How the horizon changes the answer
Equities are risky over a year and much less risky over twenty, not because the bad years stop but because they get averaged with good ones. It is a statement about averages, and the caveats matter.
- Compute the probability of a loss over one year and over T years
- Find the bad-case average return over a horizon
- Solve for the horizon a probability limit needs
- Say what a long horizon does and does not reduce
The intuition
A 60% equity portfolio expects about 5.4% a year with 11% volatility. In any one year there is nearly a one-in-three chance it loses money. Over ten years the chance that its average annual return is negative is about 6%; over twenty, under 2%. The bad years did not go away. They got averaged with good ones, and the noise in an average shrinks with the square root of the number of years while the expected return does not shrink at all.
That is the honest case for more equity when the money is not needed for decades, and against equity for a house deposit in eighteen months. It is a statement about averages, though. A long horizon lowers the chance of losing, but not the size of the worst outcomes, and the dollars at risk grow with time. The model here also assumes independent normal returns, which understates fat tails and runs of bad years.
Portfolio μ = w μ_eq + (1 − w) μ_bond, σ from the two-asset formula. Over T years the average annual return has mean μ and standard deviation σ ÷ √T. P(average < 0) = N(−μ√T ÷ σ). Bad case (5th percentile) = μ − 1.645 σ ÷ √T. Horizon for P(loss) ≤ p: T = (z_p σ ÷ μ)².
Why it works
- The conventions here: annual returns are independent and normal with constant mean and volatility for an equity/bond mix with one correlation. Over T years the average annual return keeps the mean and has volatility σ ÷ √T. Compounding and fat tails are ignored.
- The z-score does the work. μ ÷ σ over one year, times √T over T years. The probability of a loss is the normal tail below minus that.
- The bad case moves toward the mean as the horizon lengthens, because 1.645 σ ÷ √T shrinks. What it does not do is shrink the spread of ending wealth in dollars, which widens with time.
- Solve for T by setting μ√T ÷ σ equal to the z for the limit: 1.645 for 5%, 1.282 for 10%. T = (z σ ÷ μ)². A lower return or a higher volatility lengthens the horizon needed as a square.
- Less equity lowers the chance of a loss and the return with it. Whether that trade is worth making depends on what a shortfall would cost the client, which is the capacity question from the first chapter.
- Short horizons with fixed dates get cash. Eighteen months is not long enough for the averaging to matter, and the deposit is needed on a day, not on average.
| Expected return: 60% × 7% + 40% × 3% | 5.40% |
| Volatility: √((60% × 18%)² + (40% × 6%)²) | 11.06% |
| One-year z and P(loss): 5.40 ÷ 11.06 = 0.488 | 31.3% |
| Ten-year z and P(loss): 0.488 × √10 = 1.543 | 6.1% |
| Bad-case 10-year average: 5.40% − 1.645 × 11.06% ÷ √10 | −0.36% a year |
| Horizon for a 5% limit: (1.645 × 11.06 ÷ 5.40)² | 11.4 years |
Over twenty years the probability is about 1.5%. An all-equity portfolio has about a 35% chance of a loss in any single year.
The formulas
The mix's return and volatility.
The normal tail below a z that grows with the root of time.
The fifth percentile of the T-year average.
z = 1.645 for 5%, 1.282 for 10%.
Worked example
The one-year z first, then multiply by the root of the years. Each probability is the normal tail below minus the z. The follow-up says what this does not mean.
See it move
Same client and the same bonds. Change the equity weight, the equity return and volatility, the bond volatility, the correlation, the horizon and the probability limit.
- Lengthen the horizon. The probability of a loss falls and the bad-case average rises toward the mean; the one-year probability does not move.
- Raise the equity volatility. Volatility rises, the probability of a loss rises, the bad case falls, and the years the limit needs rise as a square.
- Raise the correlation. Volatility and the probability of a loss rise, unless the portfolio is all equity, where there is nothing to correlate.
- Raise the equity weight. The expected return rises; watch the probability of a loss, which depends on the return and the volatility together.
Run it backwards
The client names the probability of a loss they will accept. How many years must they be able to leave the money invested?
Set μ√T ÷ σ equal to the z for the limit and solve: T = (z σ ÷ μ)². The horizon grows as the square of volatility over return.
The follow-up compares with when the money is needed: long enough, or not, in which case the equity weight comes down or the client accepts more risk of a loss.
Traps
Say it in the interview
“Why should a client with a long horizon hold more equity, and what is the catch?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- P(average < 0) = N(−μ√T ÷ σ); the z grows with the root of time.
- Bad case = μ − 1.645 σ ÷ √T; it climbs toward the mean.
- Horizon for a limit = (z σ ÷ μ)².
- A long horizon lowers the chance of a loss, not the size of the worst one.