## Why the rate of return on insurance is low

I’m currently doing this course on Asset Pricing at Coursera, offered by John Cochrane of the University of Chicago Booth School of Business. I’m about a fourth of the way into the course and the beauty of the course so far has been the integration of seemingly unrelated concepts. When I went to business school (IIM Bangalore) about a decade ago, I was separately taught concepts on utility functions, discount rates, CAPM, time series analysis and financial derivatives, but these were taught as independent concepts without anybody bothering to make the connections. The beauty of this course is that it introduces us to all these concepts, and then shows how they are all related.

The part that I want to dwell upon in this post is the relationship between discount factors and utility functions. According to one of the basic asset pricing formulae introduced and discussed as part of this course, the returns from an asset is a positive function of the correlation between the price of the asset and your expected consumption growth. Let me explain that further.

The basic concept is that one’s utility function is concave. If you were to plot consumption on the X axis and utility from consumption on the Y-axis, the curve would look like this:

In other words, let us say I give you a rupee. How much additional happiness would that give you? It depends on what you already have! If you started off with nothing, the additional happiness out of the rupee that I gave you would be large. However, if you already have a lot of money, then the happiness you would derive out of this additional rupee would be much lower. This is known in basic economics as the law of diminishing marginal utility, and is also sometimes called the “law of diminishing returns”.

So, let us say that tomorrow you will either have Rs. 80 or Rs. 120 (the reason for this difference in payoff doesn’t matter). Let us call these as states “A” and “B ” respectively. Now, suppose I’m a salesman and I offer you two products. Product X  pays you Rs. 20 if you are in state A but nothing if you are in state B. Product Y pays you Rs. 20 if you are in state B and nothing if you are in state A. Assuming that you can end up in states A or B with equal probability, which product would you pay a higher price for?

The naive answer would be that you would be indifferent between the two products and would thus pay the same amount for both. However, rather than looking at just the payoffs, you should also look at the utility of the payoffs. Given the concave utility function, you would derive significantly higher happiness from the additional Rs. 20 when you are in State A rather than in State B (refer to appendix below). Hence, you would pay a premium for product X relative to product Y.

Now, from a purely monetary perspective, the payoffs from X and Y are equal. However, you are willing to pay more for product X than for product Y. Consequently, the expected returns from product X will be much lower than the expected returns from Y (define returns as $frac {payoff}{price} - 1$. Hence, for the same payoff, the higher the price the lower the returns). Keep this in mind.

Now let us come to insurance. Let us take the example of car insurance. Most of  the time this doesn’t pay off. However, when your car gets smashed, you are compensated for the amount you spend in getting it fixed. What should be your expected return from this product?

Notice that when your car gets smashed, you will need to spend money to get it repaired. So at the time of your car getting smashed, the amount of money (and consequently consumption) is going to be lower than usual. Hence, the marginal utility of the insurance payout is likely to be higher than the marginal utility of a similar payout at a point in time when your consumption is “normal”. This is like product X above – which gives you a payoff at a time when your consumption level is low! And remember that you were willing to expect lower returns from X. Similarly, you should be willing to expect a lower rate of return from the insurance product!

Technical Appendix

A standard utility function used in finance textbooks is parabolic. Let us assume that for a consumption of $C$, the utility is $- (200-C)^2$. The following table shows the utility at various levels of consumption:

Consumption          Utility
80  (A)                  -14400
100                        -10000
120  (B)                 -6400
140                        -3600

Notice from the above table that getting the payoff of 20 when you are at A increases your utility by 4400, whereas when you are at B, the payoff of 20 increases your utility by only 2800. Hence, your utility from the payoff is much higher when you are at A than at B. Hence, you would pay a higher price for product X (which pays you when your consumption is low) than product Y (which pays you when your consumption is already high)