Investments: formula sheet

Key formulas

Order of the investment process

Objectives → Risk profile and constraints → Asset allocation → Security selection → Implementation → Monitoring and rebalancing

Allocation comes before choosing individual securities. Objectives come before everything.

Risk-return tendency

Higher expected return ↔ higher risk

A general tendency, not a promise. Higher risk does not guarantee higher return.

Broad liquidity ranking

Cash and bank deposits > listed equity and listed debt > gold > real estate and unlisted alternatives

Use this as a rough guide. Liquidity depends on the specific instrument and market conditions.

Return components

Total return = Income (interest, dividend, rent) + Capital gain or loss

Gold gives only capital gain or loss. Debt is mostly income. Equity gives both.

Absolute return

Absolute return (%) = (Ending value − Beginning value + Income received) ÷ Beginning value × 100

Ignores the time period. Without income, it is just the change in value over the starting value.

CAGR

CAGR = (Ending value ÷ Beginning value)^(1 ÷ n) − 1

n is the number of years. Valid when there are no interim cash flows.

Total risk

Total risk = Systematic risk + Unsystematic risk

Diversification removes only the unsystematic part.

Average (mean) return

Mean = Σ returns ÷ number of observations

First step in finding standard deviation.

Variance and standard deviation (sample)

Variance = Σ (return − mean)² ÷ (n − 1); Standard deviation = √Variance

Use n instead of n − 1 if the question says population or treats the data as complete.

Beta

Beta = Covariance (asset, market) ÷ Variance (market)

Beta of 1 moves with the market; above 1 is more volatile; below 1 is less volatile.

Expected market-linked move

Expected change in portfolio ≈ Beta × Change in market

A simple approximation used in questions.

Earnings per share (EPS)

EPS = Profit after tax (available to equity shareholders) ÷ Number of equity shares

Preference dividend is deducted from profit first.

Price-to-earnings (P/E)

P/E = Market price per share ÷ EPS

Equals market capitalisation ÷ net profit. It shows how many rupees you pay for ₹1 of earnings.

Book value per share

Book value per share = Net worth ÷ Number of equity shares

Net worth = share capital + reserves, for equity shareholders.

Price-to-book (P/B)

P/B = Market price per share ÷ Book value per share

Often used for banks and asset-heavy companies.

Dividend yield

Dividend yield = Dividend per share ÷ Market price per share × 100

Uses market price, not face value.

Dividend payout ratio

Payout ratio = Dividend per share ÷ EPS × 100

The rest is retained in the business.

Return on equity (ROE)

ROE = Net profit ÷ Shareholders' equity × 100

Measures profit earned on shareholders' money.

Earnings yield

Earnings yield = EPS ÷ Price × 100 = 1 ÷ P/E × 100

Inverse of P/E.

Bond price

Price = Σ [C ÷ (1 + y)^t] + [F ÷ (1 + y)^n]

C is coupon per period, y is yield per period, F is face value, n is number of periods.

Current yield

Current yield = Annual coupon ÷ Market price × 100

Ignores capital gain or loss and time to maturity.

Approximate YTM

YTM ≈ [C + (F − P) ÷ n] ÷ [(F + P) ÷ 2]

Shortcut estimate. C is annual coupon, P is price, n is years to maturity. The exact YTM needs trial and error.

Modified duration

Modified duration = Macaulay duration ÷ (1 + y ÷ k)

y is YTM, k is coupons per year. For annual coupons, divide by (1 + y).

Price change from duration

% change in price ≈ − Modified duration × change in yield

Valid for small yield changes. Convexity corrects for larger changes.

Price versus par

Coupon rate > YTM: premium. Coupon rate = YTM: par. Coupon rate < YTM: discount.

Ordering holds for any bond at a point in time.

Zero coupon bond price

Price = F ÷ (1 + y)^n

Macaulay duration of a zero coupon bond equals its time to maturity.

Portfolio expected return

E(Rp) = w1 × R1 + w2 × R2 + ...

Weighted average of asset returns. Weights add up to 100%.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2 × w1 × w2 × ρ12 × σ1 × σ2

ρ12 is the correlation. Portfolio standard deviation is the square root of this.

Covariance and correlation

ρ12 = Cov(1,2) ÷ (σ1 × σ2)

Correlation lies between -1 and +1.

CAPM

E(Ri) = Rf + βi × (E(Rm) − Rf)

(E(Rm) − Rf) is the market risk premium. Only systematic risk is rewarded.

Portfolio beta

βp = w1β1 + w2β2 + ...

Beta of the market is 1. Beta above 1 means more volatile than the market.

