 7.47%
sigmaC = 0.4960×22.78% = 11.30%
Sharpe C = (7.47%−3%)/11.30% = 4.47%/11.30% = 0.395 ≈ 0.396 ✓
49.60% in Portfolio P, 50.40% in risk-free. Sharpe identical to P confirms both lie on same CAL.
Part (d) — All 4 Performance Measures[5 marks]
1yr later: rC=9.5%, sigmaC=11.30%, betaC=0.497, sigma(eC)=7%. Market: rM=10%, sigmaM=18%, rf=3%. Compute Sharpe, Treynor, Jensen's alpha, Info Ratio. Which is most appropriate? Interpret.
Model Answer — Part (d)
Sharpe C = (9.5%−3%)/11.30% = 0.575 | Sharpe M = 7%/18% = 0.389 → C OUTPERFORMS
Treynor C = (9.5%−3%)/0.497 = 13.08% | Treynor M = 7% → C OUTPERFORMS
Jensen alpha = 9.5% − [3% + 0.497×7%] = 9.5% − 6.479% = +3.02%
IR = 3.02%/7% = 0.432
Treynor C = (9.5%−3%)/0.497 = 13.08% | Treynor M = 7% → C OUTPERFORMS
Jensen alpha = 9.5% − [3% + 0.497×7%] = 9.5% − 6.479% = +3.02%
IR = 3.02%/7% = 0.432
Most appropriate: SHARPE — C is Rajan's entire portfolio. Total sigma (11.30%) matters. Sharpe 0.575 > Sharpe M 0.389 → C significantly outperformed. M² = 18%×(0.575−0.389) = +3.35% confirms outperformance.
SAQ 2
[12 marks]
OceanTrade Exports — FX + Interest Rate Hedging
FX EXPOSURE: Receive JPY 500,000,000 in 3 months. Spot S$0.0120/JPY. Futures S$0.0116/JPY. Contract: JPY 10,000,000.
IR EXPOSURE: S$15M bond portfolio, D*=7yr. Rates expected to rise 60bps. T-bond futures: par=$100K, price=$94/$100, D*_futures=8.5yr.
IR EXPOSURE: S$15M bond portfolio, D*=7yr. Rates expected to rise 60bps. T-bond futures: par=$100K, price=$94/$100, D*_futures=8.5yr.
Part (a) — FX Hedge Direction[2 marks]
Are they long or short JPY? Should they go LONG or SHORT JPY futures? Explain the economic logic of how the futures position offsets their exposure.
Model Answer — Part (a)
OceanTrade will RECEIVE JPY → they are LONG JPY (happy if JPY appreciates vs S$, hurt if JPY depreciates).
Core hedging rule: take the OPPOSITE position → SHORT JPY futures.
Economic logic: If JPY depreciates (S$0.0116 → S$0.0110), OceanTrade loses on the spot conversion. BUT the short futures position gains: they locked in selling JPY at S$0.0116, so profit = (S$0.0116−S$0.0110) per JPY. The futures gain OFFSETS the spot loss — locking in the conversion rate at the futures price regardless of spot movement.
Core hedging rule: take the OPPOSITE position → SHORT JPY futures.
Economic logic: If JPY depreciates (S$0.0116 → S$0.0110), OceanTrade loses on the spot conversion. BUT the short futures position gains: they locked in selling JPY at S$0.0116, so profit = (S$0.0116−S$0.0110) per JPY. The futures gain OFFSETS the spot loss — locking in the conversion rate at the futures price regardless of spot movement.
Part (b) — FX Hedge Calculation[3 marks]
(i) Number of JPY futures contracts to SHORT. (ii) Total hedged S$ proceeds. (iii) Verify hedged outcome if spot at maturity = S$0.0110/JPY.
Model Answer — Part (b)
(i) Contracts = 500,000,000 / 10,000,000 = 50 contracts SHORT
(ii) Hedged S$ = 500,000,000 × S$0.0116 = S$5,800,000
(iii) Spot = S$0.0110: Spot receipt = S$5,500,000. Futures profit = (0.0116−0.0110) × 500M = S$300,000. Total = S$5,500,000 + S$300,000 = S$5,800,000 ✓
(ii) Hedged S$ = 500,000,000 × S$0.0116 = S$5,800,000
(iii) Spot = S$0.0110: Spot receipt = S$5,500,000. Futures profit = (0.0116−0.0110) × 500M = S$300,000. Total = S$5,500,000 + S$300,000 = S$5,800,000 ✓
Part (c) — Interest Rate Hedge[4 marks]
(i) Dollar loss from 60bp rise (duration approx). (ii) PVBP of portfolio. (iii) Contracts to SHORT. (iv) Verify futures gain offsets loss.
