## Why buy bonds?

18-02-2021

Greg Tell

*How risk-adjusted returns drive outcome oriented solutions.*

Imagine the following investment: you invest $1 and with 50% probability you get back $1.30 while with 50% probability you get back $0.75. Does this sound like a good investment to you? Well, if you are reading this, you likely have at least a passing interest in finance and math so you probably recognize that the expected return is positive 2.5%. That is:

Expected return = (0.5)(1.3) + (0.5)(0.75) = 1.025 – 1 = 2.5%

So, for a risk-neutral investor, in theory, this should be an attractive opportunity. But consider this, imagine making this investment twice in succession. Reinvesting the entire proceeds from period one the second time. Without calculating, what does your intuition tell you about the probability of having a positive cumulative gain after the second period? Given that the expected value is positive, you might think the probability should be quite good. You might be surprised to learn that there is actually a 75% probability you will be worse off. That is, after two rounds there are four equally likely possible outcomes:

(1.3)(1.3) = 1.69

(1.3)(0.75) = 0.975

(0.75)(1.3) = 0.975

(0.75)(0.75) = 0.5625

Note that the expected value is behaving in line with intuition at 1.05 or up 5%.

Now you might think that we just need to run this a few more times before the positive expected value will take control and push the probability to positive outcomes higher. However, even after 20 trials, the probability of having a cumulative loss is still almost 59%. So what is going on? As you may recall, the cumulative results of these repeated trials approach a log-normal distribution. This is characterized by a long right tail where are few outsized gains are offsetting many smaller losses. In technical terms, because of the high volatility relative to the expected return, the median is much lower than the mean (see chart).

So getting back to our investment, determining whether this is appropriate really depends on your investing goal. Let’s say you are saving for a new car, house, education, or retirement, the real focus should be not simply which investment has the highest expected return, but rather, what investment provides the highest probability of achieving that goal.

This is why we stress the concept of strong risk-adjusted returns when the focus is outcome oriented investing. This is why high-quality fixed income can play a role in a diversified portfolio. For example, imagine you have an investing goal and need to generate a cumulative return of 10% over the next 5 years to achieve it. Assume you have two investment options, a portfolio of securitized assets that has an expected annual return of 2.5% and volatility of 2.5% or a small cap equity portfolio that has an annual expected return of 5% and volatility of 20%. Which is the better choice for achieving that goal. Well in this case it turns out that the probability of attaining your goal is actually higher with the lower yielding securitized portfolio because the risk-adjusted returns are significantly stronger.

Now, before the more mathematically astute readers ask, “doesn’t this assume returns are normally distributed?” and point out that if you buy a single bond and hold to maturity, the horizon return is not normally distributed (you either get back your purchase yield or it defaults). While this is true, it actually makes the case for bonds even stronger if you have a specific investment goal that can be achieved by holding bonds. For this analysis I am considering an actively managed portfolio.

The bottom line is, when investing, start with your goals and then determine the best way of achieving those goals. Often this will mean a diversified portfolio that includes the relatively strong risk-adjusted returns that bonds may provide.