Measuring Risk & Return and Modern Portfolio Theory
Chapter 4 turns risk and return into numbers you can compare. You measure return (holding-period, expected, arithmetic and geometric mean, real versus nominal, required return) and risk (variance, standard deviation, coefficient of variation, range, beta), classify where risk comes from, then combine assets using Markowitz's mean-variance framework and price risk with the CAPM. Exam questions are mostly short calculations and 'which is correct' items on correlation, the efficient frontier and the security market line, so you need the formulas exact and the traps clear.
12 min read·8 sections
✓Checked against the SCI M8 syllabus (exam details page, Contents and Objectives), Markowitz (1952) Journal of Finance 7(1); Sharpe (1964) Journal of Finance 19(3); Fisher (1930) The Theory of Interest; MoneySense 'Managing investment risk'; MAS SGS T-bill pages; standard finance textbook definitions, checked 13 Sep 2026. Unofficial prep, not endorsed by MAS or SCI.
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Must-know for the exam
✓Holding-period return = (ending price - beginning price + income) / beginning price. Both capital gain yield and income yield are divided by the BEGINNING price.
✓Expected return = sum of (probability x scenario return). Probabilities must add up to 1, so infer any missing one.
✓Geometric mean <= arithmetic mean; equal only if every period's return is the same. +50% then -50%: arithmetic 0%, geometric -13.4% a year.
✓Required return = risk-free rate + risk premium. The usual risk-free proxy is short-term government bills such as SGS T-bills (6-month and 1-year, issued at a discount, as at 13 Sep 2026).
✓Fisher relation: (1 + nominal) = (1 + real) x (1 + inflation). Shortcut: real = nominal - inflation. 8% nominal, 3% inflation gives 4.85% exactly, 5% by shortcut.
✓Variance = sum of probability x (return - expected return)². Standard deviation = square root of variance, in the same units as return.
✓Normal distribution: about 68% of outcomes within 1 standard deviation of the mean, about 95% within 2, about 16% below mean minus 1.
✓Coefficient of variation = standard deviation / expected return. Lower means less risk per unit of return.
✓Beta measures systematic risk only. Portfolio beta = weighted average of betas. Expected move = beta x market move.
✓Portfolio expected return = weighted average of expected returns, always.
✓Two-asset variance = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. Covariance = ρ x σ1 x σ2. Correlation lies between -1 and +1.
✓Portfolio standard deviation equals the weighted average only when ρ = +1. With ρ = -1, zero risk at w1 = σ2 / (σ1 + σ2).
✓Efficient frontier: highest expected return for each level of risk. A dominated portfolio has another with at least the same return and lower risk (or more return, same risk).
✓CML: return against standard deviation, efficient portfolios only. SML: return against beta, any asset or portfolio.
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Traps that cost marks
→Candidates forget that income counts towards the holding-period return, so the price gain needed is smaller than the full target.
→Dividing by the ending price instead of the beginning price is the classic slip in return splits.
→Taking a simple average, or the most likely outcome, instead of weighting each scenario by its probability.
→Leaving out the scenario whose probability is not stated, instead of inferring it from the fact that probabilities sum to 1.
→Reading expected return as a promised or most-likely return rather than a probability-weighted average.
→Assuming a +50% year and a -50% year cancel out; compounding means the investor has lost a quarter of the money.
→Compounding the arithmetic mean return, which overstates the wealth actually achieved when returns are volatile.
→Quoting the arithmetic mean, or dividing the cumulative return by the number of years, instead of taking the nth root.
Measuring return: holding-period, expected and average returns
•The exam gives you prices, income or scenario tables and asks for a return. Every formula here is short, but each has a slip examiners build distractors around.
•Holding-period return (HPR) = (P1 - P0 + income) / P0. Solve for any unknown: P1 = P0 x (1 + HPR) - income.
•Expected return = sum of probability x return across scenarios. It is a weighted average, not the most likely outcome and not a promise.
•Arithmetic mean = simple average of period returns. Geometric mean = [(1 + r1)(1 + r2)...(1 + rn)]^(1/n) - 1, the compound rate actually achieved.
•Trap: examiners give a +50% year and a -50% year and offer 0% as the average return. The arithmetic mean is 0%, but S$1 has become S$0.75. The geometric mean is always at or below the arithmetic mean, and the gap widens as returns become more volatile.
•Worked example: boom (probability 0.2, 30%), normal (0.5, 12%), recession (0.3, -10%). Expected return = 6 + 6 - 3 = 9%. A simple average of 10.67% ignores the probabilities.
•Worked example: Mr Lim buys at S$20.00, expects a S$0.80 dividend and wants a 12% HPR. Required selling price = 20.00 x 1.12 - 0.80 = S$21.60.
•Takeaway: Weight scenarios by probability, divide by the beginning price, and use the geometric mean for compound growth.
Nominal, real and required return
•A return means little until you set it against inflation and against what the risk deserves. Two relationships matter.
•Required rate of return = risk-free rate + risk premium. The risk premium is the extra return an investor demands for bearing risk; it is required, not guaranteed.
•The risk-free rate is proxied by short-term government bills. SGS T-bills are issued in 6-month and 1-year tenors at a discount to face value (MAS, as at 13 Sep 2026).
•Trap: when a question says 'exact', the shortcut answer sits among the options. The shortcut overstates real return when inflation is positive, and its error grows as inflation and returns rise. When nominal equals inflation, both methods give exactly 0%.
•Worked example: nominal 8%, inflation 3%. Exact real = 1.08 / 1.03 - 1 = 4.85%. For a 4% real target with 2.5% inflation, nominal = 1.04 x 1.025 - 1 = 6.60%.
•Worked example: a fixed deposit at 1.5% with 3% inflation gives a real return of 1.015 / 1.03 - 1 = -1.46%. The principal is safe, but its purchasing power falls.
•Takeaway: Divide to get real return, multiply to get nominal. Required return is the risk-free rate plus a risk premium.
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