Portfolio Performance Evaluation and Portfolio Revision: formula sheet

Key formulas

Holding period return (single period)

R = (P₁ − P₀ + D) ÷ P₀

P₀ is the opening value, P₁ the closing value, D the income received. Use only when there are no interim cash flows.

Excess return over benchmark

Excess return = Portfolio return − Benchmark return

Positive means outperformance, but it is meaningful only if the benchmark has similar risk.

Risk premium

Risk premium = Portfolio return − Risk-free return

The reward earned for taking risk; the base for risk-adjusted measures.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

Rp = portfolio return, Rf = risk-free return, σp = standard deviation of portfolio returns. Uses total risk.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

βp = portfolio beta. Uses systematic risk only. Meaningful for ranking when betas are positive.

Market benchmark Sharpe

Sharpe (market) = (Rm − Rf) ÷ σm

Compare the portfolio's Sharpe with this. Higher means it beat the market on a risk-adjusted basis.

Market benchmark Treynor

Treynor (market) = (Rm − Rf) ÷ 1 = Rm − Rf

The market beta is 1, so the market's Treynor ratio equals the market risk premium.

Decision rule

Higher ratio = better risk-adjusted performance

Rank portfolios in descending order of the ratio.

Jensen's alpha

α = Rp − [Rf + βp(Rm − Rf)]

Use average returns over the same period for Rp, Rm and Rf. Positive α means the portfolio beat its CAPM required return.

CAPM required return

Rβ = Rf + βp(Rm − Rf)

This is the benchmark return in Jensen's measure. It is the point on the SML at the portfolio's beta.

Return on CML at portfolio's total risk

Rx = Rf + (σp ÷ σm)(Rm − Rf)

Used in Fama's net selectivity. It is the return a passive mix of market and risk-free asset would earn at risk σp.

Fama net selectivity

Net selectivity = Rp − Rx = Rp − [Rf + (σp ÷ σm)(Rm − Rf)]

This is the return from the manager's skill after allowing for lack of diversification.

Fama diversification

Diversification = (σp ÷ σm − βp)(Rm − Rf)

This is the extra return needed to compensate for incomplete diversification. It is zero or positive, since σp ÷ σm is at least βp.

Link between Jensen and Fama

Jensen's α = Net selectivity + Diversification

Use this to check your working.

M-squared

Rp* = Rf + (σm ÷ σp)(Rp − Rf); M² = Rp* − Rm

Rp* is the portfolio's return after adjusting its risk to equal the market's. Positive M² means it beat the market at equal risk.

Information ratio

IR = (Rp − Rb) ÷ Tracking error

Rb is the benchmark return. Tracking error is the standard deviation of (Rp − Rb) over the period, not σp.

Treynor and Sharpe (for comparison)

Treynor = (Rp − Rf) ÷ βp; Sharpe = (Rp − Rf) ÷ σp

Treynor uses systematic risk. Sharpe uses total risk.

Revision worth-doing test

Revise only if: Expected gain from revision > Transaction costs + Tax on gains + Cost of any lost opportunity

A decision rule, not a statutory formula. Compare on an after-cost, after-tax basis.

Net proceeds from selling a security

Net proceeds = Sale value − Brokerage and other charges − Tax on capital gain

Reinvest only the net amount. Capital gain = Sale price − Cost of acquisition, using the tax rules applicable to the holding period.

Current weight of a security

Weight = Market value of the security ÷ Total market value of the portfolio

Compare current weights with target weights to see how far the portfolio has drifted.

Amount to trade for rebalancing

Trade amount = (Target weight − Current weight) × Total portfolio value

Positive means buy; negative means sell.

Constant rupee value plan

Stock part = fixed ₹ amount; Action = Current stock value − Fixed value

If positive and the trigger is met, sell that amount and add it to bonds. If negative, buy stock from bonds.

Constant ratio plan

Target stock value = Stock % × (Stock value + Bond value)

Action = Target − Current stock value. Bond target is the remaining percentage of total. Rebalance only when drift crosses the stated tolerance.

Variable ratio plan

Target stock value = Stock % for the new market level × Total portfolio value

Take the stock % from the table given in the question. It falls as the market rises and rises as it falls.

Rupee cost averaging: average cost

Average cost per unit = Total amount invested ÷ Total units bought

This is the harmonic mean of the purchase prices. It is never above the simple average of the prices.

Units bought each period

Units = Fixed amount ÷ Price (NAV) in that period

Add units across periods before dividing.

Quick revision

• Sharpe ratio = (Rp − Rf) ÷ σp, using total risk.
• Treynor ratio = (Rp − Rf) ÷ βp, using systematic risk.
• Jensen's alpha = Rp − [Rf + βp × (Rm − Rf)].
• A positive alpha means the portfolio beat the return expected for its beta.
• Sharpe suits a whole portfolio; Treynor suits one portfolio among many held by a diversified investor.
• For a well-diversified portfolio, Sharpe and Treynor rankings tend to agree.
• Always subtract the risk-free rate from the portfolio return before dividing.
• Portfolio revision means changing holdings to keep the portfolio aligned with the investor's objectives.
• Constant rupee value plan keeps the value of the aggressive portfolio fixed and trades the difference.
• Constant ratio plan keeps a fixed ratio between aggressive and conservative portfolios.
• Formula plans need a trigger point and ignore forecasts.
• State your ranking and recommendation in words at the end.

Common mistakes

• Judging a portfolio on return alone. Fix: Always link return to the risk taken. A higher return from much higher risk is not necessarily better.
• Leaving out dividends or interest from return. Fix: Add all income received to the capital gain before dividing by the opening value.
• Dividing the portfolio return by risk without subtracting the risk-free rate. Fix: Always write Rp − Rf as the first step, before any division.
• Using beta in the Sharpe ratio or standard deviation in the Treynor ratio. Fix: Remember: Sharpe starts with S, for standard deviation. Treynor uses beta.
• Using the market return Rm instead of the market risk premium (Rm − Rf) in the CAPM term. Fix: Always write Rf + β(Rm − Rf) in full. Compute (Rm − Rf) as a separate line first.
• Treating a positive alpha as proof of skill. Fix: Say that alpha is positive and then add the qualification. A high σp with a low β can create alpha that is just a diversification component. Fama's net selectivity separates the two.
• Treating revision as the same as performance evaluation. Fix: Evaluation measures past results. Revision changes the portfolio. Say that evaluation feeds revision.
• Listing only market changes as the need for revision. Fix: Include investor circumstances, security-specific changes, weight drift and new opportunities.
• Mixing up constant rupee value and constant ratio plans. Fix: Remember what is fixed. Rupee value plan: the stock amount in ₹. Ratio plan: the percentage split, so the stock amount moves with the total.
• Computing the constant ratio target on the old total. Fix: Add the new stock value and the bond value first. Apply the percentage to that new total.

Exam tips

• Write the stages of evaluation as a short numbered list; examiners reward a clear sequence.
• In a theory answer, always include the problems: return measurement, risk measurement and benchmark choice.
• In case questions, state your benchmark choice and its reason in one line before comparing.
• Link the conclusion to revision: say what the investor should do next.
• Show return workings fully, including income, even for 2-mark MCQs.
• Check what the question gives. If only standard deviation is given, the answer is Sharpe. If only beta is given, it is Treynor.
• Show the excess return column explicitly. Marks are given for method even if the final division has a slip.
• Always end with a ranking and a one-line recommendation. Numerical questions in SFM expect a decision.