TIME VALUE OF MONEY #6: TREASURY BOND. Professor Peter Harris Mathematics by Dr. Sharon Petrushka. Introduction


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1 TIME VALUE OF MONEY #6: TREASURY BOND Professor Peter Harris Mathematics by Dr. Sharon Petrushka Introduction This problem assumes that you have mastered problems 15, which are prerequisites. In this problem, we calculate the value of a traditional bond, a thirty year U.S. Treasury bond. A U.S. Treasury Bond gives the investor two forms of returns; interest and principal. Interest is paid semiannually at a fixed rate over the life of the bond and the principal is paid at the end (maturity date) of the life of the bond. The interest rate is known as the coupon rate and the principal amount returned to the investor is denominated in $1,000 per bond. The coupon payment remains fixed over the life of the bond, and is coupon interest rate times the face value of the bond. However, the current interest rate in an economy is always changing due to demand and supply factors. As such, if the market rate exceeds the coupon rate, the price of the bond falls (discount). Conversely, the price increases (premium) if the market interest rate is below the coupon rate. When pricing a bond, it is necessary to use the market rate as the interest rate in the calculations. The cash flow is determined by the coupon payments and the repayment of principal.
2 Mathematics: The bond s value is basically composed of the Present Value of cash flows the bond pays. Mathematically it equals the Present Value of interest payments and Present Value of the principal payment. Price of Bond = PV cash payments = PVA + PVs PVA = Present Value of annuity (from the coupon payments) PVs = Present Value of single payment (from the principal) Coupon Payment i Principal (1+i) Price of Bond = [6]! = Interest rate per period N = Number of periods N * (1(1+i ) + N The interest payments are ordinary annuities as they are paid at the end of each semiannual period. Example: A $1,000 Treasury Security paying at an annual interest rate of 6% and has a five (5) year maturity, will be selling at what price today assuming a 8% market rate. Assume payments are semi annual. 6% The security is paying a fixed interest payment of 2 * $1000, or $30 per period for 10 periods (5 * 2), at a market rate of 8%/2 = 4% per period (1+i) 10 Price of Bond = * (1  (1 +.04) ) + Price of Bond = $
3 Using the TI83: You have a $1,000 Treasury Bond which matures in five years. The coupon rate is 10%, and the market rate of interest is 8% per year. Calculate the price of the bond. We enter the following: N = 5 * 2 There are 5 * 2 = 10 coupon payments. I% = 8 PV = 0 This is the value we are solving for. 10% PMT = 50 ( 2 ) * 1,000. Divide by 2 since the bond has a semiannual coupon payment. FV = 1000 (principal value of the bond at maturity) P/Y = 2 Payments are made semiannually. C/Y = 2 The market rate is a semiannually compounded rate Solution: PV = The price of the bond is $1,081.11
4 Business Problem: An investor expects the long term interest rates to decrease by 1% within a short period of time The current 30 year interest rate on a Treasury bond is 8%. Advise the investor as to whether she should purchase a traditional 30 year US Treasury Bond with a coupon rate of 8%, or a 30 year Treasury Zero coupon bond. Analysis. In order to answer this question, we need to compute the price of each bond given the above scenario. Solution: The price of the zero coupon bond was calculated in Time Value of Money Problem # 5: Zero Coupon Bond. You can refer to Table II in that problem to obtain the price of the Zero Coupon Bond, and the effect of changes in interest rate on its price. In pricing the traditional 30 years US Treasury Bond above, we enter the following: N = 30 * 2 There are 60 coupon payments. I% = 8 PV = 0 This is the value for which we are solving. 8% PMT = 40 ( 2 ) * 1,000. Divide by 2 since semiannual coupon payment are based on 8% annual coupon rate. FV = 1000 (principal value of the bond at maturity) P/Y = 2 Payments are made semiannually. C/Y = 2 The market rate is a semiannually compounded rate Solution: PV = The price of the bond is $1,000. This makes sense because the market rate and the bond s coupon rate are the same. The following table illustrates the effects of changing interest rates on the price of the bond.
5 TABLE I 30 Year Treasury Bond with 8% Coupon Rate Market Rate, I% Price ($) of Bond Change in Price ($) % Change in Price 11% % 10% % 9% % 8% This bond % 7% % 6% % 5% % Comparing this to Table II of Time Value of Money Problem #5, we see that the percent change in price for a 30 year Zero Coupon Bond is much higher than for a traditional Treasury bond.. Thus the traditional bond has less risk and volatility than the Zero Coupon Bond. If the investor believes interest rates will soon drop, she should invest in the Zero Coupon bond, as this will give her a better return. However, she should be aware that she is taking a greater risk of loss if rates go the other way.
6 Additional Problems: 1. Calculate the price of a bond which has a 10% coupon rate, a 10 year maturity and pays interest semiannually. The market interest rate is 8%. 2. Calculate the price of a bond which has a 10% coupon rate, a 10 year maturity and pays interest semiannually. The market rate is 12%. 3. Go to the Wall Street Journal and find the interest Research Problem: rate on a thirty (30) year Treasury Note. Let s assume you purchase a 30 year Treasury Bond for $1,000. If you expect interest rates to decrease by 1% during the next year, what will be the value of your bond be? Will you make money or lose money? How about if interest rates go up 1%. What will the bond be worth?
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