CA Final · Advanced Financial Management

Portfolio Management: formula sheet

Full chapter guide

Key formulas

Expected return (with probabilities): E(R) = Σ pi × Ri; Probabilities must add up to 1. Use percentages or decimals consistently.
Average return (historical data): Average R = ΣR ÷ n; Use this when the question gives past returns and no probabilities.
Variance and standard deviation: σ² = Σ pi × (Ri − E(R))²; σ = √σ²; For historical data with no probabilities, divide the sum of squared deviations by n or by n − 1 as the question directs. If unclear, state your assumption.
Covariance: Cov(A,B) = Σ pi × (RA − E(RA)) × (RB − E(RB)); For historical data, sum the products of deviations and divide by n (or n − 1 if the question treats the data as a sample).
Correlation coefficient: ρAB = Cov(A,B) ÷ (σA × σB); Lies from −1 to +1. Hence Cov(A,B) = ρAB × σA × σB.
Portfolio return: Rp = wA × RA + wB × RB; Weights are proportions of total investment and add up to 1.
Two-asset portfolio variance: σp² = wA²σA² + wB²σB² + 2 × wA × wB × Cov(A,B); Same as using 2 × wA × wB × ρ × σA × σB for the last term. Portfolio SD = √σp².
Minimum-variance weight (two assets): wA = (σB² − Cov) ÷ (σA² + σB² − 2 × Cov); wB = 1 − wA; Gives the mix with the lowest portfolio variance.
n-asset portfolio variance: σp² = Σi Σj wi × wj × Cov(i,j); Cov(i,i) = σi². Each pair appears twice in the double sum.
Coefficient of variation: CV = σ ÷ E(R); Risk per unit of return. Useful for comparing securities with different expected returns.
Total risk: Total risk = Systematic risk + Unsystematic risk; Diversification reduces only the unsystematic part.
Portfolio expected return (two assets): Rp = w1 × R1 + w2 × R2, with w1 + w2 = 1; A simple weighted average. Correlation does not affect it.
Covariance and correlation: Cov12 = ρ12 × σ1 × σ2, so ρ12 = Cov12 ÷ (σ1 × σ2); If the question gives covariance, use it directly. Do not multiply by ρ again.
Portfolio variance (two assets): σp² = w1²σ1² + w2²σ2² + 2 × w1 × w2 × ρ12 × σ1 × σ2; Standard deviation σp = √σp². Square the weights in the first two terms.
Minimum variance weight (two assets): w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2 × Cov12); w2 = 1 − w1; Gives the lowest-variance mix. Works for any ρ. A negative weight means short selling.
Perfect negative correlation (ρ = −1): w1 = σ2 ÷ (σ1 + σ2); w2 = σ1 ÷ (σ1 + σ2); σp = 0; Risk-free portfolio is possible only in this special case.
Bounds on portfolio risk: σp ≤ w1σ1 + w2σ2, with equality only when ρ12 = +1 (for non-negative weights); Shows that diversification benefit exists whenever ρ < +1.
CAPM required return: E(Ri) = Rf + βi × (Rm − Rf); Rm − Rf is the market risk premium. Use the result as the required return (cost of equity).
Beta of a security: β = Cov(i, m) ÷ σm² = ρim × σi ÷ σm; Cov is covariance with the market; σm² is market variance. ρim is the correlation coefficient.
Beta of a portfolio: βp = Σ(wi × βi); Weights are market-value proportions and must add up to 1. Include the risk-free asset with β = 0.
Security Market Line: Required return = Rf + slope × β, where slope = (Rm − Rf); Intercept is Rf. Straight line with beta on the x-axis.
Capital Market Line: E(Rp) = Rf + [(Rm − Rf) ÷ σm] × σp; Uses total risk σp. Applies only to efficient portfolios.
Alpha / valuation test: Alpha = Expected return − Required return; Alpha > 0: undervalued. Alpha < 0: overvalued. Alpha = 0: fairly priced.
Total risk split: σ² = β² × σm² + unsystematic variance; Systematic variance is β²σm². Only this part is priced by CAPM.
Single index model: Ri = αi + βi Rm + ei; Taking expected values: E(Ri) = αi + βi E(Rm), because E(ei) = 0.
Security variance: σi² = βi² σm² + σei²; First term is systematic risk, second is unsystematic risk.
Covariance between two securities: Cov(i, j) = βi × βj × σm²; Holds in the single index model because errors are assumed uncorrelated.
Portfolio alpha and beta: αp = Σ wi αi ; βp = Σ wi βi; Weights wi are market-value proportions and sum to 1.
Portfolio variance (single index): σp² = βp² σm² + Σ wi² σei²; Unsystematic part shrinks as the number of securities rises.
