Desk Puzzle No. 2: The Fortieth Strategy

By EC Assets · Published 2 September 2026 · Updated 2 September 2026

Puzzle No. 2 is below; the solution to Desk Puzzle No. 1 follows it.

Puzzle No. 2: The Fortieth Strategy

Two multi-strategy funds are pitching for the same allocation.

Fund A runs 4 strategies. Fund B runs 40, and that is precisely its pitch: ten times the research effort, ten times the diversification, and a fee to match.

Assume everything in B's favour. Every strategy in both funds is equally good — Sharpe ratio 0.40, identical volatility, equally weighted — and every pair of strategies, inside either fund, is correlated 0.25. Both funds run the portfolio at 10 percent volatility.

The questions

1. What Sharpe ratio does each fund achieve? Commit to a guess before computing: is B's more than twice A's?

2. Suppose B keeps hiring. What is the best Sharpe ratio it can ever reach, with unlimited strategies of this quality? And how many strategies does it take to capture 90 percent of that?

3. B charges 1.5 percentage points more per year than A. On these numbers, is B worth it? And if an allocator insisted on a portfolio Sharpe of 1.00, what would have to change — and what could never deliver it, no matter how much of it you bought?

Ground rules

Pen, paper and one square root are enough. You need exactly one piece of machinery: for n equally weighted assets of equal volatility σ with a common pairwise correlation ρ, the portfolio variance is

σ²ₚ = (σ² / n) × [1 + (n − 1)ρ]

Derive it in two lines if you prefer — it is only the sum of n variances and n(n−1) covariances, divided by n². Nothing in the puzzle depends on the particular value 0.40, and question 3 needs no more than the definition of the Sharpe ratio.

Fair hints

The honest places to look: Correlation, Diversification, Sharpe Ratio, and the Correlation bench, which shows the two-asset version of this effect with a dial. The puzzle asks what is left of that effect when you keep adding assets.

The solution will be published here in next month's note, alongside Puzzle No. 3.

Solution: The Coin-Flip Fund

1. The expected return is exactly what it looks like

Each year a fair coin decides between +30 percent and −25 percent:

E[r] = 0.5 × (+30%) + 0.5 × (−25%) = +2.5 percent per year

Nothing is hidden. The manager was honest, the odds are as stated, and the expected return really is positive.

2. And the typical investor still loses money

The probability that she is above water after 40 years is 43.7 percent — below the coin flip you were asked to guess against.

The two questions do not contradict each other, because they ask about different things. An expected return averages across outcomes. Wealth compounds across time. What you multiply, year after year, are not the returns but the growth factors 1.30 and 0.75.

One head and one tail, in either order, leaves you with 1.30 × 0.75 = 0.975. Not 1.05. A balanced pair of years destroys 2.5 percent of capital, so the rate at which the typical path actually compounds is:

sqrt(0.975) − 1 = −1.26 percent per year

Over 40 years, the median investor — 20 heads, 20 tails — holds 100 × 0.975^20 = 60.27. She has lost 40 percent of her money in a fund with a positive expected return and no fees.

To get the exact probability, note that you finish above 100 only if 1.30^k × 0.75^(40−k) > 1. Taking logs, that needs k > 20.92: at least 21 heads out of 40. From the binomial distribution,

P(k ≥ 21) = 43.73 percent

So 56 percent of investors end below where they started, and the single most likely outcome of all — exactly 20 heads, which happens on 12.5 percent of paths — lands on 60.27.

The mean is real. Almost nobody receives it. E[W₄₀] = 100 × 1.025⁴⁰ = 268.51, and only 21.5 percent of paths finish above that number. The average is carried by a thin tail of very lucky sequences, and the arithmetic is brutally sensitive: each flip is a lever of 1.30 / 0.75 = 1.733, which is why 20 heads leaves you at 60.27 and 21 heads at 104.47.

3. Cash repairs it, because rebalancing is a trade

Half in the fund, half in cash, rebalanced to 50/50 every year, turns the portfolio's year into +15 percent or −12.5 percent. Now:

1.15 × 0.875 = 1.00625 → sqrt(1.00625) − 1 = +0.31 percent per year

The compound growth rate has changed sign. After 40 years the median holding is 113.27 instead of 60.27, and the probability of being above water is 56.3 percent — the exact mirror image of the undiluted fund's 43.7.

The arithmetic mean was halved, from 2.5 percent to 1.25 percent. The drag fell by a factor of four, from 3.76 percentage points to 0.94. That is the entire mechanism: drag scales with variance, and variance scales with the square of exposure. Halve the exposure and you keep half the expected return while shedding three quarters of the drag.

The rebalancing itself is not bookkeeping. Every year it sells the fund after it rises and buys it after it falls, which is what converts a wild multiplicative process into a tamer one. The cash earns nothing and still pays for itself.

What the puzzle was actually about

Sizing. If you hold a fraction f of your wealth in the fund and rebalance annually, the growth rate is maximised at f* = 1/3 — the Kelly fraction for this bet.

The line worth keeping is the third one. Growth reaches exactly zero at two thirds exposure. Holding the fund outright is not the aggressive version of a decent bet — it is fifty percent past the point where the bet stops compounding money at all.

Everything the manager told you was true. The error was never in his arithmetic; it was in reading an expected return as a description of a path.

Further reading: Volatility Drag, Rebalancing, Volatility Targeting, and the Volatility drag calculator, which takes an average return in and gives a compounded return out.