Question

Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are 2%, 2.1%, 2.3%, 2.5%, and 2.7% per annum with continuous compounding, respectively. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months and pays a coupon of 4% per annum semiannually.

Please give me the process, thank you!

Answer 1

Given that, 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are 2%, 2.1%, 2.3%, 2.5%, and 2.7% per annum with continuous compounding, respectively. These rates will be used to discount each coupon received by bond to the present value.

Given about bond,

Face value = $100

Maturity = 30 months

Coupon rate = 4% paid semiannually,

So, semiannual coupon = (4%/2) of 100 = $2

Price of the bond is sum of PV of its coupon and face value discounted at respective maturity rate using continuous compounding.

PV using continuous compounding = FV*e^(-r*t)

where r is rate in decimal and t = time of maturity in years.

So, price of bond = C*e^(-r0.5*0.5) + C*e^(-r1*1) + C*e^(-r1.5*1.5) + C*e^(r2*2) + C*e^(r2.5*2.5) + FV*e^(r2.5*2.5)

=> Price = 2*e^(-0.02*0.5) + 2*e^(-0.021*1) + 2*e^(-0.023*1.5) + 2*e^(-0.025*2) + 2*e^(-0.027*2.5) + 100*e^(-0.027*2.5)

=> Price = $103.12

So, price of the bond = $103.12