Model Answer — Part (c)
(i) ΔP = −7 × 0.006 × $15M = −$630,000
(ii) PVBP = $15M × 7 × 0.0001 = $10,500 per bp
(iii) F_contract = ($94/$100)×$100,000 = $94,000. PVBP_futures = $94,000×8.5×0.0001 = $79.90/bp. Contracts = $10,500/$79.90 = 131 SHORT
(iv) Futures gain = 131×$79.90×60 = $628,014 ≈ $630,000 ✓
(ii) PVBP = $15M × 7 × 0.0001 = $10,500 per bp
(iii) F_contract = ($94/$100)×$100,000 = $94,000. PVBP_futures = $94,000×8.5×0.0001 = $79.90/bp. Contracts = $10,500/$79.90 = 131 SHORT
(iv) Futures gain = 131×$79.90×60 = $628,014 ≈ $630,000 ✓
SAQ 3
[10 marks]
CAPM vs APT vs FF3 — Conceptual Analysis
Portfolio P achieved rP=14.5%. Market: rM=10%, rf=3%. SIM regression: beta=1.1, R²=0.55, sigma(eP)=9%. FF3 analysis adds: s_P=0.4 (SMB), h_P=−0.2 (HML). SMB risk premium=3%, HML risk premium=4%.
Part (a) — CAPM alpha vs FF3 alpha[4 marks]
Compute (i) CAPM alpha and (ii) FF3 alpha for Portfolio P. Explain why the two alphas differ.
Model Answer — Part (a)
(i) CAPM E(rP) = 3% + 1.1×7% = 3% + 7.7% = 10.7%. CAPM alpha = 14.5% − 10.7% = +3.8%
(ii) FF3 E(rP) = 3% + 1.1×7% + 0.4×3% + (−0.2)×4% = 10.7% + 1.2% − 0.8% = 11.1%. FF3 alpha = 14.5% − 11.1% = +3.4%
(ii) FF3 E(rP) = 3% + 1.1×7% + 0.4×3% + (−0.2)×4% = 10.7% + 1.2% − 0.8% = 11.1%. FF3 alpha = 14.5% − 11.1% = +3.4%
Alphas differ because FF3 includes two additional risk factors (SMB, HML) that explain some of the return CAPM labels as alpha. The SMB loading (s=0.4) earns an extra 1.2% risk premium; the negative HML loading costs 0.8%. FF3 alpha (3.4%) is smaller than CAPM alpha (3.8%) because FF3 attributes 0.4% of the "alpha" to measurable risk exposures. If FF3 is the correct model, the true skill alpha is only 3.4%.
Part (b) — Risk Decomposition & R²[3 marks]
Using SIM, compute: (i) systematic risk, (ii) firm-specific risk, (iii) what R²=0.55 implies about the portfolio's exposure to the market factor.
Model Answer — Part (b)
sigma²(eP) = (0.09)² = 0.0081 (given as firm-specific)
R² = systematic / total → total = systematic / R²
Systematic = beta²×sigmaM² = (1.1)²×(0.07)² ... wait, sigmaM not given.
Instead use: sigma²(eP) = (1−R²)×sigmaP² → sigmaP² = 0.0081/0.45 = 0.018
Systematic = R²×sigmaP² = 0.55×0.018 = 0.0099
Firm-specific = 0.0081
R²=0.55 means 55% of return variance is explained by the market factor; 45% is unexplained firm-specific risk.
R² = systematic / total → total = systematic / R²
Systematic = beta²×sigmaM² = (1.1)²×(0.07)² ... wait, sigmaM not given.
Instead use: sigma²(eP) = (1−R²)×sigmaP² → sigmaP² = 0.0081/0.45 = 0.018
Systematic = R²×sigmaP² = 0.55×0.018 = 0.0099
Firm-specific = 0.0081
R²=0.55 means 55% of return variance is explained by the market factor; 45% is unexplained firm-specific risk.
