# Know the return on your investments

Oct 29, 2006 by

I once went to a community bank to inquire about a CD special: 5.5% APY for six months. I asked the sales associate if I could borrow a calculator with a power (^) function, so that I could figure out how much interest I’d be earning. She told me that she didn’t have one. I asked her if she knew how much I would be making in interest, and she couldn’t tell me either. Unfortunately, many people don’t know how to calculate the return on their investments before they make them. As a result, they may make poor investing decisions, or may tie-up their money in investments with far lower returns than they realize.

To make matters worse, interest can be reported in two different ways: annual interest rate and APY (Annual Percentage Yield). Annual interest rate and APY are in different units, and APYs are always bigger numbers than annual interest rates. As a result, things involving payments, such as loans and credit card debt, tend to be reported in terms of annual interest rates, to make them look smaller. Things in which the consumer is paid, such as CDs and savings accounts, tend to be reported in terms of APY, which looks bigger. In actuality, one can convert between the two. The difference between annual interest rate and APY is that the interest is compounded in APY, but is not compounded in the annual interest rate.

If you put \$100 in the bank, at 5% APY, you will get \$105 at the end of the year. At the end of two years, you will get \$100*1.05*1.05 = \$110.25. To calculate the amount of money you get from an investment with known APY, simply take:
r = the APY in terms of a percentage in the range 0 to 1 (so, 3% would be r=.03)
d = the amount of dollars you begin the investment with
t = the number of years you will be investing (r=1 for one year, r=1.5 for eighteen months)
Once you have written down r, d, and y, calculate the following:
d*(1+r)^t

If you put \$100 in the bank, at a 5% annual interest rate, compounded monthly, you will get \$105.12 after one year, and \$110.49 after two. As you can see, this amount is greater than what you would have gotten with a 5% APY, as a 5% APY corresponds to an annual interest rate below 5%. The annual interest rate does not include compounding (money made from re-investing the returns on one’s investments), while the APY does.
To calculate the amount of money that you get from an investment with known annual interest rate, simply take:
r = the annual interest rate in terms of a percentage in the range 0 to 1 (so, 3% would be r=.03)
d = the amount of dollars you begin the investment with
t = the number of years you will be investing
n = assuming you compound at the end of each period, the number of period units in one year (n=1 if the period is in years, n=12 if the period is in months, n=365 if the period is in days)
Once you have written down r, d, t, and n, calculate the following:
d*(1+(r/n))^(n*t)

Now that we understand the math, what would I have gotten had I bought a \$100 CD from the community bank, with 5.5% APY for six months? The answer is \$100*(1.055)^.5 = 102.71. So, I basically ended up at the end of six months with 2.71% more money than when I bought the CD. Note that if the bank had let me hold the CD for a year, I would have ended up with 5.50% more money than when I had bought the CD. Why do I get less than half the benefit of the full year CD by only holding for six months (2.71%*2 = 5.42% < 5.50%)? The answer is that I lose the opportunity for the money to compound. In the second six months that I own the CD, is is as if I had sold the first CD, and then reinvested \$102.71 in a new six month CD paying 5.50% APY. Since I start the second six month period with more than I started the first with, due to compounding, I end up with more than twice the interest I would have gotten from holding the CD for six months. If this math is confusing you, you can use an online CD return calculator, such as the one at bankrate.com
http://www.bankrate.com/brm/calc/cdc/CertDeposit.asp

The same principles that apply to calculating the return on CDs apply to calculating the return on savings accounts. Given that interest rates are rising, I would advise my readers to not buy any CDs at the moment, and instead keep unriskable money in high-yield savings accounts, as previously discussed on this blog.

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1. Anne S.

Another wrinkle to throw into the mix is when you are trying to compare CD rates to regular interest to securities such as T-Bills. I still find myself getting confused and when the reported rates are so similar it is hard to figure out which is a better deal. Are there any calculators out there that will help me compare the return on a taxable CD account versus a T-bill?

2. The current investment rates of T-bills are posted at http://www.treasurydirect.gov/RI/OFBills

The “Investment Rate %” is essentially the APY you would get if you held the bill for a year. Note that in most cases, you won’t be holding the bill for a year. The “Discount Rate %” is the annual interest rate.

FatWallet.com has some advice on T-bills. According to http://beta.fatwallet.com/wiki/index.php?a=a_view&title=Treasury_Bill_FAQ&type=FatWallet

9. How are the purchase price and the investment rate (APR) related?
If you consider the purchase price as the “principal” of the investment and the difference between face value and purchase price the “interest,” if you calculate (interest/principal)*(365/days of bill) you get the APR. The Treasury does this calculation for us to 1/1000 basis point, however the price is actually more accurate to 1/1000 of a cent for a \$1000 bill.

So, you can just compare the APR on a T-bill with the APR on a CD. As T-bills are more tax-advantaged than CDs, all things being equal, go for the T-bill.

If you need to know the formula for calculating the rate of return (yield) on T-bills, check out

If you want a calculator, check out:
http://www.investopedia.com/calculator/TBillPrice.aspx
This calculator takes a rate of return and the amount of days until maturity, and then tells you what the price of the T-bill would have to be.

3. I think dividing an annual rate by 12 to get the monthly rate is mathematically incorrect. It should be (1+r)^(1/12)-1, isn’t it?
However, if banks do it this way, you must follow them, otherwise your numbers will not match the bank’s.
By the way, I come to you blog from the Mechanical Turk. 🙂

4. Forest,

The difference between my formula and your formula, is that my formula takes a given annual interest rate, and then tells you the APY. Your formula takes the APY, and then tells you the per-month annual interest rate that you would need to get that APY. Both are correct, but do slightly different things.

5. If i am not mistaken, your APY is similar to what our country, or some circles call the effective interest rate.

I don’t know in the US, but in our country, in order to guide the consumer to make effective decisions, they are supposed to publicize or answer what the effective interest rate would be.

So as in your example, they are bound to have to tell consumers that when they advertise that they offer an interest rate of 12% compounded monthly, the consumer has a right to know that the effectiveyield or the APY is 12.68%.