When we invest in the financial markets we get paid to take risks. The more risks we take, the more we expect to be paid. The corollary to getting paid to take risks is the realization that we may take significant losses. The higher the probability of loss and the greater the loss severity the more we expect to be paid. There is no free lunch in the financial markets. Contrary to what many people believe you can’t expect big returns for taking small risks.

I’m a Laker fan. Kobe Bryant was recently asked whether the elation that comes with winning an NBA championship more than offsets the devastation caused by losing one. Kobe responded by saying that the pain of losses far outweighs the euphoria of wins. In economics this is known as risk aversion. For a risk-averse investor the negative effects of losing x dollars in the market outweighs the positive effects of making x dollars in the market.

Imagine that we have a fair coin toss game where there is a 50% chance of a head and a 50% chance of a tail. If heads pay $10 and tails pay nothing then the expected payout on this game is…

Expected payout = $10 x 50% + $0 x 50%= $5

A risk-neutral player would pay $5 to play the game because that is the expected payout. A risk-preferred player would pay more than $5 to play the game even though the expected payout is less than the price to play. An example of a risk-preferred player would be a gambler in Las Vegas where the expected payout on a $1 slot machine is only 96 cents on the dollar. A risk-averse player would pay less than $5 to play the game because the negative effects of getting a zero payout (and losing money) outweighs the positive effects of getting a $10 payout (and making money). If the risk-averse player pays $4 to play the game then the return profile for this player is…

Expected return = expected payout / price to play - 1 = $5/$4 – 1 = 25% gain
Maximum return = $10/$4 – 1 = 150% gain
Minimum return = $0/$4 - 1 = 100% loss

Investors as a rule are risk-averse. The risk-averse investor’s return profile is similar to what we see in the example above. The historical annual rate of return on the stock market is approximately 10% and the historical volatility (i.e. standard deviation) is approximately 18%. The investor’s annual return profile on a diversified stock portfolio assuming that past returns are indicative of future returns is…

Expected return = 10% gain
A one standard deviation gain = 10% + 18% = 28% (16% probability of gains greater than 28%)
A one standard deviation loss = 10% - 18% = -8% (16% probability of losses greater than 8%)

Note that the probabilities above assume a normal distribution. Returns are usually modeled as coming from a normal distribution but in the real world large losses are more frequent than what a normal distribution would imply. This is known as the distribution having a fat tail.

The caveat to the return profile above is that the markets are fairly priced at any given point in time. If the markets are over-valued then the probability of realizing losses is far greater than the probability of realizing gains. Next time we will talk about current market valuations.