How confident should you be about your investment goals?<\/strong><\/p>\n

What is the probability that your investments will fall short of a particular investment goal?\u00a0 How is this probability affected by uncertainty about the equity risk premium?<\/p>\n

In this post, I plan to look at a simple method for estimating the probability that\u00a0a risky\u00a0investment will fall short of\u00a0a particular\u00a0return goal, and I\u2019ll then extend this analysis to allow for uncertainty about the equity risk premium.<\/p>\n

This analysis is a very simplified\u00a0example of the\u00a0concept presented by Lubos Pastor and Robert F. Stambaugh in their paper \u201cAre Stocks Really Less Volatile in the Long Run?<\/a>\u201d.<\/p>\n

For starters, if we assume stock prices follow a random walk, then the probability of a risky investment underperforming a risk free investment, such as a T-bill, is given by this equation:<\/p>\n

<\/p>\n

Where:
\nr<\/em> = continuously compounded (log return) t-bill rate
\n = continuously compounded (log) equity return
\n\u00a0= annual standard deviation of equity returns
\nT<\/em> = number of years<\/p>\n

This shortfall probability can be calculated by putting everything to the right of the greater than sign into the \u201cnormsdist\u201d function in Excel or Google Docs.<\/p>\n

Notice that as long as\u00a0 is greater than r<\/em>, the probability of shortfall will approach zero for large values of T<\/em>.\u00a0 So, for this model, the risk of falling short of an investment goal will approach zero if the horizon is long enough and the mean return is greater than the t-bill return.<\/p>\n

Now, if\u00a0we add equity premium uncertainty, the shortfall probability is given by this equation:<\/p>\n

<\/p>\n

The additional\u00a0 term is the standard deviation of our estimate of the equity premium.\u00a0\u00a0Again, the probability can be calculated using the \u201cnormsdist\u201d function.<\/p>\n

Notice that in this model, the probability will not approach zero for large values of T.\u00a0 There is a limit to how small the probability can become which is dependent on our uncertainty about the equity premium.<\/p>\n

I plotted both shortfall probabilities vs. horizon for an assumed equity return of 8% and 16% standard deviation, and risk-free rate of 2%. I also used 2% as the sigma for the equity premium uncertainty.<\/p>\n

Here is the plot for shortfall probability vs. horizon.<\/p>\n

<\/a><\/p>\n

Clearly, the risk of a shortfall is elevated when we include the equity premium uncertainty.\u00a0 However, the difference\u00a0does\u00a0not appear to be\u00a0huge.\u00a0\u00a0 The probability of underperforming the risk free rate is still small at long horizons.<\/p>\n

We can also substitute a minimum return goal for the risk free rate to calculate the probability of not meeting a particular return goal.\u00a0 For example, if I need a return of at least 5% to meet my goals for retirement, then, keeping the other parameters the same as above, the shortfall probability is shown by this plot.<\/p>\n

<\/a><\/p>\n

In this example, the effect of the equity premium uncertainty is larger.\u00a0 Over a 40-50 year horizon, there is a very meaningful difference between the two methodologies.<\/p>\n

These examples are only intended to illustrate the effect of equity premium uncertainty as compared to the pure random walk case.\u00a0 The examples don’t consider the potential for mean-reversion or fat tailed return distributions. \u00a0Also, it is very important to note that they don\u2019t consider the magnitude of the shortfall<\/em>.\u00a0 For example, if your probability of falling short is 10%, the magnitude of this shortfall might be very small or it might be huge.\u00a0 This method of analysis ignores this consideration, but you can get a sense of the expected shortfall by experimenting with equations using a variety of return thresholds.<\/p>\n

My spreadsheet<\/a> for generating these graphs can be accessed using Google Docs.\u00a0 If you save a copy of this document, it can be modified to incorporate your own assumptions and the effect of equity premium uncertainty can be evaluated under a variety of scenarios.<\/p>\n","protected":false},"excerpt":{"rendered":"

How confident should you be about your investment goals? What is the probability that your investments will fall short of a particular investment goal?\u00a0 How is this probability affected by uncertainty about the equity risk premium? In this post, I plan to look at a simple method for estimating the probability that\u00a0a risky\u00a0investment will fall […]<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/posts\/1429"}],"collection":[{"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/comments?post=1429"}],"version-history":[{"count":107,"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/posts\/1429\/revisions"}],"predecessor-version":[{"id":4039,"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/posts\/1429\/revisions\/4039"}],"wp:attachment":[{"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/media?parent=1429"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/categories?post=1429"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/tags?post=1429"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}