Nominal Profits

Before we move further with options, we need to provide a sturdy foundation for understanding them. That is, in the last post we talked about how much money various option strategies made, but what we didn't consider what the opportunity cost of the strategy was. By that I mean there is a cost of the funds. Consider that you could have left the 9,400 dollars in your bank account earning 5% interest. It took one month for the strategy to play out. Thus, we need to consider how much money we could have earned. The formula is A(1 + R/m)^mn where

* A equals the deposit
* R equals the interest rate
* m equals the compounding frequency
* n equals the period of time the investment is held

Therefore, 9,400*(1+(.05/12))^(12*1/12)= $9,439.17 that is we could have earned $39.17 on the investment risk-free. Conversely, we should take the profit we earned and discount it backwards by this same discount rate, 1.004167 to find out what we made in real terms.

In the real world though we would need to find a "better" or more suitable risk-free rate. While it is true that your deposit at the bank is FDIC insured, it is more than likely that the bank is using that subsidy to achieve funding versus seeking funds in the capital markets. Most people would point to the US Treasury rates or LIBOR. The latter consisting of the rate at which one bank will lend funds to another bank. Thus, it is considered the opportunity cost of capital in the specified time period, 1, 3 6 and 12 month durations. There is also a super-short duration that is sometimes used which is the Repo rate. Basically, one bank sells a security to another bank with an agreement to buy the security back at a slightly higher price. A quick example, I sell a MBS to JP Morgan for 99.98 dollars of its face value and buy it back for 100 or par two days later. Thus, JPM lent me 99.98 dollars for two days. I paid two cents for the privilege. This is basically what the Federal Reserve did during the credit crunch in 2008, they would extend these loans at very low rates, so that banks could remain liquid and meet their liabilities. Now the Fed is actually buying securities, which is an entirely different matter.

However, in derivative securities when using a rate like OIS, LIBOR or Treasury rates they are continuously compounded. So in the bank account example, the bank only credits my account at month end. So I end up with slightly more than 5% more on my account at the end of the year. That is I receive 1.004167 more each month, at the end of two months I would have 1.008351 until I ended up at 1.051162 at the end of the year. If instead it compounded only twice or four times a year I would have less money the less amount of compounding terms. Conversely the more times the money is compounded the more money I would have at year-end. So when a financial asset is continuously compounded it reaches the maximum it could possibly grow at a stated rate of interest. We can then use the exponential factor e^x so that the equation can be written Ae^xn or substituting x for the interest rate the equation is Ae^Rn. Thus, continuous compounding brings up the factor to 1.051271.

However, in the financial markets quotes are given in all variants of time, quarterly, monthly and we need to be able to go back and forth between them.

Ae^(Rn)=A(1+ r/m)^(mn) which you can subtract n from both sides and A as well to end up with e^R=(1+r/m)^m.

Now we can go back and forth. So if a broker quotes you 10% interest compounded semiannually that gives you a m of two and a r of .1. Therefore, e^R=(1+r/2)^2 so take the natural log of both sides so that R=2 ln (1+.1/2) finding that R equals 0.09758

A basic rule is that when you go from the various compounding to continuous compounding the continuous compounding will always be at a lower rate because it compounds more often. In the reverse going from continuous to quarterly you should expect that quarterly rate would be higher.

Homework problem:

Your savings account pays 12% annually continuously compounded but pays out the interest to you in quarterly installments. How much interest will be paid on your 1,000-dollar deposit each quarter?