## Cash & Carry, #3: the Price of “Peso” Risk

In our previous post in this series, we highlighted a line of research by Burnside et al. that explains the profitability of the carry trade as a “peso” problem. That is, the historical data doesn’t completely reflect all of the bad outcomes that may arise. The as yet unobserved bad outcomes are known as “peso events”. The historical profitability of the carry trade, isn’t the complete story. It’s a biased sample. The few bad outcomes not yet observed resolve the puzzle. Investors are worried about these “peso events” and price the currencies accordingly.

In this post we want to delve a bit more into this issue of how investors may be pricing “peso events.”

There are two different aspects to pricing “peso events.” First, we must assess the probability of an extreme negative event. Second, we must assess the discount we want to apply to that negative event. This second aspect is what we want to focus on.

In finance all dollars are not created equal. Cash received in states when the investor otherwise has plenty cash is not worth as much as cash received in states when the investor otherwise is short cash.

The standard models for pricing risk – such as the CAPM, the Fama-French and the Carhart factor models, among others – are based on what is called a linear stochastic discount factor across all states. They assume a linear relationship between the discount factor and some underlying risk factor. In the CAPM, that would mean a linear relationship between the discount factor and the payoff on the market portfolio. But there isn’t any natural presumption in favor of linearity in the discount factor.

The argument made by Burnside et al. is that the discount factor is not linear. Instead, a very high discount factor is applied to the as yet unobserved “peso events.” And this very high discount factor is essential in making sense of the historic profitability of the carry trade strategy.

In order to make the argument more accessible, we have constructed a simple numerical example: In this example there are 3 possible future states corresponding to three different payoffs on the market portfolio. The table shows the probability of each state and the discount factor associated with each state. The discount factor is linear in the market payoff. The value of a dollar earned when the market payoff is high is less than the value of a dollar earned when the market payoff is low. The table also shows the payoff to a carry trade investment. At the bottom of the table we see the expected payoff, calculated by taking the ‘sumproduct’ of the probabilities and payoffs. We also see the present value of the payoff, which is calculated by taking the ‘sumproduct’ of the probabilities, payoffs and the discount factors. The carry trade portfolio assumed here requires a zero net investment, and nevertheless, the portfolio has a positive expected value of 0.43. This is the original carry trade puzzle.

The next table shows the “peso event” solution to the puzzle. In addition to the 3 possible future states derived from the historical data, we have added a fourth possible state, the “peso event” with a very small probability of occurring. The “peso event” is shown in red. Therefore, we have had to adjust the probabilities of the 3 original states accordingly. The discount factors and payoffs for the 3 original states remain the same as before. The “peso event” has a very negative payoff. It also has a very high value for the discount factor, one that is far above what would be implied from a linear stochastic discount factor model. Now, we see that the present value of the payoff, taking into account the “peso event,” is zero. This means that the risk of the “peso event,”  including both the probability and the discount factor, outweighs the value of the average profits earned in the other 3 states.

This example illustrates how critical it is to think carefully about the discount we apply in the case of extremely bad outcomes—the price of “peso risk.” It can make a big difference to how companies evaluate investment strategies such as the carry trade, and many other corporate financial decisions.