Standard Deviation, Beta and Sharpe Ratio for NISM V-A

Understand Measures of Risk: Standard Deviation, Beta and Sharpe Ratio

Key formulas to remember

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.

How to solve Measures of Risk: Standard Deviation, Beta and Sharpe Ratio questions

1. 1Identify what the question asks: total risk, market risk, risk-adjusted return or interest rate risk.
2. 2Match the measure: standard deviation for total risk, beta for market risk, Sharpe or Treynor for risk-adjusted return, modified duration for bond price sensitivity.
3. 3Write down the given values and convert percentages carefully. Keep all returns on the same time basis (annual with annual).
4. 4For Sharpe or Treynor, first compute excess return = fund return − risk-free return.
5. 5Divide by standard deviation (Sharpe) or beta (Treynor).
6. 6For duration, multiply modified duration by the yield change and attach the correct sign.
7. 7Compare and interpret: a higher ratio is better, a beta above 1 is more aggressive than the market, a longer duration means more rate risk.
8. 8Check that your answer matches one option exactly and that the direction (rise or fall) makes sense.

Quickest way: Match the measure, then do one division

Common mistakes in Measures of Risk: Standard Deviation, Beta and Sharpe Ratio

• Dividing the fund return, not the excess return, by standard deviation or beta.

  Students forget that both ratios are built on return above the risk-free rate.

  Fix: Always write Rp − Rf first, then divide.

• Saying Treynor uses total risk and Sharpe uses market risk.

  The two names and formulas look alike, so they get swapped.

  Fix: Remember: Sharpe goes with standard deviation (total risk), Treynor goes with beta (market risk).

• Treating beta as a measure of total risk.

  Beta is often quoted next to standard deviation, so they seem interchangeable.

  Fix: Beta covers only market risk. A fund with a low beta can still have high fund-specific risk.

• Believing a higher standard deviation means higher returns.

  Risk and return are linked in theory, so students assume it holds for each fund.

  Fix: Standard deviation only shows volatility. It does not promise or measure return.

• Confusing modified duration with YTM or maturity.

  All three relate to time and bond yields.

  Fix: Modified duration is price sensitivity to yield changes. YTM is the yield. Maturity is the date principal is repaid.

• Getting the sign wrong on bond price changes.

  Students focus on the arithmetic and forget the inverse link between price and yield.

  Fix: If yield rises, price falls. If yield falls, price rises.

Worked examples