Risk, Return and Performance of Funds: formula sheet

Key formulas

Interest rate and bond price

Interest rates ↑ ⇒ bond prices ↓; interest rates ↓ ⇒ bond prices ↑

The relationship is inverse. Longer maturity or duration means a bigger price change.

Real return (approximate)

Real return ≈ Nominal return − Inflation rate

Used to explain inflation risk. A positive nominal return can still be a negative real return.

Risk to fund type map

Equity: market, concentration, liquidity | Debt: credit, interest rate, reinvestment, liquidity

Use this as a starting match. Inflation risk affects all funds, especially low-return ones such as liquid funds.

Credit vs liquidity

Credit risk = issuer may not pay | Liquidity risk = holding may not be sellable at a fair price

A frequent exam contrast.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

Rp is fund return, Rf is risk-free return, σp is standard deviation of fund returns. Measures excess return per unit of total risk.

Treynor ratio

Treynor = (Rp − Rf) ÷ β

Uses beta as the risk measure, so it captures only market risk.

Beta

β = Covariance(fund, market) ÷ Variance(market)

Beta of the market is 1. Beta is also the slope of the regression of fund returns on market returns.

Standard deviation

σ = √variance

Variance is the average of squared deviations from the mean. Higher σ means higher volatility.

Expected price change using modified duration

Approx. % change in price ≈ −Modified duration × change in yield (in %)

The minus sign shows that price falls when yield rises. It is an approximation that works best for small yield changes.

Modified duration from Macaulay duration

Modified duration = Macaulay duration ÷ (1 + YTM)

For annual compounding. YTM is entered as a decimal, e.g. 0.08.

Absolute return

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

Ignores time. Ending value includes any income received, if the question says so.

CAGR

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

n is the number of years. Use for a single lump sum held for more than one year.

Future value from CAGR

Ending value = Beginning value × (1 + CAGR)^n

Rearranged form of the CAGR formula. Useful to check your answer.

Simple annualised return

Annualised return = Absolute return ÷ n

Simple, not compounded. Gives a different number from CAGR for periods over one year.

XIRR

Σ [Cash flow ÷ (1 + XIRR)^(days from first date ÷ 365)] = 0

Investments are negative, redemption value is positive. Solved with software, not by hand.

Rule of 72 (approximation)

Years to double ≈ 72 ÷ annual return (%)

A rough estimate only, not an exact result.

NAV

NAV per unit = (Market value of assets − Liabilities) ÷ Units outstanding

Liabilities include accrued expenses. NAV is already net of expenses.

NAV return (absolute)

Return % = (Ending NAV − Beginning NAV) ÷ Beginning NAV × 100

Use only when no dividend is paid, such as the growth option.

Units from a dividend reinvestment

New units = Dividend per unit × Units held ÷ Ex-dividend NAV

Reinvestment is done at the NAV after the payout, not before.

Total return with reinvestment

Return % = (Total units at end × Ending NAV − Initial investment) ÷ Initial investment × 100

Total units = original units + all units added by reinvestment.

Total return without reinvestment

Return % = (Ending NAV − Beginning NAV + Dividend per unit) ÷ Beginning NAV × 100

Dividend is treated as cash received and not reinvested.

Annualised return (more than one year)

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

Apply after you have the total ending value.

Tracking difference

Tracking difference = Fund return − Index (TRI) return

Measured over the same period. Usually negative for index funds, close to the expense ratio.

Tracking error

Tracking error = Standard deviation of (Fund return − Index return) over the periods

Measures consistency of tracking, not the size of the average gap. Lower is better for index funds.

Excess return over benchmark

Excess return = Fund return − Benchmark return

For active funds, positive means outperformance. Use TRI as the benchmark.

Benchmark rule

Compare fund NAV return (dividends included) with TRI, not PRI

TRI includes reinvested dividends, so it is the like-for-like comparison.

Excess return

Excess return = Fund return − Risk-free return

This is the numerator of both Sharpe and Treynor ratios.

Sharpe ratio

Sharpe ratio = (Rp − Rf) ÷ σp

Rp is fund return, Rf is risk-free return, σp is the fund's standard deviation. Measures return per unit of total risk.

Treynor ratio

Treynor ratio = (Rp − Rf) ÷ β

Uses beta, so it measures return per unit of market (systematic) risk. Suits well-diversified portfolios.

