Quantitative investment concepts and measures of investment returns

Standard deviation, variance, beta, correlation and covariance; holding period, arithmetic, geometric, time-weighted, dollar-weighted, real and after-tax returns; Sharpe, Treynor, Jensen's alpha and information ratio; and choosing the right measure for a stated evaluation.

Modern portfolio theoryCapital asset pricing model

Practice question for this objective

Free sampleInvestment Planninghard

A planner is reviewing an equity manager for a client whose investment policy statement requires managers to be judged on return above the return their systematic risk warranted. Over the review period the manager's portfolio returned 12.0 percent, the benchmark market portfolio returned 9.0 percent, the risk free rate was 3.0 percent, and the portfolio's beta measured against that market portfolio was 1.20. Using the Capital Asset Pricing Model to set the required return, what was the manager's Jensen's alpha over the period?

A3.0 percent, obtained by subtracting the 9.0 percent market portfolio return from the 12.0 percent portfolio return, on the basis that the market return is the return the manager had to beat over the period.
B9.0 percent, obtained by subtracting the 3.0 percent risk free rate from the 12.0 percent portfolio return, on the basis that alpha measures the reward earned for accepting risk instead of holding the risk free asset.
C4.8 percent, obtained by multiplying the 6.0 percent market risk premium by the beta of 1.20 to give 7.2 percent, and subtracting that from the 12.0 percent portfolio return without adding the risk free rate back.
D1.8 percent, obtained by setting the required return at 3.0 percent plus 1.20 times the 6.0 percent market risk premium, giving 10.2 percent, and subtracting that required return from the 12.0 percent portfolio return. Correct

Jensen's alpha is the realised return less the Capital Asset Pricing Model required return computed from the portfolio's own beta. The market risk premium is the market return less the risk free rate, 9.0 minus 3.0, which is 6.0 percent. The Capital Asset Pricing Model required return for a beta of 1.20 is 3.0 plus 1.20 times 6.0. That product is 7.2, so the required return is 3.0 plus 7.2, or 10.2 percent. Jensen's alpha is the realised 12.0 percent less the required 10.2 percent, which is 1.8 percentage points. Checking the arithmetic a second way, the portfolio earned 12.0 less 3.0, or 9.0 percentage points above the risk free rate, while a passive position with the same beta would have earned 1.20 times 6.0, or 7.2 points, and 9.0 less 7.2 is again 1.8. Because beta prices systematic risk alone, alpha here is the return that the manager's systematic exposure does not account for.

Why A is wrong: Tempting because raw outperformance against the market is the figure most often quoted in a client meeting. It is wrong because it ignores the beta of 1.20, so it gives the manager credit for the extra return that simply came from carrying more systematic risk than the market.

Why B is wrong: Tempting because return above the risk free rate is the numerator of both the Sharpe and the Treynor ratios, so the subtraction is familiar. It is wrong because that quantity is the excess return, and alpha is what remains after the required return for the beta taken is also deducted.

Why C is wrong: Tempting because it applies beta to the market risk premium correctly, which is the step candidates most often miss. It is wrong because the Capital Asset Pricing Model required return is the risk free rate plus that product, and omitting the 3.0 percent understates the hurdle by exactly that amount.

Why D is correct: Correct because Jensen's alpha is the realised return less the Capital Asset Pricing Model required return for the portfolio's beta. The required return is 3.0 plus 1.20 times 6.0, which is 10.2 percent, and 12.0 less 10.2 leaves 1.8 percentage points of return the beta does not explain.

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Exam traps in Investment Planning

Answers that look right on this material and are not. Each one is a distractor from a different question in the CFP bank for this domain.

Approximately 11.26 per cent, obtained by omitting the covariance term altogether
Why it is wrong: This is the square root of the two weighted variance terms alone, 0.011664 plus 0.001024, which would apply only if the two funds were uncorrelated at zero, so it understates the risk of a pair that moves together to a degree of 0.25.
Sharpe ratio of 0.71
Why it is wrong: This picks the right measure but divides the raw 11.4 percent return by the 16.0 percent standard deviation, omitting the subtraction of the 3.4 percent risk-free rate, so it rewards the fund for return an investor could have earned without risk.
The 8.0 percent time-weighted figure is distorted by the September contribution and therefore overstates what the manager actually achieved over the year.
Why it is wrong: This inverts the mechanism, and it is tempting because a large late deposit clearly did affect the account's growth in dollars; the time-weighted method breaks the record at that cash flow, so the contribution cannot distort it.

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