Long-Run Risk





Kerry Back

Is there risk in the long run?

  • A common belief is that the law of averages will rule in the long run, eliminating risk.
  • It’s true that the average outcome from a gamble should not be risky in the long run.

But the gain or loss from gambling is average gain per gamble \(\times\) number of gambles.

Betting on the stock market

  • Based on history, the bet is definitely in our favor.

  • Play for a long time \(\Rightarrow\) almost certainly come out ahead.

  • But how far ahead is quite uncertain.

    • In worst 20-year period in U.S. stock market since 1926, $1 \(\rightarrow\) $1.73, a geometric average return of 2.8% per year (1929-1948).
    • In best 20-year period since 1926, $1 \(\rightarrow\) $24.65, a geometric average return of 17.4% per year (1980-1999).

Simulate returns

  • Mean and std dev of U.S. market return 1970-2021 was 12.5% and 17.4%.
  • Simulate 20-year compounded returns.
import numpy as np

mn = 0.125
sd = 0.174
nyears = 20

r = np.random.normal(loc=mn, scale=sd, size=nyears)
comp_ret = np.prod(1+r)

Repeat and evaluate the distribution

nsims = 1000

r = np.random.normal(loc=mn, scale=sd, size=nyears*nsims)
r = r.reshape((nyears, nsims))
comp_ret = np.prod(1+r, axis=0)
mean     34.66
std      35.44
min       2.14
10%       7.56
25%      13.53
50%      25.19
75%      43.05
90%      70.67
max     433.06
dtype: float64

Retirement Planning Simulation


Uncertainty about long-run returns \(\Rightarrow\) uncertainty about retirement plans.


  • Revisit the retirement plan
  • Generate random returns and simulate many lifetimes.