Rate, rate who's got the rate?

If this all seems confusing and that you've entered into a gag in Mr. Wonka's factory, you need not worry trained professionals find this baffling as well. In the last post we used a discount rate to discover what the true benefit of a security was in present value terms. Now, we need to add a bit of complexity to obtain a further degree of accuracy. So today will be about zero-rates on those risk-free securities.

First we will just assume that we know what the zero-rate is and then we can later show how to figure out the implied zero rate. The zero rate is basically what we calculated yesterday. However, coupons and dividends do not all come at the end of the security's life. So we need various zero-rates to find out what the correct price of the security is.

Given:
Zero Rate - Maturity
5.0% - 0.5 (Years)
5.8% - 1.0
6.4% - 1.5
6.8% - 2.0

Now we want to price a 2 year Treasury bond with a principal of $1,000 and a 6% coupon paid semiannually.
0.0 - ?
0.5 - 30
1.0 - 30
1.5 - 30
2.0 - 1,030. That is 1,000 in principal and 30 for the coupon payment.
We utilize the formula from yesterday and use the spot/zero-rate for each maturity.
So we find that:
30e^(-.05*.5)+30e^(-.058*1.0)+30e^(-.064*1.5)+1030e^(-.068*2.0)= 983.85
29.2593+28.3095+27.2539+899.0279=983.85

This result in and of itself is rather interesting. See the security pays 6% annual via two coupons of $30. However, it sells at a discount. This is because the majority of the bond is paid in the final payment where the interest rate is greatest and is discounted by the most periods as well. We can figure out the bond's yield maturity by utilizing a financial calculator. Entering 4 for the period, Present value of -983.85, Future Value of 1000 and Payments of 30. Then solving for the Yield to Maturity it shows 3.439% and multiplying this by two gives the annual rate of 6.89%

What happens when there is not a corresponding Treasury? That is the million dollar question.

Say you have these 5 Treasury securities:We now employ the bootstrap method.
So to find the quarterly zero-rate we know that you will be repaid your principal in 1/4 a year. Thus you gain 25 on your 975 worth of investment. If we annualize that figure you would have:
(4 x 25)/975= 10.26% Which we should change into continuous compounding.
So 4 x ln(1 +(.1026/4)= 10.13% We do the same with the other non-coupon bearing instruments and come out with continuous compounding rates of 10.47% and 10.54%. Now is where the bootstrapping begins in earnest. So the 1 1/2 year bond will pay out coupons semiannually( all Treasury securities do) so it will pay 40, then 40 and finally 1,040. So we can use the zero-rates we have already applied to find the zero-rate for 1.5 years.

40e^(-.1047x0.5)+40e^(-.1054*1.0)+1040e(-R*1.5)=960
37.9599+35.9986+1040e^(-R*1.5)=960
e^(-R*1.5)=886.0415/1040
e^(-R*1.5)=.85196
-1.5R=ln.85196
R=10.68%

Then the last one:
60e^(-.1047*0.5)+60e^(-.1054*1.0)+60e^(-.1068*1.5)+1060e^(-R*2.0)=1016
56.9398+53.9979+51.1184+1060e^(-R*2.0)=1016
e^(-R*2.0)=853.9439/1060
-2R=ln(.8056)
R=10.81%

Now we have a zero yield curve

Homework:
Calculate the zeroes for 1/2 year, 1 year, 1 and 1/2 year and 2 years.