Question

6. (20 points) Suppose months, maturity in 12 months, and maturity in 18 months. Suppose the 6 month bond is a zero-coupon bond and has a theoretical price of $101. Suppose the 1 year bond pays a coupon every 6 months at an annual rate of $6, and has a theoretical price of $97. Suppose the 18 month bond pays a coupon every 6 months, at an annual rate of $8 and has a theoretical price of $96. The face value of each type of bond is $100 a bond is available in three types: maturity in 6 (a) (8 points) Calculate the zero rates for the 6-month, 12-month, and 18-month bonds. (b) (7 points) What is the forward rate from 12-months to 18-months? (c) (7 points) What is the theoretical price today of a zero coupon bond with face value $100 expiring in 1 year?

Answer 1

Answer a

Using excel the YTM rate is calculated

fe RATE(D6,D7,D5,D4) D8 A C D E Yield to maturity for zero coupon bond 1 Maturity on 6 months Amount(S) 2 Amount(S 2 4 Face value 100 FV 100 101 PV 5 Price -101 6 Maturity 0.5 0.5 7 Coupon 0 -2% 8 Yield to maturity for 6% coupon bond 10 Maturity on 1 year Amount($) 11 Amount($) 12 13 Face value 100 FV 100 97 PV -97 14 Price 15 Maturity 16 Coupon 1 2 6% 3 5% 17 18 Yield to maturity for 8% coupon bond 19 20 Maturity on 1.5 year Amount($) Amount($) 21 22 Face value 100 FV 100 96 PV 23 Price -96 24 Maturity 25 Coupon 1.5 3 8% 4 5% 26

Answer b

Forward rate = (1+yield rate)^n1(1+yield)^n2 -1

(1+yield rate) ^n1 (1+yield) ^n2

    Assume where n1 is the term of bond 1 and having a certain yield rate

n2 is the term of bond2 having certain yield rate

In case of 1.5 year bond the yield rate is 5%

and for the 1 year bond the yield rate is also 5%

Forward rate = (1+yield rate)^n1(1+yield)^n2 -1

(1+yield rate) ^n1 (1+yield) ^n2

                      = (1+.05)^1.5(1+.05)^1 -1

(1+.05)^1.5 (1+.05)^1

                         1.0761.05 -1

                      = 1.025‐1

                      = .025

                      =2.5%

Answer c

                                           

Theoretical price of a zero coupon bond after 1 year is shown below applying the formula of excel.

f =PV(C4,C5,(C3*C6),C3) C7 A B E 1 Theoretical price of zero coupon bond After 1 year 3 Face value 100 -2% 4 Discount rate 5 Period 1 6 Coupon rate 0% 7 -$102.04 8

The theoretical price is $102.04