An endowment can mix a risky portfolio with 8% expected excess return and 16% volatility with cash. It wants 10% volatility. What allocation does it hold, and what excess return should it expect?

Oshmira holds 62.5% in the risky portfolio, Rs 312.5 crore, and 37.5% in cash, and should expect about 5% a year above cash. Cash adds no volatility, so the risky weight is 10/16, and the excess return scales with it: 0.625 x 8%. Every mix on the line has the same Sharpe ratio of 0.50. In a typical year the excess return lands somewhere between about -15% and 25%.

Dilute a strong cordial with water: half cordial is half the flavour and half the sugar, and nothing about the ratio of flavour to sugar changes. Cash has no volatility and no excess return, so a portfolio that is a fraction w in the risky portfolio has exactly w times its volatility and w times its excess return, and the ratio of the two, the Sharpe ratio, is the same at every w. That is the capital allocation lineThe straight line of risk and expected excess return traced out by mixing one risky portfolio with cash in different proportions.: it starts at cash, passes through the risky portfolio at 16% and 8%, and continues beyond it where the endowment would be borrowing to hold more than it owns.

Set w x 16% = 10%, so w = 0.6250: 62.5% in the risky portfolio and 37.5% in cash. The expected excess return is 0.625 x 8% = 5.0% a year, Rs 25 crore on Rs 500 crore, with a Sharpe ratio of 0.50, the same as the risky portfolio's. In rupees, Rs 312.5 crore goes into the risky portfolio and Rs 187.5 crore stays in cash. Tell the trustees the range as well as the centre: with 10% volatility, two standard deviations is 20 points, so roughly 19 years in 20 the excess return lands between -15% and 25%, if returns are close to normal, which in a bad year they are not.

Three things, and each one changes the number. First, 16% is an estimate: if the risky portfolio's volatility is really 20%, the same 62.5% weight runs at 12.5%, over the policy cap, and the weight that honours the cap is 50%. Second, the line is straight only while the endowment is lending; the moment it wants more than 16% volatility it must borrow, at one point over cash here, so the slope above the risky portfolio drops, which is why the 20% row's Sharpe ratio is lower. Third, the weights drift. After a 20% rise in the risky portfolio and flat cash, the 62.5% becomes about 67%, so holding the target means selling after rises and buying after falls, which costs trading and needs a rule for how far to let it drift. The limitation to state: the line prices risk by volatility alone; a portfolio with 10% volatility and a fat left tail is not what the trustees mean by a 10% risk budget.