Part (c) — Limits to Arbitrage[3 marks]
Even if Portfolio P has genuine positive alpha, explain THREE reasons why this alpha might persist without being fully arbitraged away.
Model Answer — Part (c)
1. Fundamental risk: Even if P appears mispriced, there is no perfect substitute that has the same risk characteristics. Arbitrageurs who buy P and short a substitute still bear basis risk — the substitute may not move identically to P. If the mispricing worsens before correcting, the arbitrageur may be forced out of their position (Keynes: "markets can remain irrational longer than you can remain solvent").
2. Implementation costs: Exploiting a 3.4% FF3 alpha requires substantial transaction costs (bid-ask spreads, brokerage), potential short-selling constraints, and margin requirements. After costs, the net profit may be zero or negative, making the trade unattractive. The alpha must exceed the cost of the strategy.
3. Model risk: If neither CAPM nor FF3 is the true asset pricing model, the computed "alpha" may simply reflect a missing risk factor that the models don't capture. Arbitrageurs who believe the alpha is genuine risk mis-measurement rather than mispricing would not trade on it. The persistence of alpha may simply reflect model mis-specification rather than true opportunity.
2. Implementation costs: Exploiting a 3.4% FF3 alpha requires substantial transaction costs (bid-ask spreads, brokerage), potential short-selling constraints, and margin requirements. After costs, the net profit may be zero or negative, making the trade unattractive. The alpha must exceed the cost of the strategy.
3. Model risk: If neither CAPM nor FF3 is the true asset pricing model, the computed "alpha" may simply reflect a missing risk factor that the models don't capture. Arbitrageurs who believe the alpha is genuine risk mis-measurement rather than mispricing would not trade on it. The persistence of alpha may simply reflect model mis-specification rather than true opportunity.
3 Drills
Calculation Drills
Step-by-step worked problems — futures, bonds, duration, performance
Drill 1
Mark-to-Market & Margin Calls
LONG 2 T-bond futures. Opening price: 97'08 (=97.25). IMR=$2,530/contract, MMR=$2,300/contract.
Monday settle: 96'16 (=96.50) | Tuesday settle: 95'24 (=95.75) | Wednesday settle: 96'08 (=96.25)
Contract multiplier = $1,000 (=$100,000/$100)
Monday settle: 96'16 (=96.50) | Tuesday settle: 95'24 (=95.75) | Wednesday settle: 96'08 (=96.25)
Contract multiplier = $1,000 (=$100,000/$100)
Part (a)[3 marks]
Calculate daily P&L for Monday, Tuesday, Wednesday (for 2 contracts). Total cumulative P&L?
Model Answer
Monday: (96.50−97.25) × 1,000 × 2 = −0.75 × 2,000 = −$1,500
Tuesday: (95.75−96.50) × 1,000 × 2 = −$1,500
Wednesday: (96.25−95.75) × 1,000 × 2 = +$1,000
Cumulative: −$1,500 − $1,500 + $1,000 = −$2,000
Tuesday: (95.75−96.50) × 1,000 × 2 = −$1,500
Wednesday: (96.25−95.75) × 1,000 × 2 = +$1,000
Cumulative: −$1,500 − $1,500 + $1,000 = −$2,000
T-bond prices use 32nds: 97'08 = 97 + 8/32 = 97.25. Not 97.8!
Part (b)[3 marks]
Track the margin account balance for 2 contracts. Starting balance = IMR = $5,060. On which days are there margin calls? How much must be deposited?
Model Answer
After Monday: $5,060 − $1,500 = $3,560 < MMR ($4,600) → MARGIN CALL: deposit $1,500 → restore to $5,060
After Tuesday: $5,060 − $1,500 = $3,560 < $4,600 → MARGIN CALL: deposit $1,500 → restore to $5,060
After Wednesday: $5,060 + $1,000 = $6,060 > $4,600 → no call
After Tuesday: $5,060 − $1,500 = $3,560 < $4,600 → MARGIN CALL: deposit $1,500 → restore to $5,060
After Wednesday: $5,060 + $1,000 = $6,060 > $4,600 → no call
After a margin call, restore to IMR ($5,060), not just to MMR ($4,600). Very common mistake.
Drill 2
Bond Valuation, YTM & Duration
Bond: 8-year maturity, 7% coupon rate, semiannual payments, par=$1,000, current YTM=9% APR.