APT expected return: E(R) = Rf + β1 λ1 + β2 λ2 + … + βn λn; λ is the risk premium of each factor, that is, the expected return of the factor over Rf.
Arbitrage rule: Expected return > APT return: underpriced, buy. Expected return < APT return: overpriced, sell.; Fair price today = (expected price + dividend) ÷ (1 + APT return).
Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; Uses total risk. Higher is better. Related to the Capital Market Line.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; Uses systematic risk. Higher is better. Related to the Security Market Line.
Jensen's alpha: α = Rp − [Rf + βp(Rm − Rf)]; Positive means outperformance. Zero means in line with CAPM. Negative means underperformance.
CAPM expected return: Expected Rp = Rf + βp(Rm − Rf); This is the benchmark return used inside Jensen's alpha.
Excess return: Excess return = Rp − Rf; The numerator of Sharpe and Treynor.
Fama: selectivity: Selectivity = (Rp − Rf) − (σp ÷ σm)(Rm − Rf); Return for the investor's risk = (σp ÷ σm)(Rm − Rf). Selectivity is the excess return left after deducting it.
Fama: selectivity split: Diversification = (σp ÷ σm − βp) × (Rm − Rf); Net selectivity = Selectivity − Diversification; Two separate relations. Jensen's alpha = Selectivity + Diversification (in the worked example, 1.4 = 0.5 + 0.9). Net selectivity = Selectivity − Diversification (0.5 − 0.9 = −0.4), so it is not equal to Jensen's alpha. A negative net selectivity means the selection return was less than the extra return required for incomplete diversification.
Constant ratio plan target: Target equity = Total portfolio value × equity ratio; Target debt = Total portfolio value × debt ratio; Revise when the actual ratio moves beyond the trigger. Equity rises: sell equity and buy debt. Equity falls: buy equity and sell debt.
Constant rupee value plan: Target equity = fixed rupee amount; Debt = Total portfolio value − fixed equity amount; Any equity gain above the fixed amount moves to debt. Any shortfall is topped up from debt.
Rupee cost averaging: Average cost per unit = Total amount invested ÷ Total units bought; With equal amounts invested each period, this is the harmonic mean of prices, which is never above the arithmetic mean of the prices.
CPPI cushion: Cushion = Portfolio value − Floor; The floor is the value you want to protect. If a question gives a guaranteed amount at a future date, the floor is usually its present value at the risk-free rate.
CPPI equity exposure: Equity exposure = Multiplier (m) × Cushion; If this exceeds the portfolio value, a no-borrowing question caps equity at 100% of the portfolio.
CPPI safe asset: Investment in safe asset = Portfolio value − Equity exposure; Use this to check that the two parts add back to the total.
Payoff shape: Constant ratio plan: concave payoff. CPPI: convex payoff.; Constant ratio does better in oscillating, range-bound markets. CPPI does better in strongly trending markets and protects against steep falls, but does badly in whipsaw markets.
Net Asset Value (NAV) per unit: NAV = (Market value of investments + Receivables + Accrued income + Cash − Liabilities − Accrued expenses) ÷ Number of units outstanding; Use market value, not cost. Deduct all liabilities and expenses due before dividing.
Return on investment (single period): Return % = [(NAV₁ − NAV₀) + Distributions] ÷ NAV₀ × 100; NAV₀ is opening NAV, NAV₁ is closing NAV. Include dividends and capital gains distributions.
Return with loads: Return % = [(Redemption proceeds + Distributions) − Amount invested] ÷ Amount invested × 100; Amount invested includes entry load. Redemption proceeds are after exit load.
Annualised return (simple): Annual return % = Return for the period % × (12 ÷ months held); Use only if the question asks for simple annualisation.
Units after reinvestment: Units purchased = Amount ÷ Purchase price per unit; Use NAV (plus load if any) as the purchase price.
Strategic vs tactical allocation: Strategic = long-term target weights; Tactical = temporary deviation based on a market view; Say this in one line each in theory answers.

