Inflation

The quantity theory of money describes inflation and has a long history dating back to thinkers in the Renaissance. Coepernicus among others noted that as gold arrived from the New World the price of goods rose. Thus the theory expounds that there is a positive relationship between increases in the money supply and prices of goods and services.

An easy example would be coffee. Imagine a Dunkin Donuts as it started in the 1950s, it charged a nickel. Now to buy that same cup of joe it costs 2 pieces of silver ($2.00.) Is it because the coffee is better, or because people enjoy it more? Nope, according to the theory, your dollar's value is less, so before a 20th of a dollar got you some warm goodness now it costs two. So how did the dollar come to be valued less? Isn't monetary policy off the gold standard?

It wouldn't be economics if supply and demand were not involved. The money supply is based on the Federal Reserve through its operations including its open market operations. Again, it buys bonds to increase the money supply and it sells them to decrease the money supply. For the next example it is assumed that the Federal Reserve keeps the money supply constant. The money demand however is controlled by people. Some factors that affect how much money people want to hold are: that availability of credit through credit cards, the availability of withdrawing funds either through an ATM or from a bank's teller. Underlying this idea though is the cost of goods/services because as they are priced higher than people must carry more cash for daily transactions. Since the people carry higher balances with higher prices than the demand for money is greater.

Here is a chart.

The blue line is the fixed money supply. The Red line is the Money Demand. The demand line curves downward because as the value of money falls or conversely the price level is higher people desire a larger quantity of money. The equilibrium is shown by the dotted line it is where the price level and the value of money intersect.

Now as the Fed performs open market operations, say buying treasury bonds, it will increase the money supply. The next chart shows the money supply parallel shifting out (from blue to orange.) This cause the value of money to fall and thus price levels to rise.

It is important to note that the fundamental economy has not been affected by this injection of money. The capital level has not changed, the labor productivity has not changed, knowledge is static, everything in short is held constant. However, because there is now more money chasing the same amount of goods then prices must therefore rise.

This brings up another point, that prices should not matter or restated that there exists a monetary neutrality. If prices double, then so should wages because the inputs for the good/services include labor, thus everything should be equal in the long run.

Now back to MV=PY or V=(PY/M) so the velocity of money should equal the price level times real output divided by the quantity of money. During most times V is stable, however, during financial crises this does not have to be so.

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?