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When you're earning interest, your money increases in time. How long does it take for your money to double? A quick rough calculation can be made with the rule of 72. This rule says that the (interest rate) * (number of years for your money to double) = 72. So if you're earning 4%, your money will double (before taxes) every 72/4 = 18 years. If you're earning 10%, your money will double every 72/10 = 7 years and 10 weeks. This is not exact, but it's very close and good for making quick estimates of your retirement nest egg. Einstein is credited with discovering the rule of 72. He watched his retirement fund too.

Where did the rule of 72 come from? If your yearly interest rate is **x**%, then the number of
years **n** it takes your money to double is found from:

(1+(x/100))^{n} = 2

n ln( 1 + x/100 ) = n x/100 = ln 2

n x = 100 ln 2 = 69.3

It seems at a glance that it should be the rule of 69.3, not the rule of 72. However, our formula is off by a little bit. It's not precisely true that ln(1+a) = a, this is just an approximation. The error is about 2.5% when the interest rate is 5%, and it's about 5% when the interest rate is 10%. We compromise and fix the formula to be correct at an interest rate of 8%, which increases the 69.2 to about 72. 72 has the added benefit of being divisible by 12. That's it, that's what Einstein figured out.

There are government securities called T-Notes. These are much more complicated to analyze, but we're going to learn some important things working on them. T-Notes come in 2, 3, 5, and 10 year notes. They pay interest every six months. It's these interest payments that make analyzing them interesting, and have resulting in some financial inventions.

Suppose you are offered a note. This note pays you an interest payment of $150 in six months and then a payment of $10,000 in one year. The current interest rate is 3% per year. What is the note worth?

First, we need the interest rate for six months. If we knew the interest rate for six months, call it x, we could calculate the interest rate for one year, call it i:

1+i = (1+x)*(1+x)

√(1+i) = 1+x

Since i = 3%, x = 1.489%. Now we can work our problem. All we need is the current value of the payments. The current value of a single payment is just the payment divided by the appropriate interest rate. If the interest rate is 3%, then the value of this bond is $150 / (1.01489) + $10,000 / (1.03) = $147.80 + $9708.74 = $9856.54. We would expect these bonds to sell for $9856.54 at auction.

Let's consider a more complicated bond. This bond pays $150 each six months three times, then pays $10,000 after two years. It's just like our bond above except it's a two year bond, not a one year bond. If the interest rate is 3%, then the six month rate is 1.489%. The value of the bond is just the sum of the values of the payments:

value = $150 / (1.01489) + $150 / 1.03 + $150 / 1.01489 * 1.03 + $10,000 / 1.03 * 1.03

value = $147.80 + $145.63 + $143.49 + $9,425.96 = $9,862.88

If the interest rate is 3%, we expect these bonds to sell at auction for $9,862.88.

T-Notes have more risk than T-Bills. This is because when you receive the interest payments, you would most likely want to reinvest them. However, the interest rate will most likely be different at that time than at the time you bought the bond. This interest rate risk means T-Bonds are not as risk-free as a T-Bill.

A major complication with bonds is, 'what if the interest rate changes?' We all know that due to inflation and changes in the economy, interest rates also change. But when we buy these bonds we are being forced to forecast interest rates for the next two years. Plus, we're also having to forecast interest rates every six months for the next two years. People are not very good at forecasting, so this makes people nervous. Nervous people pay less. Bonds with coupons include interest rate risk, as you will get back money to reinvest at some unknown future interest rate.

The actual US government bond I described above would be a piece of paper that you can redeem for $10,000 on or after a particular date. Attached to this piece of paper would be three coupons; each of the coupons can be redeemed for $150 on or after particular dates. Because of the uncertainty in interest rates, and the complicated way we calculate the value of the bond and the coupons, a long time ago someone figured out that the coupons and the bond should be worth more if split apart. They are worth more because without the coupons, the bond is simply worth $10,000 / 1.03 * 1.03, less just a little bit of money perhaps due to the uncertainty in the interest rate over two years.

People started buying a whole bunch of government bonds, stripping off the coupons, and selling the coupons and the bonds separately. They made a few cents profit each time they did this due to lowering the uncertainty and risk on each piece of paper. Bonds which have been stripped of their coupons like this are called zero coupon bonds. The original bond sold for $9,862.88 less a little bit due to interest uncertainty; the zero coupon bond sells for $9,425.96. However, the dealer also got to sell the three coupons for about $147.80 + $145.63 + $143.49. As I said, they made a few cents on the entire deal. However, this few cents profit was almost guaranteed by the market place, so this process of stripping coupons and selling them off piece by piece remains very popular to this day.

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