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LE Paper 12 · Topic 3
This is the topic where most of Paper 12's calculations sit. You must work out holding period, annualised, effective, after-tax and real returns; measure risk with standard deviation and the normal curve; combine two assets using correlation; and read a chart of the efficient frontier and indifference curves. The second part adds the capital asset pricing model and beta, the arbitrage pricing theory, the P/B-ROE valuation model and the efficient market hypothesis. Learn each formula with one worked number: examiners build the wrong options from the usual slips.
23 min read16 sections
Standard investment mathematics and statistics; H. Markowitz, 'Portfolio Selection', Journal of Finance (March 1952), for portfolio variance, the efficient frontier and the indifference-curve choice; every worked number computed by script. Scope follows the HKSI LE Paper 12 syllabus v3.7 (effective 1 November 2024), Topic 3 headings 1-2, as at October 2026. Standard finance theory, checked October 2026: Sharpe (1964) and Lintner (1965) on the CAPM; Jensen (1968) alpha; Roll (1977) critique; Ross (1976) arbitrage pricing theory; Chen, Roll and Ross (1986) macroeconomic factors; the constant-growth dividend model and Wilcox (1984) P/B-ROE valuation model; Fama (1970) efficient market hypothesis. Scope: HKSI LE Paper 12 syllabus v3.7 (effective 1 November 2024), Topic 3 headings 3-6, learning outcomes (e)-(g). Every worked figure computed by script, as at October 2026. Independent prep, not endorsed by HKSI Institute or the SFC.
Return is what you gain from holding an investment: income (dividends, coupons, rent) plus the change in its price. The holding period return (HPR) states it as a percentage of what you paid: HPR = (P1 − P0 + D) ÷ P0 (standard investment mathematics). It covers the whole time you held the investment, however long.
For several yearly returns, the arithmetic average is their simple mean. The geometric average is [(1 + r1)(1 + r2)…(1 + rn)]^(1/n) − 1: the constant yearly rate that gives the same ending value. To compare holding periods of different lengths, annualise with compounding: (1 + HPR)^(1/n) − 1, where n is the number of years.
| Measure | Formula | Use it for | Worked example |
|---|---|---|---|
| Holding period return | (P1 − P0 + D) ÷ P0 | Total return over the period held | Buy at HK$25, dividend HK$0.50, sell at HK$27: 2.50 ÷ 25 = 10% |
| Arithmetic average | (r1 + … + rn) ÷ n | An estimate of one future period's expected return | +30% then −20%: 5% |
| Geometric average | [(1 + r1)…(1 + rn)]^(1/n) − 1 | Compound growth actually achieved | +30% then −20%: √(1.30 × 0.80) − 1 = 1.98% |
| Annualised return | (1 + HPR)^(1/n) − 1 |
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| Comparing periods of different lengths |
| 21% over 2 years: √1.21 − 1 = 10% a year |
The geometric average is never above the arithmetic average, and the gap widens as returns become more volatile. A 50% gain followed by a 50% loss averages 0% arithmetically but leaves you 25% down (HK$100 → HK$150 → HK$75).
Trap: dividing a multi-year return by the number of years. That ignores compounding: 21% over two years is 10% a year, not 10.5%. Also, never leave the dividend out of the HPR.
Takeaway: HPR is price change plus income over the price paid. Annualise and measure past growth geometrically, never by simple division.
A nominal (quoted) annual rate ignores compounding within the year. The effective annual rate (EAR) is what you actually earn once interest is compounded: EAR = (1 + r ÷ m)^m − 1, where r is the nominal rate and m the number of compounding periods a year. With continuous compounding, EAR = e^r − 1, the upper limit. To go back from an effective rate to a nominal one: r = m × [(1 + EAR)^(1/m) − 1].
| 4% nominal, compounded | m | Effective annual rate |
|---|---|---|
| Annually | 1 | 4.00% |
| Semi-annually | 2 | 4.04% |
| Quarterly | 4 | 4.06% |
| Monthly | 12 | 4.07% |
| Continuously | — | 4.08% |
Compare deposits, loans or products quoted with different compounding frequencies on their effective annual rates, never on their quoted rates. A rate quoted per month (say 1% a month) is a nominal 12% a year; compounded monthly it is an effective 12.68%.
Trap: treating the quoted rate as the return. At the same nominal rate, more frequent compounding always gives a higher effective rate; only annual compounding makes the two equal.
Takeaway: EAR = (1 + r ÷ m)^m − 1. More compounding, higher EAR. Compare quotes on EAR only.