FNCE30001 Investments
Portfolio Performance Evaluation
A raw return is meaningless without its risk. Every performance measure is a reward-to-risk ratio; they differ only in which risk sits in the denominator — total risk (σ), systematic risk (β), or active/residual risk. The five measures are the Sharpe ratio (excess return per unit of total σ), the Treynor ratio (per unit of beta), Jensen's alpha (abnormal return versus the CAPM/SML), the information ratio (alpha per unit of residual risk), and M² (the Sharpe gap re-expressed as a percentage return at benchmark risk). The whole exam distinction is the denominator, and the one rule for which to use is the diversified-versus-standalone fork: a fund that is the investor's whole risky holding is judged on total risk (Sharpe or M²); a fund held as one sleeve of an already-diversified portfolio is judged on systematic risk (Treynor or Jensen); an active bet bolted onto the index is judged on the information ratio. Jensen's alpha reads off the SML as a vertical distance, and performance attribution splits out-performance into asset-allocation and security-selection contributions. The AU framing uses Morningstar ratings, managed-volatility funds and ASX-200 ETF attribution.
What this chapter covers
- 01The five risk-adjusted measures and their denominators
- 02Sharpe and Treynor — the denominator is the whole distinction
- 03The diversified-vs-standalone fork: which measure, when
- 04Jensen's alpha as a vertical distance off the SML
- 05Performance attribution: asset allocation vs security selection
- 06AU fund framing (Morningstar, managed-volatility, ETF-vs-index)
- 07Resolving a Sharpe–Treynor ranking conflict
Worked example: rank two funds and resolve the conflict
- +1Identify. We have total risk σ and systematic risk β for each fund, so we can compute both ratios; which we act on depends on how the fund is held.
- +1Sharpe S = (r_p − r_f)/σ_p. A: (11 − 3)/18 = 0.444; B: (8 − 3)/12 = 0.417. Both beat the market's S_M = (9 − 3)/16 = 0.375. Sharpe rank: A > B.
- +1Treynor T = (r_p − r_f)/β_p. A: (11 − 3)/1.10 = 7.27; B: (8 − 3)/0.60 = 8.33. Both beat T_M = 6.00. Treynor rank: B > A — the two swap.
- +1Why they disagree: B's low beta (0.60) makes its modest return very efficient per unit of market risk, even though its total-risk efficiency is slightly lower than A's.
- +1Stand-alone investor (the fund is their whole risky holding): total risk matters → use Sharpe → pick A.
- +1Diversified investor (the fund is one sleeve added to a diversified portfolio): only systematic risk matters → use Treynor → pick B.
Key terms
- Sharpe ratio
- Reward per unit of total risk: S = (E[r_p] − r_f)/σ_p — the slope of the CAL. Use it when the fund is the investor's whole risky holding (stand-alone). It ranks funds, and a fund beats the market if its Sharpe exceeds the market's.
- Treynor ratio
- Reward per unit of systematic risk: T = (E[r_p] − r_f)/β_p. Same numerator as Sharpe, but the denominator is beta. Use it when the fund is one sleeve added to an already-diversified portfolio, so only systematic risk survives. Two funds with the same mean and σ but different beta tie on Sharpe yet differ on Treynor — the lower-beta fund wins.
- Jensen's alpha
- Abnormal return versus the CAPM: α = r_p − [r_f + β_p(r_M − r_f)]. On the SML it is the vertical distance from the fund's (β, realised return) point up to the line. Above the line = positive alpha (beat CAPM); on the line = zero; below = negative. Positive alpha means beating what CAPM required for the fund's risk, not necessarily beating r_M.
- Information ratio
- Alpha per unit of residual (active) risk: IR = α_p / σ(e_p) ≈ (r_p − r_M)/σ(r_p − r_M). It answers a distinct question — is a manager's active bet worth its tracking error? — and is the right measure when bolting a small active overlay onto the index.
- M² (Modigliani–Modigliani)
- The Sharpe gap re-expressed as a percentage return at the benchmark's risk: M² = (S_p − S_M)σ_M. It gives the same ranking as the Sharpe ratio but in intuitive return units — how much a fund, levered or de-levered to the benchmark's risk, would out- or under-return the benchmark.
Portfolio Performance Evaluation FAQ
Which performance measure should I use?
Match the measure to how the fund is held — this is the one rule. If the fund is the investor's whole risky holding (stand-alone), total risk matters, so use Sharpe or M². If the fund is one sleeve added to an already-diversified portfolio, only systematic risk survives, so use Treynor or Jensen's alpha. If you are bolting a small active bet onto the index, use the information ratio. Stating all five ratios is only half the marks; the decision marks come from choosing the right one for the context.
Why do Sharpe and Treynor sometimes rank funds differently?
Because they use different denominators. Both have numerator (r_p − r_f), but Sharpe divides by total σ while Treynor divides by beta. Two funds with the same mean and same σ but different beta tie on Sharpe yet differ on Treynor — the lower-beta fund wins Treynor, because it delivers more reward per unit of market risk. A Sharpe–Treynor disagreement is a deliberate exam trap: explain why they disagree (a low-beta fund can win Treynor while losing Sharpe) and which one fits the holding context.
Does positive alpha mean the fund beat the market return?
No — it means the fund beat what the CAPM required for its risk. A fund can return less than r_M yet still have positive alpha if its beta is low (it beat the SML-required return for that beta). Conversely, a high raw return with a high beta can be negative alpha. Always compare to the SML-required return, not to r_M. On the SML, alpha is the vertical gap from the fund's point up to the line.
What does performance attribution split out-performance into?
Two distinct moves. Asset allocation is over- or under-weighting whole asset classes versus the benchmark; security selection is picking better-than-index stocks within each class. Attribution decomposes the bogey-relative excess r_p − r_B into an allocation contribution (weight differences applied to benchmark class returns) plus a selection contribution (managed weights applied to within-class return differences). It tells you whether a manager's edge came from being in the right classes or from picking the right stocks.
Exam move
Memorise the denominator of each of the five measures — that is the whole distinction — and the diversified-vs-standalone fork that tells you which to use: stand-alone → Sharpe/M², added to a diversified portfolio → Treynor/Jensen, active overlay → information ratio. Practise computing all five for a small table of funds, then resolve a deliberate Sharpe–Treynor ranking conflict by naming the holding context and explaining why a low-beta fund can win Treynor while losing Sharpe. Read Jensen's alpha as a vertical distance off the SML and never confuse beating CAPM with beating r_M. For attribution, be able to split out-performance into allocation versus selection. The marker is testing whether you pick the right measure and turn the number into a decision — a number with no decision earns no decision marks.
