Meera Menon: Beta-Constrained Alpha and a Two-Fund Ranking Conflict
Meera Menon runs a ₹50 crore discretionary portfolio for a family office. Her mandate caps portfolio beta at 1.25 and requires an expected return of at least 17%. She is fully invested, with no cash, in three stocks: A (weight 40%, beta 1.4, expected return 20.0%), B (weight 35%, beta 0.8, expected return 12.5%) and C (weight 25%, beta 1.1, expected return 16.5%). Her research desk uses a risk-free rate of 7% and an expected market return of 15%. Separately, the trustees are choosing between two external managers for a satellite allocation: Fund P returned 19% with a beta of 1.1 and a standard deviation of 22%; Fund Q returned 15% with a beta of 0.85 and a standard deviation of 14%. One trustee wants a single ranking; another points out that the two funds carry very different risk.
1. Compute the portfolio's beta, expected return, CAPM required return and Jensen's alpha, and assess compliance with the mandate. Also compute each stock's alpha and reconcile.
Read the worked answer
Market risk premium = 15% − 7% = 8%
Portfolio beta
- 0.40(1.4) + 0.35(0.8) + 0.25(1.1) = 0.56 + 0.28 + 0.275 = 1.115
Portfolio expected return
- 0.40(20.0) + 0.35(12.5) + 0.25(16.5) = 8.00 + 4.375 + 4.125 = 16.50%
CAPM required return
- 7 + 1.115(8) = 7 + 8.92 = 15.92%
Jensen's alpha
- 16.50 − 15.92 = +0.58%
Stock-level alphas (expected return − required return)
| Stock | Beta | Required | Expected | Alpha |
|---|---|---|---|---|
| A | 1.40 | 18.20% | 20.00% | +1.80% |
| B | 0.80 | 13.40% | 12.50% | −0.90% |
| C | 1.10 | 15.80% | 16.50% | +0.70% |
Weighted alpha check: 0.40(1.80) + 0.35(−0.90) + 0.25(0.70) = 0.720 − 0.315 + 0.175 = +0.58% ✓
Mandate compliance
- Beta: 1.115 ≤ 1.25 → compliant, with 0.135 of unused risk budget
- Expected return: 16.50% < 17.00% → breach of the return floor by 50 bps
Reconciliation. The portfolio is not badly built - it earns a positive alpha - but it is under-risked relative to the mandate. Stock B is the only holding with a negative alpha (−90 bps) and is consuming 35% of capital; at beta 0.8 it contributes low return without using the risk budget the mandate permits.
2. Can the 17% floor be met within the beta cap using only these three stocks, with no cash and no leverage? Find the allocation that maximises expected return subject to the beta constraint, and comment on whether Menon should implement it.
Read the worked answer
Framing it as a constrained problem
Maximise 20wA + 12.5wB + 16.5wC subject to 1.4wA + 0.8wB + 1.1wC ≤ 1.25 and wA + wB + wC = 1.
The optimum will sit on the beta constraint, so the question is which pair of stocks best uses the beta budget. Compare each stock against B, the low-return, low-beta anchor:
- Trading B for A: +0.6 beta buys +7.5% return → 12.5% return per unit of beta
- Trading B for C: +0.3 beta buys +4.0% return → 13.33% return per unit of beta
C is the more beta-efficient upgrade, so exhaust C first, then top up with A.
All C first: beta 1.10, return 16.50%. Beta budget remaining = 0.15.
Add A: 1.4wA + 1.1(1 − wA) = 1.25 → 0.30wA = 0.15 → wA = 0.50, wC = 0.50, wB = 0.
- Beta = 0.5(1.4) + 0.5(1.1) = 1.25 ✓
- Expected return = 0.5(20.0) + 0.5(16.5) = 18.25% ✓ (exceeds 17%)
- Portfolio alpha = 18.25 − [7 + 1.25(8)] = 18.25 − 17.00 = +1.25%, versus +0.58% today
Compare with the A/B corner (75% A, 25% B): return 18.125%, beta 1.25 - dominated.
Should she implement it? Three caveats. (i) B is eliminated, so 50% of the portfolio sits in one stock and 100% in two - a concentration that the mandate does not forbid but that a fiduciary should disclose. (ii) The alphas are research estimates: A's +1.80% is within any reasonable standard error, and the entire optimisation rests on point estimates. (iii) The mandate's 17% is an expected return, not a floor guarantee; satisfying it by pushing beta to the ceiling raises the dispersion of outcomes. A defensible compromise is 45% A / 10% B / 45% C, which delivers roughly 17.9% at beta 1.235 - compliant on both counts with room to absorb estimation error.
3. Rank Fund P and Fund Q using the Sharpe ratio, the Treynor ratio and Jensen's alpha. Explain why the rankings differ and recommend which measure the trustees should use.
Read the worked answer
Computations (Rf = 7%, Rm = 15%)
| Measure | Fund P | Fund Q | Winner |
|---|---|---|---|
| Return | 19.00% | 15.00% | P |
| Beta | 1.10 | 0.85 | - |
| Std deviation | 22.00% | 14.00% | - |
| Sharpe = (Rp − Rf)/σp | 12 ÷ 22 = 0.545 | 8 ÷ 14 = 0.571 | Q |
| Treynor = (Rp − Rf)/βp | 12 ÷ 1.10 = 10.91 | 8 ÷ 0.85 = 9.41 | P |
| Jensen α = Rp − [Rf + β(Rm − Rf)] | 19 − 15.80 = +3.20% | 15 − 13.80 = +1.20% | P |
| M² = Rf + Sharpe × σm − Rm (σm = 18%) | 7 + 0.545(18) = 16.82 → +1.82% | 7 + 0.571(18) = 17.28 → +2.28% | Q |
Why the rankings conflict
Sharpe and M² use total risk; Treynor and Jensen use systematic risk only. Fund P generates more alpha per unit of beta but also carries 22% standard deviation against 14%. Part of P's dispersion is diversifiable, and the two measures implicitly disagree on whether that diversifiable risk should be penalised.
Recommendation
Ask what the mandate is. If P or Q would be the trustees' only holding, the investor bears the unsystematic risk, and the Sharpe/M² ranking - Fund Q - is the correct one. If the fund is a satellite inside the family office's larger, already diversified book, the unsystematic risk is washed out and the relevant risk is beta; then the Treynor and Jensen ranking - Fund P - governs.
Given the ₹50 crore core portfolio is a three-stock, undiversified book with substantial idiosyncratic risk of its own, the family office is not in a position to diversify away P's extra volatility, so Q is the safer recommendation. The trustees should also note that Treynor and Jensen both assume the CAPM holds; if the market proxy or the risk-free rate is mis-specified, both ratios are biased, whereas Sharpe requires no benchmark at all.