In my last two posts I made arguments against the idea that time diversifies a portfolio. This goes against the conventional wisdom that if you hold risky assets, like stocks, for long enough you are sure to get the good returns you are hoping for. Today we will look at option pricing theory and see what that has to tell us about time diversification. Zvi Bodie a professor at Boston University did just this in a 1995 paper titled “On The Risk Of Stocks In The Long Run”. Today I will summarize this paper. Obviously the option prices and interest rates he uses are not current, but the logic in his argument still holds true today. If you would like to read his original paper click HERE.

**The Set Up **

First let’s start by defining risk. Risk can be defined as the probability of a shortfall. A shortfall would occur if the value of a stock portfolio was below that of a portfolio holding only risk free investments. The best risk free investment is a zero coupon Treasury bond maturing at the target date. Remember that conventional wisdom tells us that as time horizon increases, the probability of a short fall decreases. Basically, the longer we hold our stock portfolio the less the chance we will fail to beat a risk free investment. Like I mentioned in the first post, this is true. The probability of a short fall does in fact decrease with time horizon. However, it ignores the magnitude of losses by treating all losses equally.

**Insurance Cost**

Let’s pretend I own an investment in the ETF SPY which tracks the S&P 500. It has done pretty well lately, but I am concerned the good run is over. I am not ready to sell my position, but I would like to buy some insurance just in case the market corrects. In fact I would like to insure it against earning less than the risk-free rate of return over the next year. Luckily this type of insurance exists — put options. Owning one SPY put option means I have the right to sell one share of SPY stock at a predetermined price, called a strike price. My “insurance” would effectively be purchasing the number of put options equal to the number of shares I own in SPY; at a put option strike price with a value equal to the SPY price earning a risk-free rate in one year; the put options would expire in one year. By holding this option I am guaranteed not to lose any money besides the cost of the put option. In one year from today, if SPY is below the strike price I can exercise the options. I can purchase the SPY in the marketplace and then immediately have the right to sell it at the higher strike price. If SPY is above the strike price the options would expire worthless, *but* I will enjoy the extra returns.

How much does this “insurance” cost? Market makers decide how much this option cost using various option pricing models. Zvi Bodie used a very common option pricing formula called the Black-Sholes formula for European options. Today’s option pricing models are much more complicated and incorporate variables not found in the original option formulas (such as skew), but the end result for our discussion would be the same.

When Zvi Bodie calculated how much it would cost for this put option it was 8% of the portfolio value. So if the value of my position in SPY was $100,000 it would cost me $8,000 to make sure it grew at least at the risk free rate of return over the next year. If you are the type of person who believes time alleviates the risk of stocks surely you would expect the cost of this insurance to go down the longer you held your position. If for example you wanted to insure that your $100,000 position grew at least at the risk free rate over 10 years the cost of that insurance should go down, right? Maybe something like $4,000. In fact the opposite is true. Zvi Bodie calculated that the cost of this put option would be 25% of your position or $25,000 in our example! This result is counter to what most people would expect.

Many hold the view that stock prices follow a mean reverting process causing more even and less risky returns. However, arbitrage based option models like Black-Scholes are valid regardless of the process of the mean. They are based on the law of one price and the condition of no arbitrage profits. If two investors disagree about the mean rate of return, but agree on the variance of the mean, they will agree about the option price, but that is for another post.

Next week we will explore the idea of your human capital, or the present value of your future earning power, and how that fits into your asset allocation decisions.