# Finding Investor's Surplus

Investor’s surplus is equal to the risk premium squared divided by (2 times risk aversion times the standard deviation squared), where the risk premium   and standard deviation refer to the return of the risky mutual fund.  The risk premium is also sometimes called the mean excess return.

Notation:

sigma = standard deviation

A = risk aversion

Investor’s Surplus = mu^2 / (2 A sigma^2)

Tricky Units. The trickiest aspect is handling the percentages and the time units right (though if everything is on an annual basis, you won’t make a mistake with the time units).

Besides indicating a context, % is the same thing as multiplying by .01.  So for example, 10% is the same thing as .1, so 10% squared is .1 squared which is .01 which is 1%. Similarly, 20% squared is 4%.

Since the formula for investor’s surplus has mu squared on top and sigma squared on the bottom, if both mu and sigma are expressed as percentages, the % squared on top and the % squared on the bottom cancel. BUT, then you will want to express investor’s surplus as a percentage, so you need to multiply by 100%.  The 100 you are multiplying by is going to make the arithmetic look easier if you are doing it by hand. Anyway, you need to practice to get this right. A good benchmark to help get this right is that if the risk premium is 1% per year, the standard deviation is 10% over a year, and risk aversion is 1, then Investor’s surplus is .5% per year.

Why do I say the standard deviation is something like 10% or 20% “over a year” instead of “per year”? This is a very interesting technical thing. With uncorrelated shocks, it is the variance that adds nicely over time. 10% standard deviation over a year is a 1% variance per year, a 10% variance per decade, a .25% variance per quarter, etc.  Strangely enough, if you want to express it with “per,” the standard deviation in this example is 10% per square root of year, since that is what squares to yield the variance of 1% per year. This is one of the fascinating technical things I learned from Greg Mankiw.

What are reasonable values?

The Risk Premium. Historically, the risk premium for a broad stock index has been about 6% per year in the postwar era. Since it has been a very good 80 years, many people argue it is too much to expect 6% again in the coming years. But I think a 3% risk premium is pretty reasonable to count on, unless people start acting a lot less risk averse in the future than they do now (which would drive up the price and drive down the risk premium).

The Standard Deviation. I think of 20% over a year as a normal level of the standard deviation of a broad stock index. During the Great Moderation, it looked like it was down to 15% over a year.

As I said in class, we get a much better read on the standard deviation and how it evolves over time than the risk premium. Why? The standard deviation of returns is so big that it creates a huge amount of noise in trying to figure out what the risk premium is, without decades and decades of data. By contrast, every day that passes gives a good read on the standard deviation for that day. (And in the formal theory, even much smaller chunks of time give a good read on the standard deviation for whatever length of time it is.)

Risk Aversion. I argued in class (and hope to back it up in a blog post) that by the Raj Chetty argument, the appropriate risk aversion for most people is between 1 and 2.  This is much less risk averse than most people act. I was the instigator for a risk aversion measure in the Health and Retirement Survey that found that the average risk tolerance is something like 1/6, which would be a risk aversion of 6 for the representative investor. But many many people make choices on the survey that indicate a risk aversion considerably higher than 6.

Problems. Calculate the investor’s surplus in percent per year for the following situations. Express it in percent per year.