Alpha return

A fund manager reports 25% in a year when the market did 15%. Was he good?

Not necessarily. If he ran a portfolio 50% more volatile than the market, a 25% return is only what the extra risk entitled him to. The manager did nothing except take a bigger bet on the same market.

Alpha separates those two things. Using CAPM, it works out what return the portfolio's market risk required, and calls everything above that the alpha. Alpha return is the reward for bearing non-market risk; beta return is the reward for bearing market risk. Beta return can be bought for a few basis points from an index fund. Alpha cannot be bought at all.

CAPM splits risk in two: a systematic component, which is non-diversifiable, and an unsystematic (idiosyncratic or unique) component, which is diversifiable. Since unsystematic risk can be diversified away, an investor should expect no compensation for bearing it. Investments are therefore priced only for market risk, and market risk is measured by beta.

That gives the required return, and the portfolio's actual return decomposes into three pieces:

1. the risk-free return, earned for waiting;
2. the beta return, earned for carrying market risk — beta x (market return - risk-free return);
3. the alpha return, whatever is left.

The workbook notes a second usage worth knowing: some professionals distinguish alpha from Jensen alpha, using "alpha" for the plain excess return over the market benchmark return. In the worked case below, that plain excess is 25% - 15% = 10%, against a Jensen alpha of 5%. Both numbers describe the same fund. Read which one the question wants.

Required return = Rf + B x (Rm - Rf)

Alpha return    = Portfolio return - [ Rf + B x (Rm - Rf) ]

Rf = risk free return
B  = beta of the security or portfolio
Rm = return on the market portfolio (a broad market index is a good proxy)

The workbook's own decomposition. A portfolio returns 25%. The market benchmark returns 15% over the same period. The portfolio beta is 1.5. The Treasury bond yield is 5%.

Risk free return  = 5%
Beta return       = 1.5 x (15% - 5%)      = 15%
Required return   = 5% + 15%              = 20%
Alpha return      = 25% - 20%             =  5%

The three components sum back to the reported figure: 5% + 15% + 5% = 25%.

In rupees, on a Rs 1,00,00,000 portfolio:

Rs 20,00,000 of the Rs 25,00,000 was available to anyone willing to take that much market risk. The manager's contribution is Rs 5,00,000, and that is the number an advisory fee should be judged against.

Change one input and the verdict flips. Suppose the beta is 2.0 rather than 1.5:

Required return = 5% + 2.0 x 10% = 25%
Alpha return    = 25% - 25%      = 0

Same 25% return, same market, zero alpha. The entire performance was leverage on the index.

Chapter 16 (Portfolio Performance Measurement and Evaluation), section 16.2.9 on alpha and beta return, which sits with the Sharpe and Treynor measures and the return decomposition that follows. This is one of the most computational sections in the paper. Expect to be given portfolio return, benchmark return, beta and a risk-free rate and asked for alpha — and to be given the same four numbers and asked which component rewards market risk.

• Alpha is not the portfolio return minus the benchmark return in the CAPM sense. That plain excess is 10% here; the Jensen alpha is 5%. The workbook flags both usages, so read the wording of the question.
• Use (Rm - Rf) inside the bracket, not Rm. Multiplying beta by the whole market return instead of the market risk premium is the standard arithmetic error.
• Alpha rewards non-market risk; beta return rewards market risk. Reversing those two sentences is a favourite distractor.
• A high return with a high beta can carry zero alpha. Raising beta raises the required return, so it raises the bar the manager has to clear.
• Alpha can be negative, and a negative alpha with a positive return is common — it means the portfolio underperformed what its own risk demanded.
• The risk-free return is a third component, not part of alpha. Return decomposition has three pieces, and they must add back to the reported return.
• Unsystematic risk earns nothing under CAPM. It is diversifiable, so no compensation is expected for bearing it — which is why alpha is defined against systematic risk alone.