A second 8-year 7% semiannual bond has YTM=7% (priced at par).
A second 8-year 7% semiannual bond has YTM=7% (priced at par).
Part (a)[3 marks]
Compute the price of the first bond (YTM=9%). Is it a premium, discount, or at-par bond?
Model Answer
CPN=$35, N=16, r=4.5%
PVIFA(4.5%,16) = [1−(1.045)^−16]/0.045 = [1−0.4945]/0.045 = 11.234
PVIF(4.5%,16) = (1.045)^−16 = 0.4945
P = 35×11.234 + 1,000×0.4945 = 393.19 + 494.50 = $887.69
PVIFA(4.5%,16) = [1−(1.045)^−16]/0.045 = [1−0.4945]/0.045 = 11.234
PVIF(4.5%,16) = (1.045)^−16 = 0.4945
P = 35×11.234 + 1,000×0.4945 = 393.19 + 494.50 = $887.69
DISCOUNT bond (price $887.69 < par $1,000). Coupon rate 7% < YTM 9% → discount. Ordering: Coupon rate < Current Yield < YTM.
Part (b)[4 marks]
For the first bond (YTM=9%), compute Macaulay Duration by calculating PV weights for ALL 16 cashflows (or show the process for the first 3 and last 1, then state the final result). Then compute Modified Duration D*.
Model Answer
Process: For each period t (t=0.5 to 8.0): PV(CFt) = CF/(1.045)^(2t). Weight = PV(CFt)/887.69. Contribution = t × weight.
Sum of [t × PV(CFt)] ≈ 5,284 (in half-year weighted PV terms)
Macaulay Duration ≈ 5,284 / 887.69 ≈ 5.95 years
Modified Duration: D* = D / (1+r) = 5.95 / 1.045 = 5.70 years
Sum of [t × PV(CFt)] ≈ 5,284 (in half-year weighted PV terms)
Macaulay Duration ≈ 5,284 / 887.69 ≈ 5.95 years
Modified Duration: D* = D / (1+r) = 5.95 / 1.045 = 5.70 years
If yields rise by 1%: ΔP/P ≈ −5.70 × 0.01 = −5.70%. Dollar change ≈ −$50.60 per bond.
Drill 3
Full Performance Evaluation
Fund A: rA=35%, sigmaA=42%, betaA=1.2, tracking error=18%
Fund B: rB=22%, sigmaB=25%, betaB=0.9, tracking error=9%
Market: rM=28%, sigmaM=30%, rf=6%
Fund B: rB=22%, sigmaB=25%, betaB=0.9, tracking error=9%
Market: rM=28%, sigmaM=30%, rf=6%
Full Drill[6 marks]
Compute Sharpe, Treynor, Jensen's alpha, and M² for both funds and the market. State which fund is preferred when: (i) it's the investor's only holding, (ii) it's one sub-portfolio among many, (iii) selecting active fund to mix with index.
Model Answer
Sharpe A=(35−6)/42=0.690 | Sharpe B=(22−6)/25=0.640 | Sharpe M=(28−6)/30=0.733
Treynor A=(35−6)/1.2=24.2% | Treynor B=(22−6)/0.9=17.8% | Treynor M=22.0%
Jensen alpha A=35%−[6%+1.2×22%]=35%−32.4%=+2.6% | alpha B=22%−[6%+0.9×22%]=22%−25.8%=−3.8%
M² A: y=30/42=0.714; rP*A=0.714×35%+0.286×6%=24.99%+1.72%=26.71%; M²=26.71%−28%=−1.3%
M² B: y=30/25=1.2; rP*B=1.2×22%−0.2×6%=26.4%−1.2%=25.2%; M²=25.2%−28%=−2.8%
IR A=2.6%/18%=0.144 | IR B=−3.8%/9%=−0.422
Treynor A=(35−6)/1.2=24.2% | Treynor B=(22−6)/0.9=17.8% | Treynor M=22.0%
Jensen alpha A=35%−[6%+1.2×22%]=35%−32.4%=+2.6% | alpha B=22%−[6%+0.9×22%]=22%−25.8%=−3.8%
M² A: y=30/42=0.714; rP*A=0.714×35%+0.286×6%=24.99%+1.72%=26.71%; M²=26.71%−28%=−1.3%
M² B: y=30/25=1.2; rP*B=1.2×22%−0.2×6%=26.4%−1.2%=25.2%; M²=25.2%−28%=−2.8%
IR A=2.6%/18%=0.144 | IR B=−3.8%/9%=−0.422
(i) Entire portfolio → SHARPE: A wins (0.690 > 0.640), but BOTH underperform market (Sharpe M=0.733). Neither is a good standalone choice!
