# Return, Risk and Probability of Success

### By: Frank Armstrong

By: Frank Armstrong, CFP, AIFA

**Why manage risk?**

Investors focus with a laser-like intensity on a single factor of investment success: rate of return. It’s simple to understand and reduces the entire multidimensional problem to a single number. A good strategy, or investment manager, has a high rate of return. A bad strategy or investment manager has a low rate of return. If only life were that simple!

By focusing on rate of return and ignoring risk, investors may actually torpedo themselves. High-risk strategies, even those that have high returns, may actually decrease the chance of an investor having a successful experience. Within a wide band investors may be far better served to focus on managing risk than stretching for additional return.

Few investors have an intuitive feel for the impact of risk. Just mention standard deviation and most of them will his zone right out. It would be helpful if we replaced the term standard deviation with relative risk rating. It also should be required that fund managers place equal emphasis on risk rating along with rate of return and publish relevant comparative data for the appropriate indexes.

Managing risk isn’t nearly as glamorous as generating excess returns. But, excess returns are elusive while basic risk management is easily achieved.

In a previous article we demonstrated that diversification reduces risk without compromising rate of return. An investor choosing between the two strategies with equal expected returns would certainly prefer the one with a lower risk. Lower risk not only reduces the dispersion of returns it increases both median and average returns. High-risk strategies may produce a few winners with outsized returns, but many more investors will experience substandard and unsatisfactory results. Investors are concerned with the certainty of results. After all, if you are a dead broke it’s a small consolation that somewhere else is an investor who struck it rich.

**An accumulation example**

Here’s a table showing the distribution of returns at different theoretical risk levels. We assume a 10% average return, a one hundred thousand dollar beginning balance, 30 year time frame, standard deviations of 10%, 20% and 30% respectively. Feeding these assumptions into a Monte Carlo simulator shows just how important managing risk is.

Standard Deviation |
Average Return |
Best Case |
Worst Case |
Median |

10% | $1,730,329 | $7,569,806 | $186,925 | $1,534,396 |

20% | 1,699,584 | 25,313,829 | 16,146 | 1,060,438 |

30% | 1,641,217 | 65,208,720 | 1,171 | 585,919 |

As you can see, as risk levels increase while holding rate of return constant, results become skewed. The average returns are virtually identical. But, both the best results and worst become more extreme. A few trials yield mega results, balancing out the trials that fall under the average. Importantly, the median result decreases precipitously as risk increases. More and more trials fall below the average result. This lower median return is the “cost” of the higher risk strategy.

This finding is consistent with the widely understood concept of variance drag. Because of variance drag, average (arithmetic) returns are always above compound (geometric) returns by an amount which increases as the volatility of the portfolio increases. Only in the case of no volatility are they the same. Volatility reduces the returns that investors care about, the compound return that ends up in their pockets.

Because there are no withdrawals in the above illustrations, none of the portfolios crash and burn. But, when we introduce systematic withdrawals the probability of portfolio failure increases with the withdrawal rate. During down markets so much capital is consumed at depressed prices to fund disbursements that the portfolio may self liquidate. So, it’s essential that wherever there are systematic withdrawals from a portfolio, risk should be vigorously controlled. Higher risk leads to predictably higher portfolio failures. Retirees, charitable institutions, endowment funds, and defined benefit pension plans must exercise prudence when managing their funds or risk portfolio blow out.

**Risk is amplified with systematic withdrawals**

Here’s a table showing the results at various risk levels for a portfolio taking systematic withdrawals. We assume a $100,000 beginning capital, $6000 dollar a year withdrawal beginning year one, 30 year time horizon, and 10%, 20% and 30% standard deviation. We will call successful any portfolio with $1 remaining after 30 years.

Standard Deviation |
Average Remaining Capital |
Best Case |
Worst Case |
Median Remaining Capital |
Probability of Failure |

10% | $749,508 | $ 4,801,855 | $0 | $615,244 | 1% |

20% | 759,208 | 18,387,065 | 0 | 318,072 | 21% |

30% | 782,699 | 50,637,405 | 0 | 52,577 | 43% |

**A real case postmortem**

Recently I analyzed a case where a retiree age 49 rolled over a single stock from her qualified retirement plan and failed to diversify it. For the ten years preceding the rollover that stock had had a standard deviation of 37.97%. Her broker estimated the future returns at 10% to 12% and commented that it was a great company. Additionally, the brokerage had a very high “target price” estimate over the near future. The value of the portfolio at rollover was $1,365,383, and an $80,000 annual withdrawal was agreed on from the IRA.

Poorly diversified portfolios pick up a boat load of uncompensated risk. But, in this case, the totally undiversified portfolio had a supertanker load of additional risk. The broker’s estimate approximated the return of the S&P 500 but the risk was a large multiple of the index’s risk. The difference is by definition uncompensated. The goal of any investment manager is to ruthlessly eliminate any uncompensated risk.

In this case, the entire value of the portfolio was lost. But, how predictable was that result? Was it a bolt from the blue, or a highly likely outcome of a totally unsuitable portfolio design?

Using Monte Carlo analysis we can get a very good indication of the range of possible outcomes for such a risky strategy. Because the retiree was so young, we believed that a minimum of 40 years was the appropriate time horizon. We accepted for purposes of the analysis the broker’s 11% average expected return assumption for the stock.

Year |
Probability of Success |
Median CapitalRemaining |

10 | 86 | $1,062,257 |

20 | 59 | $547,294 |

30 | 48 | $0 |

40 | 42 | $0 |

It’s highly unlikely that an informed investor would have opted for a strategy that offered a 58% chance of being dead broke before their projected life expectancy. Easily obtainable portfolios would reduce those dreadful possibilities to insignificance. In this case, the Monte Carlo analysis provides us with a powerful intuition that the strategy is fatally flawed. It can’t tell us whether any particular individual will succeed or fail, but it can point out that this strategy is insanely risky and highly unlikely to meet the investor’s goals.