# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Expected Utility and Valuation of Risky Payoffs

Having established that expected utility is the quantity to consider when valuing risky gambles, we are ready to start thinking about how to value risky payoffs.

The easiest way to operationalize risk is to break-up future possibilities into ‘things going up’ (optimistic) and ‘things going down’ (pessimistic).

Consider a bet that offers $X_H$ if the coin toss turns up ‘heads’ $H$, and $X_T$ if it turns up ‘tails’ $T$. Since the payoffs are arbitrary, we can assume $X_H > X_T$. Assuming the coin is unbiased, expected value of the payoff $X$ is:

\begin{aligned} \displaystyle E[X] &= \frac{1}{2}X_H + \frac{1}{2}X_T \\&= \frac{X_H + X_T}{2} \end{aligned}

Daniel Bernoulli, however, told us that the way to ‘value’ this payoff is to not to look at $E[X]$, but the utility of this payoff, i.e. $E[U(X)]$. This is easily calculated as:

\begin{aligned} \displaystyle E[U(X)] &= \frac{1}{2}U(X_H) + \frac{1}{2}U(X_T) \\&= \frac{U(X_H) + U(X_T)}{2} \end{aligned}

Now consider what will be the utility of a payoff that is quantitatively equal to $E[X]$, but instead is a sure thing? Well, that’s simple. Given our utility $U(X)$ of any payoff $X$, the payoff for the quantity $E[X]$ will be just $U(E[X])$.

So we have two quantities. $E[U(X)]$ and $U(E[X])$. Let’s compare the two. We first do it graphically, and then in a separate post later we’ll do it mathematically. (Why do it mathematically too? Well, maths remove all confusions that may arise when drawing lines on a graph, and it is good to be aware how to think about it mathematically.)

Graphically: $E[U(X)]$ vs $U(E[X])$

(Click on the graph to zoom.)

Bernoulli told us that when valuing gambles we should consider expected utility $E[U(X)])$ of the risky payoff, and not expected payoff $E[X]$. The utility of the sure amount (quantitatively speaking) $E[X]$ is $U(E[X])$ which, as is apparent from the graph (and as we also show mathematically later), is greater than expected utility $E[U(X)]$ .

Further, it is clear that the utility of the sure payoff $X^* < E[X]$ is same as that for the gamble – both are equal to $U(X^*)$.

To reiterate, both utility being quantitatively same means that people might as well take a sure $X^*$ rather than indulge in a gamble that exposes them to the risk of not even getting $X^* - X_T$. Diminishing marginal utility implies that people worry more about the lost $X^* - X_T$ than they are excited by the gain $X_H - X^*$ . That is, concavity implies risk aversion.

This is a powerful result and forms the foundation for most of the classical finance theory.

The utility of ‘sacrificed’ amount / utility when taking up the sure $X^*$ instead of the gamble $U(E[X]) - U(X^*$) corresponds to the risk premium (and we’ll have a chance to talk more about this later). An important consequence of the existence of risk premium is that people value risky gambles less than their expected values. The utility of the sure thing $X^*$ is same as the expected utility of the risky $E[X]$ – i.e. the risky gamble is valued at $X^*$ and not at $E[X]$.

We saw in the St. Petersburg Paradox that the expected utility rule tells us that the bet posed by Nicolaus Bernoulli is worth only $4$ bucks – much less than its expected value. Now we can claim that concavity / diminishing marginal utility / risk aversion (all mean the same thing) implies that risky payoffs are worth less than their expected values.

This means that the PV of the expected value of a risky gamble would be less than what would it be for a sure thing. So if a rate $r$ applies to the risk-free payoff ($X^*$), for a risky payoff ($X$, with expected value $E[X]$) we’ll have:

$\displaystyle PV(X) < \frac{E[X]}{1 + r}$

Alternatively,

$\displaystyle PV(X) = \frac{E[X]}{1 + r^*}$

$; r^* > r$.

Note that we have not defined $r^*$ yet (this we’ll do it later in the course). We are just saying that $r^*$ corresponds to the distance $E[X] - X^*$ (the risk premium), and in general, higher the value of the distance $E[X] - X^*$, higher the $r^*$, i.e. higher the risk premium.

A corollary to this result is that since the utility from $X^*$, $U(X^*)$, and the expected utility $E[U(X)]$ is the same, the following will hold:

$\displaystyle \frac{X^*}{1 + r} = \frac{E[X]}{1 + r^*}$

That is we can now compare risky and risk-free payoffs in PV terms as long we use the right discount rates! This means we now have a way to evaluate and compare all kinds of investments – not just the ones with risk-free cash flows but also risky cash flows. Now we are talking!

We are now ready to state our final results:

1. Risky payoffs are discounted at a rate higher than the risk-free rate. This rate is called the opportunity cost of capital.
2. We can compare all investments by the PV rule as long as we choose the right discount rate commensurate with the risk of the investments.

Please note that this says nothing about what any individual would consume. You may as well choose the gamble $E[X]$ and I may choose the sure thing $X^*$. This is not about our individual choices. Go back to the PV rule which said that we can evaluate all investments in PV terms. Our results on valuation of risky payoffs – as a consequence of concavity of utility curves – tell us that we can still use the PV rule as long as we discount our risky payoff at the right discount rate (the opportunity cost of capital).