(ii) One sub-portfolio → TREYNOR: A wins (24.2% > 22.0% > 17.8%). A adds value per unit of systematic risk.
(iii) Mix with index → INFO RATIO: A wins (0.144 > −0.422). A generates positive alpha per unit of avoidable risk. B has NEGATIVE alpha → don't add B to an index fund.
(ii) One sub-portfolio → TREYNOR: A wins (24.2% > 22.0% > 17.8%). A adds value per unit of systematic risk.
(iii) Mix with index → INFO RATIO: A wins (0.144 > −0.422). A generates positive alpha per unit of avoidable risk. B has NEGATIVE alpha → don't add B to an index fund.
Interactive Calculators
Financial
Calculators
Live calculators mirroring exact exam formulas — enter values, see full working instantly.
Capital Allocation
Optimal y* Calculator
y* = [E(rP) − rf] / [A × σP²]
E(rP) — Risky Portfolio Return (%)
rf — Risk-Free Rate (%)
σP — Portfolio Std Dev (%)
A — Risk Aversion Coefficient
Optimal y* (Weight in Risky Portfolio)
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Bond Valuation
Bond Pricer
P = CPN × PVIFA(r,n) + Par × PVIF(r,n)
Par Value ($)
Coupon Rate (% annual)
YTM (% annual)
Years to Maturity
Payment Frequency
Bond Price
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Duration & Sensitivity
Duration Calculator
D = ∑[t × PV(CFt)] / Price D* = D / (1+r)
Par Value ($)
Coupon Rate (% annual)
YTM (% annual)
Years to Maturity
Yield Change ΔR (basis points)
Macaulay Duration
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Modified Duration
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Futures Hedging
PVBP Hedge Ratio
Contracts = PVBP_portfolio / PVBP_futures
Portfolio Value ($M)
Portfolio D* (Modified Duration, years)
Target D* (0 = full hedge)
Futures Price (per $100 par)
Futures Par Value ($)
Futures D* (Modified Duration, years)
Contracts Required
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CAPM / SML
CAPM Alpha Calculator
E(ri) = rf + β × [E(rM) − rf] α = Actual − CAPM
rf — Risk-Free Rate (%)
E(rM) — Expected Market Return (%)
β — Asset Beta
Actual E(ri) — Forecasted Return (%)
CAPM Required Return
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Alpha (α)
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FX Hedging
FX Hedge & IRP Calculator
F₀ = E₀ × [(1+r_A)/(1+r_B)]^T | Contracts = Exposure / Contract size
Spot Rate E₀ (home per foreign)
Home Interest Rate r_A (%/yr)
Foreign Interest Rate r_B (%/yr)
Tenor T (years)
FX Exposure (units of foreign currency)
Contract Size (units of foreign currency)
IRP Forward Rate F₀
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Contracts to SHORT
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Instructor's Dynamic
I was taught by Professor Karen Gan, who brings prior investment banking experience into the classroom — she regularly draws on real-world examples to show how the concepts apply beyond academia. Her lectures are thorough and well-structured, making even the denser topics approachable.
That said, the tutorial practice questions were fairly straightforward, so the questions in this study guide are a better reflection of the actual difficulty tested. The mid-term was more manageable than the final assessment, which was noticeably more demanding.
For project work, there were two components: Part I was a current affairs presentation that required an interactive element, and Part II was an investor recommendation report. I've linked the interactive game I built for Part I below, just as a point of reference.
Project Work
The game is like a ripped-off version of monopoly, and it is built in three parts: Host, Board and Player.
We split the class into 6 teams, and each team will join the game, and as our presentation progress, the results will be revealed. Participants are prompted to invest in the different assets class as a response to current affairs. Feel free to explore how the game work through the links.
That's all! I hope you enjoyed my study guide of the Financial Markets & Investments module. Once again, this is based on my personal experience, and there may be errors that I did not notice. As Heraclitus wisely said, "The only constant in life is change," so the curriculum might have already evolved by the time you read this. Please take this as a guide, not a definitive account!