Quick revision

Expected return of a portfolio = Σ (weight × expected return of each security).
Two-asset risk: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2.
Covariance = ρ × σ1 × σ2. Correlation lies between -1 and +1.
Diversification removes unsystematic risk, not systematic risk.
Portfolio beta = Σ (weight × beta of each security).
CAPM: Required return = Rf + β × (Rm − Rf). The Security Market Line plots this against beta.
If expected return is above the CAPM required return, the security is underpriced; if below, overpriced.
APT uses several factors, each with its own sensitivity and risk premium, and needs no market portfolio assumption.
Sharpe ratio = (Rp − Rf) ÷ σp. Treynor ratio = (Rp − Rf) ÷ βp.
Alpha = actual return − return required by CAPM. A positive Alpha means outperformance.
Passive strategy holds a diversified portfolio with little trading; active strategy tries to beat the market.
NAV per unit = (Market value of assets − liabilities) ÷ number of units outstanding.

Common mistakes

Not squaring the weights in the variance formula. Fix: Write the formula before substituting. The weight multiplies the SD in the SD form, and its square multiplies the variance.
Taking portfolio SD as the weighted average of the individual SDs. Fix: That holds only when ρ = +1. Otherwise use the full variance formula with the covariance term.
Calculating portfolio risk as the weighted average of standard deviations. Fix: Always use the variance formula with the covariance term. The weighted average of standard deviations is only the upper limit, reached when ρ = +1.
Putting ρ in the covariance slot, or multiplying by ρ when covariance is already given. Fix: Check the data first. If covariance is given, the term is 2 × w1 × w2 × Cov12 with no further ρ or σ.
Using Rm instead of (Rm − Rf) as the multiplier of beta. Fix: Always write the premium (Rm − Rf) as a separate line before multiplying by beta.
Dividing covariance by market standard deviation to get beta. Fix: Beta = Cov(i, m) ÷ σm². If only standard deviations and correlation are given, use ρ × σi ÷ σm.
Adding the risk-free rate inside each factor premium or subtracting it twice Fix: Read the wording. If the figure is a premium, use it as is. If it is a factor's total expected return, subtract Rf first.
Taking a weighted average of standard deviations for portfolio risk Fix: Average only alpha, beta and returns. Compute portfolio risk from variances using σp² = βp² σm² + Σ w² σe².
Using Rp instead of Rp − Rf in the numerator Fix: Always write the excess return as a separate first step before dividing.
Dividing by beta in Sharpe or by standard deviation in Treynor Fix: Remember: Sharpe uses Standard deviation (S for S). Treynor uses beta, the systematic risk.

Exam tips

Show the formula, the substituted values and the result as separate lines. Marks are given for method even if the final figure is slightly off.
In case-scenario MCQs, first check whether covariance or correlation is given. Using the wrong one is the commonest trap, so keep ρ × σA × σB handy.
Do the ρ = +1 check on every two-asset answer. If your SD is above the weighted average of the SDs, you have made an error.
Always end a written answer with a one-line interpretation: which portfolio or security is better on return, risk or CV, and why.
Keep SD to two decimal places in working, and avoid rounding SDs before squaring them again. Rounding early causes small mismatches.
In case-based MCQs, read for the correlation value first. ρ = +1 means no diversification benefit, ρ = −1 allows zero risk, and anything between gives partial benefit.
In numericals, show the covariance, the three terms of the variance and the weights separately. Marks are given for each step even if the final figure slips.
For theory questions, keep a fixed structure: define systematic and unsystematic risk, explain how diversification works through covariance, then describe the efficient frontier and the Markowitz assumptions.