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Work out alpha, tracking error, the Sharpe ratio and the information ratio from data, after defining each of these four measures.

FRM® Part II · Risk Management and Investment Management · Alpha (and the Low-Risk Anomaly) · FRM2.IM.3.b

The idea

Alpha is the return earned beyond what the chosen benchmark delivers. It is the portfolio return minus the benchmark return, or the intercept of a regression once beta is accounted for. Tracking error is the standard deviation of the active return, meaning portfolio return minus benchmark return, so it measures how far the portfolio strays from its benchmark. Let Rp be the portfolio return, Rf the risk-free rate and σp the standard deviation of the portfolio's own return. Then the Sharpe ratio is the mean of Rp − Rf divided by σp. The information ratio is alpha divided by tracking error, the reward per unit of active risk. From data, compute the active return each period, then take its mean and sample standard deviation. Annualise by multiplying the mean by the periods per year and the standard deviation by the square root of that number. The usual mistakes are dividing alpha by portfolio volatility and leaving the risk-free rate out of the Sharpe ratio.

Formulas

Sharpe ratio

Sharpe ratio = (Rp − Rf) ÷ σp

Rp
the portfolio's return
Rf
the risk-free rate
σp
the standard deviation of the portfolio's return

Uses the portfolio's own total volatility and needs the risk-free rate subtracted; leaving it out is a common mistake.

Information ratio

IR = mean active return ÷ tracking error

mean active return
the average of portfolio return minus benchmark return
tracking error
the standard deviation of that active return

Active risk, not total volatility, goes in the denominator. Use the sample standard deviation when working from a short history.

Annualising an information ratio

annual IR = periodic IR × √k

periodic IR
the information ratio from monthly or other periodic data
k
the number of periods per year

The mean scales with k and the standard deviation with √k, so the ratio scales with √k, assuming independent periods.

A practice question

From the question bank, with the reason each wrong option is wrong.

How is tracking error defined?

  1. The standard deviation of the portfolio's own returns over the period.Why not: This is the portfolio's total volatility, the denominator of the Sharpe ratio, not tracking error.
  2. The standard deviation of the difference between the portfolio return and the benchmark return. - correct
  3. The average difference between the portfolio return and the benchmark return.Why not: This is the mean active return, i.e. alpha; tracking error is the dispersion of that difference.
  4. The standard deviation of the benchmark's returns over the period.Why not: This is benchmark volatility and ignores the portfolio altogether.

Tracking error is the standard deviation of active returns (portfolio minus benchmark). It is the risk denominator of the information ratio.

A harder one

A portfolio's annual returns over five years were 12%, 8%, -4%, 15% and 9%, while its benchmark returned 10%, 9%, -3%, 11% and 7%. Using the mean active return as alpha and the sample standard deviation of active returns as tracking error, which is closest to the information ratio?

  1. 0.62Why not: Uses the population standard deviation (divides by 5) instead of the sample standard deviation (divides by 4).
  2. 0.17Why not: Divides alpha by the portfolio's own return volatility instead of the volatility of the active returns.
  3. 1.24Why not: Multiplies by the square root of 5, which turns the information ratio into a t-statistic.
  4. 0.55 - correct

Active returns: 2, -1, -1, 4, 2. Mean = 1.2%, sample standard deviation = 2.168%. IR = 1.2 / 2.168 = 0.55.

The hardest tier

A portfolio's monthly active returns (portfolio minus benchmark) over six months were 0.5%, -0.3%, 0.8%, 0.2%, -0.1% and 0.4%. Using the sample standard deviation, which is closest to the annualised information ratio?

  1. 0.62Why not: Reports the monthly ratio and does not annualise by the square root of 12.
  2. 2.15 - correct
  3. 7.43Why not: Annualises the mean by 12 but leaves the standard deviation monthly, so the ratio is scaled by 12 instead of sqrt(12).
  4. 2.35Why not: Uses the population standard deviation (divides by 6) instead of the sample standard deviation (divides by 5).

Monthly mean = 0.25%, sample sd = 0.404%. Monthly IR = 0.619. Annual IR = 0.619 x sqrt(12) = 2.15. The mean scales with 12 and the standard deviation with sqrt(12), so the ratio scales with sqrt(12).

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