FINC3017 · Investments And Portfolio Management
Foundations: Returns, Statistics & Utility
Foundations: Returns, Statistics & Utility (Week 1) builds the toolkit the whole course runs on. You learn to measure returns two ways — simple net returns that compound multiplicatively and log returns that add cleanly across time — and to tell the arithmetic mean from the always-smaller geometric mean. You meet the moments (mean, variance, skewness, kurtosis), the matrix primitives (transpose, inverse, Cholesky) that scale portfolios to n assets, OLS with robust White standard errors, and expected-utility theory, where risk aversion shows up as a concave utility function (U' > 0, U'' < 0) and is summarised by CARA and CRRA coefficients.
What this chapter covers
- 01Simple net return R_t = P_t/P_{t-1} − 1 vs gross return 1 + R_t
- 02Log return r_t = ln(1 + R_t), additive across time
- 03Arithmetic mean vs geometric mean, with r_G ≤ arithmetic
- 04Moments: mean, variance, skewness, kurtosis (equity returns: negative skew, excess kurtosis)
- 05Variance of the sample mean under i.i.d.: Var(r̄) = Var(r)/T
- 06Matrix algebra: transpose, the 2×2 inverse, Cholesky A = LLᵀ
- 07OLS β̂ = (X'X)⁻¹X'Y and why White SEs change only the standard errors, not the coefficients
- 08Risk aversion: concave utility, absolute (CARA) and relative (CRRA) risk aversion, certainty equivalent and risk premium
Geometric mean return and the log-return check
- 2 marks(a) Period 1 simple return R₁ = 112/100 − 1 = +12%. Period 2 simple return R₂ = 105/112 − 1 = −0.0625 = −6.25%.
- 2 marks(b) Build the gross factors and multiply: (1 + R₁)(1 + R₂) = 1.12 × 0.9375 = 1.05. This is total wealth growth of 5% over two periods.
- 1 mark(b cont.) The geometric mean per period solves (1 + r_G)² = 1.05, so r_G = √1.05 − 1 = 1.02470 − 1 ≈ +2.47% per period.
- 1 mark(c) Log returns: r₁ = ln(1.12) = 0.11333 and r₂ = ln(0.9375) = −0.06454. Their sum is 0.04879.
- 1 mark(c cont.) The total log return is ln(105/100) = ln(1.05) = 0.04879 — identical to the sum, confirming log returns are additive across time.
Key terms
- Simple vs log return
- Simple net return R_t = P_t/P_{t-1} − 1 compounds multiplicatively across time and adds across assets in a portfolio; the log return r_t = ln(1 + R_t) adds across time. Use simple for cross-sectional portfolio sums, log for time-series aggregation.
- Geometric mean return
- The constant per-period rate that reproduces the actual end wealth: (1 + r_G)ᵀ = ∏(1 + R_t). It is always ≤ the arithmetic mean, and the gap widens with volatility, so it is the honest measure of realised compound growth.
- Robust (White) standard errors
- An OLS variance estimator that stays valid under heteroskedasticity. The slope estimates β̂ are unchanged; only the standard errors (and therefore t-statistics and significance) are corrected — essential because financial residuals rarely have constant variance.
- Risk aversion (CARA / CRRA)
- A risk-averse investor has concave utility (U' > 0, U'' < 0). Absolute risk aversion A(W) = −U''/U' (constant under exponential utility, CARA); relative risk aversion R(W) = −W·U''/U' (constant under power/log utility, CRRA).
- Certainty equivalent
- The guaranteed wealth that gives the same utility as a risky gamble: U(CE) = E[U(W)]. The risk premium an investor demands equals E[W] − CE — larger for more risk-averse investors or riskier gambles.
Foundations: Returns, Statistics & Utility FAQ
When do I use log returns instead of simple returns?
Use log returns when you need to add returns across time, because a multi-period log return is just the sum of the single-period log returns. Use simple (arithmetic) returns when you combine assets within a portfolio at one point in time, because a portfolio's simple return is the weighted sum of the assets' simple returns. Mixing the two up is a classic exam trap.
Why is the geometric mean always below the arithmetic mean?
Volatility drags compound growth: a gain and an equal-percentage loss leave you below where you started (e.g. +50% then −50% ends at 0.75 of your wealth). The arithmetic mean ignores this, while the geometric mean captures the actual compounded outcome, so r_G ≤ arithmetic mean with equality only when every period's return is identical.
What does it mean that White standard errors leave the coefficients unchanged?
Ordinary least squares still produces the same best-fit slopes; the robust correction only re-estimates how uncertain those slopes are. Because financial data are typically heteroskedastic, the plain OLS standard errors understate the uncertainty, so White SEs give more trustworthy t-statistics for deciding whether a coefficient (say a factor loading) is really non-zero.
Exam move
Make the return-and-utility mechanics reflexive: know instantly when to add log returns versus weight simple returns, and be able to compute a geometric mean and a Z-style risk premium without hesitating. Memorise the CARA-vs-CRRA distinction and the concavity definition of risk aversion, because the exam tests them conceptually as often as it does numerically.
