Utility score
Also written Investor utility
A single number that scores an investment for one particular investor: expected return minus a penalty for risk, scaled by that investor's own risk aversion.
In plain language
Two investors look at the same portfolio. One finds it attractive; the other would not touch it. Neither is wrong — they simply dislike risk to different degrees.
A utility score captures that. It starts with the portfolio's expected return and subtracts a charge for risk. How big the charge is depends on the investor's risk aversion index (A): the more risk-averse the person, the bigger the deduction.
What is left is, in effect, a certainty-equivalent return — the risk-free return this investor would treat as equally good. If it beats the actual risk-free rate, the investment is worth taking.
How it works
Higher expected return raises utility; higher variance lowers it. The ½ is a scaling convention.
The workbook's worked example: an investor with A = 3 looks at an investment with expected return 15% and risk of 10%, when the risk-free rate is 8%.
U = 0.15 − (0.5 × 3 × 0.10 × 0.10) = 0.135 = 13.5%
or, keeping percentages as whole numbers and using 0.005 in place of ½:
U = 15 − (0.005 × 3 × 10 × 10) = 13.5%
Excess utility is 13.5% − 8% = 5.5%, so the investor should invest.
This is also why utility is different from the Sharpe ratio. Sharpe treats every investor alike, so a higher Sharpe ratio is better for everyone. Utility explains why a good portfolio is not equally attractive to all investors.
The formula
U = E(r) − ½ × A × σ²
with returns as whole-number percentages:
U = E(r) − 0.005 × A × σ²
Decision rule (workbook): invest if U > risk-free rate
A worked example
Illustrative figures. A PMS offers two strategies. The risk-free rate is 7%.
| Strategy X (small-cap) | Strategy Y (large-cap value) | |
|---|---|---|
| Expected return | 14% | 10% |
| Standard deviation | 20% | 8% |
Ravi, a 32-year-old with A = 2:
- X: 14 − 0.005 × 2 × 20² = 14 − 4 = 10.00%
- Y: 10 − 0.005 × 2 × 8² = 10 − 0.64 = 9.36%
Both beat 7%; Ravi prefers X.
Meenakshi, a 64-year-old retiree with A = 6:
- X: 14 − 0.005 × 6 × 400 = 14 − 12 = 2.00% → below 7%, reject
- Y: 10 − 0.005 × 6 × 64 = 10 − 1.92 = 8.08% → above 7%, accept
On ₹25 lakh each, X's 14% expected return is ₹3.5 lakh against Y's ₹2.5 lakh — yet for Meenakshi, X is worth less than a government bond. Same menu, opposite choices, and the arithmetic explains why.
The workbook's Chapter 15 caselet runs the same test: a portfolio returning 20% with risk 25%, investor A = 5, risk-free 6% → U = 0.20 − 0.5 × 5 × 0.25² = 4.38%, so No.
Why NISM asks about it
Chapter 15 (Introduction to Modern Portfolio Theory), section 15.3, defines risk-averse investors through utility scores and works the A = 3 example; the Chapter 15 caselets test the decision rule with A = 5. Expect a direct computation and a Yes/No on whether to invest.
Common exam traps
- The workbook's own example mislabels its input. It says the investment has "a variance of 10%" but then squares 10% inside the formula (0.10 × 0.10). The arithmetic treats 10% as the standard deviation. If a question gives you a standard deviation, square it; if it genuinely gives a variance, do not square it again.
- Compare utility with the risk-free rate, not with zero and not with the expected return.
- Higher A means a bigger penalty. A risk-averse investor has a high index, not a low one.
- Keep units consistent. Use 0.5 with decimals (0.15, 0.10) or 0.005 with whole-number percentages (15, 10) — mixing them gives answers off by a factor of 100.
- Utility is investor-specific; the Sharpe ratio is not. That contrast is itself examinable.
Where this is taught
Free preparation for NISM Series XXI-BRelated terms
- Sharpe ratioReturn earned above the risk-free rate divided by standard deviation — how much reward an investment produced for each unit of total risk its holder had to live with.
- Risk premiumThe extra return an investor demands over the nominal risk-free rate as compensation for uncertainty about future cash flows — the last and largest block in the required rate of return.
- CorrelationA measure of the strength and direction of the relationship between two variables, running from -1 to +1, and the single factor that determines how much risk diversification actually removes.
- Modern Portfolio TheoryMarkowitz's framework for building portfolios on expected return and risk together, in which the co-movement between holdings — not their individual riskiness — decides the risk of the whole.
- Risk averse investorAn investor who accepts risk only when paid a positive expected risk premium for it, and who demands a larger premium the greater the risk.
- Risk Aversion IndexThe 'A' in the Markowitz utility formula U = E(r) − 0.5 × A × σ² — a number capturing how heavily an individual investor's utility is penalised for taking on variance.
- Absolute return objectiveA return goal stated on its own, such as 15% a year, with no comparison to a benchmark — the opposite of a relative return objective.
- Probability distribution of expected returnsThe list of returns an investment might deliver, each with the chance of it happening — the form in which Modern Portfolio Theory assumes an investor sees every investment choice.
- Risk neutral investorAn investor who judges an investment only by its expected return and applies no penalty at all for risk, so the certainty equivalent rate equals the expected return on the risky portfolio.
- Risk seeking investorAn investor who will engage in a fair game — one with zero risk premium — because he makes an upward adjustment to utility for risk, where a risk averse investor makes a downward one.
- Certainty Equivalent RateThe rate a risk-free investment would have to offer to be exactly as attractive to an investor as a given risky portfolio — lower than the expected return for a risk-averse investor.