Notes on T-Bills and APY
Treasury Bills, Finding APR and APY
Just a quick note on yields. We can purchase Short Term Treasury Bills. These are sold in weekly increments (4 week, 8 week, 26 week, 52 week). For a 4 week term, that is 28 days. A 365 day year can have 365/28 =13.03 periods per year. Each period provides a return amount.
Treasury Bills are sold at a discount. That means the we are buying a Future Value for less money now. For example, you request a $1,000 Bill. The action happens a few days later and you find out that the $1000 bills actually sold at a discounted price of $996.70. You pay $996.70 now to purchase the $1000 Bill. In 28 days (or 4 weeks from now), you cash-in the Bill and get paid back the face value of $1000.
In this example you invested $996.70, got $1000 back, yielding a profit of $3.30 after 28 days.
We have a periodic rate, that is the amount of money gained per dollar invested over some time period. In this case, the time period is 4 weeks. The amount of money gained per dollar invested for this period is ($3.30/$996.70) = 0.00331. That is $0.00331 dollars gained for every dollar invested is easy to use (just multiply how much money you are going to invest and it tells you the returns). People don’t like small numbers, let’s make that per 100, convert to percent by multiplying by 100 -> 0.331%
These numbers all look scary small. Now, let us assume you invest the same amount over the entire year. So, let’s multiply that 0.331 percent by the number of periods in a year. If we are using the 28 day period, that is 13 periods per year: 0.331% * 13 = 4.30% Annual Percent Return (APR). This number looks better but is less easy to understand.
We can also apply this in real dollar amounts. We collect $3.31 every time it reinvests (every 28 days). $3.31*13=$43.03 over a year. Or using that APR 4.3% * $1000 = $43.
If you want to compare to a high yield savings account. Those are typically reported in Annual Percent Yield or APY. Where they assume compounding because the money is sitting in an account and the entire balance is invested (not in discounted increments). APY is compounded over a period. At the bottom of this entry is a sample derivation of the equation. APY = (1 + Rate over the period)^(Number of Periods in a Year) – 1
In our case the rate is 0.00331 for each 28 day period. So we have: (1+0.00331)^(13)-1=0.0439 APY -> convert to percent by multiplying by 100 -> 4.39% APY
So, these are what we need to compare. The APY in a high yield savings account must be equal to or higher than the APR or APY for a Treasury Bill.
Derive the APY Equation
We need to first understand how the compounding interest works. If we have a starting value (V1) and we invest it at a given rate. The amount of interest gained is added to the initial starting value. The new higher amount is then re-invested the next period at the same interest rate. Again, the additional interest gained is added to the starting amount… This continues for as long as the money is invested. In this simple example let’s see what 3 months looks like. V1 is month One when we start, V2 is month two, V3 is month three and V4 is the start of month four when we end, R is the interest Rate. We end up with three equations, one for each month, each describes the amount of money at the end of the month.
Let’s say that today you only know the starting amount, the rate, and number of months. Yes, we can do the math for each month manually and just keep repeating this process. Or… we can try to get it all calculated at one time. To find the final amount, use the V4 equation and substitute each variable to get back to a form that only uses V1 and R – because when we start we only know the initial values. We already have the substitutions figured out, just work backwards, and then simplify the math by combining terms to make it easier to read. I put the substitutions in brackets:
For the 3 period time, we get to a single equation. It will tell us the amount of money after 3 periods (aka V4) using only the Rate for each period and the Starting amount (V1). Now, this nesting action, where the results from one month are in the next month, and the form of the final equation looks a lot like the quantity (1+R) raised to a power of 3:
Can you see that? The parts in the parentheses match! So, now we can see the form or structure of the math. The nested summations are a power function. We can simplify the equation to look better and make it easier to enter into a calculator by replacing the long string by a simple (1+R)^n.
V1 is the starting amount of money, R is the rate for the period (for example 28-days), and n is the number of 28-day periods to calculate the final value over. This format can allow us to pick any arbitrary number of periods and rates to calculate. This way, we don’t need to do as many steps if the rate stays the same for all the investment periods. For example a single month calculation:
Yields are the Returns divided by the Present Value. So, our Annual Return would be the compounded future value minus the present value; then divide by the present value. [V1*(1+R)^n – V1]/ V1 = Yield, to make this a percentage yield, divide by the starting amount, and multiply by 100.
