Risk Aversion Index

Two investors can look at the exact same portfolio — same expected return, same risk — and value it completely differently. One is thrilled with the risk-return trade-off; the other finds it far too risky for the return on offer. The risk aversion index, usually written A, is the number that captures that difference.

It is the input that turns a portfolio's raw expected return and variance into a personal utility score for a specific investor. A higher A means the same amount of variance costs that investor more utility — they need more return to accept the same risk.

The workbook's formula: "U is the utility value, A is an index of investor's risk aversion. σ2 is the variance of the returns of the investment." A higher A scales up the penalty variance imposes on utility; a lower A shrinks that penalty. Read directly off the formula: if A were 0, the variance term vanishes entirely and utility would depend on expected return alone — indifferent to risk. A positive A, as the workbook's examples always use, means risk genuinely reduces utility, which is what makes an investor risk-averse in this framework.

The workbook's own worked sample question (Chapter 15, Sample Caselet, Q3) is the cleanest illustration. A fund manager's portfolio has an expected return of 20% and a risk (standard deviation) of 25%. The investor's risk aversion index is 5, and the risk-free rate is 6%. The utility score is:

U = 0.20 − (0.5 × 5 × 0.25 × 0.25) = 0.20 − 0.15625 = 0.04375, or about 4.38%

Because this utility (4.38%) is lower than the risk-free rate (6%), the workbook's answer is that the investor should not take the portfolio — a risk-free 6% beats a risky 20%-expected-return portfolio once this investor's aversion to its 25% risk is priced in.

U = E(r) − 0.5 × A × σ²

where U is the utility score, E(r) is the portfolio's expected return, A is the investor's risk aversion index, and σ² is the variance of the portfolio's returns.

A portfolio manager presents Ms Deshmukh with an equity-heavy portfolio expected to return 20% p.a., with a 25% standard deviation. Ms Deshmukh's risk-profiling questionnaire assesses her risk aversion index at A = 5, and the prevailing risk-free rate — a government T-bill — is 6%.

Using the workbook's formula: U = 0.20 − (0.5 × 5 × 0.25 × 0.25) = 0.04375, or 4.38%. Compared against the risk-free rate of 6%, her utility from the risky portfolio (4.38%) is lower — so, by this measure, Ms Deshmukh is better off in the risk-free instrument than in the proposed portfolio, given how strongly her personal risk aversion penalises the 25% volatility.

A second client with an identical portfolio on offer, but assessed at a lower risk aversion index of A = 2, would compute U = 0.20 − (0.5 × 2 × 0.25 × 0.25) = 0.20 − 0.0625 = 13.75% — comfortably above the 6% risk-free rate, so the same portfolio that Ms Deshmukh should decline is one this less risk-averse client should accept. The portfolio has not changed; only A has.

Chapter 15 (Introduction to Modern Portfolio Theory), section 15.3, introduces the utility formula and works two numeric examples (A = 3 and, in the Sample Caselet, A = 5) through to a decision. Expect the exact utility calculation tested as a numeric question, and a conceptual question on why the Sharpe ratio — which the same section contrasts with Utility — ignores individual risk aversion while Utility does not.

• Do not confuse the Risk Aversion Index (A) with the Sharpe Ratio. The workbook explicitly distinguishes them: the Sharpe Ratio assumes all investors are alike; Utility, driven by A, explicitly allows investors to differ.
• The 0.5 scaling factor is part of the formula — dropping it, or applying A without it, gives the wrong utility score.
• Watch the decimal versus percentage form. The workbook shows both a decimal-form calculation (0.20 − 0.5×5×0.25×0.25) and a percentage-form shortcut (20 − 0.005×5×25×25) that must give the same answer — mixing the two conventions inside one calculation produces an order-of-magnitude error.
• A higher utility than the risk-free rate favours the risky portfolio; a lower one favours the risk-free asset — the comparison, not the utility number alone, is what the workbook's sample question is actually testing.