Risk at extreme correlations

ρ = +1: σp = w1σ1 + w2σ2 | ρ = -1: σp = |w1σ1 − w2σ2|

At ρ = +1 there is no diversification benefit. At ρ = -1 risk can fall to zero with the right weights.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

σp is the standard deviation of portfolio returns (total risk). Higher is better.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

Uses beta (systematic risk). Higher is better. Works best for diversified portfolios.

Jensen's alpha

α = Rp − [Rf + βp × (Rm − Rf)]

Rm is the market (benchmark) return. Positive alpha means outperformance after adjusting for beta.

Excess return over benchmark

Excess return = Rp − Rbenchmark

Simple comparison with no risk adjustment.

Time-weighted vs money-weighted

TWRR = [(1 + r1)(1 + r2)…(1 + rn)] − 1

r1 to rn are sub-period returns between cash flows. TWRR ignores cash-flow timing; MWRR does not.

Long call payoff at expiry

Payoff = max(Spot − Strike, 0); Profit = Payoff − Premium

Buyer's maximum loss is the premium. Profit is unlimited as spot rises.

Long put payoff at expiry

Payoff = max(Strike − Spot, 0); Profit = Payoff − Premium

Buyer's maximum loss is the premium. Maximum gain occurs if spot falls to zero.

Breakeven of a long call

Strike + Premium

Spot must rise above this for the buyer to profit.

Breakeven of a long put

Strike − Premium

Spot must fall below this for the buyer to profit.

Long futures profit

Profit = Spot at close − Futures price (× lot size)

Short futures profit is the reverse: Futures price − Spot at close.

Writer's position

Writer's profit = Premium − buyer's payoff

The writer's profit is capped at the premium.

Index futures hedge (beta-adjusted)

Value to sell = Portfolio value × Portfolio beta

Number of lots = Value to sell ÷ (Futures price × lot size). Beta of 1 means hedge the full value.

Quick revision

• Asset classes differ in risk, return, liquidity and tax treatment; equity is usually the most volatile of the main classes.
• Standard deviation measures total risk; beta measures sensitivity to the market and so only systematic risk.
• Diversification reduces unsystematic risk but cannot remove systematic risk.
• Lower correlation between assets gives greater diversification benefit; a correlation of +1 gives none.
• CAPM: expected return = risk-free rate + beta × (market return − risk-free rate).
• P/E = market price ÷ earnings per share; dividend yield = dividend per share ÷ market price.
• Bond prices and yields move in opposite directions.
• A bond's price is the present value of its coupons and principal at the required yield.
• Higher duration means higher sensitivity of price to interest rate changes.
• Sharpe ratio = (portfolio return − risk-free rate) ÷ standard deviation, so it uses total risk.
• Treynor ratio = (portfolio return − risk-free rate) ÷ beta, so it uses systematic risk.
• Alpha is return earned beyond what beta and market return explain.
• Derivatives derive value from an underlying; in portfolios they are used for hedging, not only speculation.

Common mistakes

• Placing security selection before asset allocation. Fix: Remember that the plan comes first. Objectives, risk profile, then allocation, and only then the choice of securities.
• Treating debt as risk-free. Fix: Debt carries credit risk and interest rate risk. It is usually less volatile than equity, not free of risk.
• Saying diversification removes all risk. Fix: Diversification removes only unsystematic risk. Systematic risk stays.
• Treating beta as total risk. Fix: Beta captures only market-related risk. Standard deviation captures total risk.
• Calculating dividend yield on face value Fix: Convert to rupees per share first, then divide by current market price.
• Treating a low P/E as always cheap Fix: A low P/E may reflect poor growth or high risk. Compare with peers and growth prospects.
• Saying price rises when yield rises. Fix: Remember that fixed cash flows are discounted at a higher rate, so present value falls. Price and yield always move opposite.
• Calculating the coupon on market price instead of face value. Fix: Coupon rupees = coupon rate × face value. Use market price only in current yield and YTM.
• Adding up standard deviations to get portfolio risk when correlation is below +1. Fix: Remember risk is a weighted average only at ρ = +1. For lower correlation it is smaller.
• Believing diversification removes all risk. Fix: Diversification removes unsystematic risk only. Systematic (market) risk remains and is measured by beta.

Exam tips

• Learn the process order cold. Questions often ask which step comes before or after another.
• Build a one-line profile for each asset class on return source, risk and liquidity, and test every option against it.
• Be wary of options with words like guaranteed, always or risk-free.
• Read the client clue in the stem: horizon, income need and risk appetite usually decide the answer.
• Remember that NISM-Series-XXI-A has negative marking of 10% of the marks for a question. If you cannot narrow down the options, think before guessing.
• Expect definition questions asking which risk can be diversified. The answer is unsystematic.
• Match the example to the risk type: interest rate change or inflation is systematic; a company's labour strike is unsystematic.
• Check if the question says total risk or market risk before choosing between standard deviation and beta.