Alpha (CAPM based)

Alpha = Rp − [Rf + β × (Rm − Rf)]

Rm is market return. Positive alpha means the fund beat its risk-justified return.

Alpha (simple benchmark form)

Alpha = Fund return − Benchmark return

Used loosely for outperformance. Use only when the question defines it this way.

Reading the ratios

Higher Sharpe or Treynor = better risk-adjusted performance

Compare ratios only among funds measured over the same period and against the same risk-free rate.

Six riskometer levels (low to high)

Low < Low to Moderate < Moderate < Moderately High < High < Very High

Learn the order and the exact names. Exam options often shuffle or rename them.

Risk components assessed

Risk level = f(credit risk, liquidity risk, market risk)

Equity portfolios are assessed mainly on market risk; debt portfolios on credit, liquidity and interest rate related risk.

Product label wording

"This product is suitable for investors who are seeking*" + objective of scheme; *Investors should consult their financial advisers if in doubt about whether the product is suitable for them

The label states the scheme's objective and the advice to consult an adviser.

Disclosure and change rule

Riskometer reviewed monthly; changes in level communicated to unitholders

The level is dynamic, not fixed at launch.

Quick revision

• Market risk affects all securities; credit risk is the issuer failing to pay; interest rate risk hits bond prices when rates rise.
• Liquidity risk is difficulty in selling without a price drop.
• Standard deviation measures how much returns vary around their average. Higher means more volatile.
• Beta measures sensitivity to the market. Beta of 1 moves with the market; above 1 is more volatile; below 1 is less.
• Sharpe ratio = (Fund return − Risk-free return) ÷ Standard deviation. Higher is better.
• Absolute return = (End value − Start value) ÷ Start value × 100.
• CAGR = (End value ÷ Start value)^(1/n) − 1, where n is years.
• XIRR is used when investments happen on different dates, such as SIPs.
• NAV per unit = (Total assets − Liabilities) ÷ Units outstanding.
• Tracking error shows how much a fund's returns deviate from its benchmark. Lower is better for index funds.
• A fund's benchmark must match its category and style.
• The riskometer shows risk levels from low to very high.

Common mistakes

• Treating credit risk and liquidity risk as the same. Fix: Ask: is the problem non-payment (credit) or inability to sell at a fair price (liquidity)?
• Saying bond prices rise when interest rates rise. Fix: Remember the inverse link. New bonds pay more, so old lower-coupon bonds fall in price.
• Dividing the fund return, not the excess return, by standard deviation or beta. Fix: Always write Rp − Rf first, then divide.
• Saying Treynor uses total risk and Sharpe uses market risk. Fix: Remember: Sharpe goes with standard deviation (total risk), Treynor goes with beta (market risk).
• Using absolute return as if it were a yearly return Fix: Always ask: over how long? Absolute return is for the whole period. Convert to CAGR for a multi-year comparison.
• Dividing absolute return by years and calling it CAGR Fix: That gives simple annualised return. CAGR needs the power of 1/n. For periods over a year, simple annualisation overstates the compounded rate.
• Using the plain NAV change for a scheme that paid a dividend. Fix: Whenever a dividend appears, add it back through reinvestment or as cash.
• Reinvesting the dividend at the cum-dividend NAV. Fix: Use the ex-dividend NAV, which is the NAV after the payout, unless the question states otherwise.
• Using a Price Return Index to judge a fund Fix: Remember the fund earns dividends, so compare against the TRI.
• Treating tracking error and tracking difference as the same thing Fix: Difference is a simple return gap. Error is the standard deviation of the gap over time.

Exam tips

• Expect scenario questions that describe an event and ask which risk applies. Find the cause first.
• Know which fund type carries which risk. Credit risk funds stress credit risk; long duration and gilt funds stress interest rate risk.
• Watch absolute words such as 'no risk' or 'eliminates'. These options are usually wrong.
• Remember the contrast pairs: credit vs liquidity, and interest rate vs reinvestment.
• NISM-Series-V-A has no negative marking, so attempt every question.
• Learn the pairings: Sharpe with standard deviation, Treynor with beta. Many questions test only this.
• Remember a beta of 1 means the fund moves with the market. Above 1 is more volatile, below 1 is less.
• Expect a duration question on direction of price change. Yield up, price down.