Question

Q3. Spot rate, forward rate, and yield to maturity One year zero priced at 5% yield. Two year 6% coupon bond priced at par. Three year 7% coupon par priced at par.

a. what is one year, two year AND three year spot rates (ie s1 s2 s3)?

b. what is the 1 year and 2 year forward rate (ie f12 f23)?

c. How much should a THREE year 10% coupon bond with face value of $1,000 be price at?

d. What is the yield to maturity for bond in part 3c (4 points)?

4. Mortgage Pricing A 30Y fixed rate mortgage is issued at 6% coupon rate. The loan fully amortizes over 30 year period. Expected payoff time is 8 Years when initially issued. Assuming $1M in loan balance.

a. Price the loan today at 5%, 6%, and 7% market yield, assuming loan termination term stays constant with interest rate (96 months at 5%; 96 months at 6%, and 96 months @ 7% ).

b. calculate numerical duration and convexity at 6% market interest rate based on pricing from

4a c. Price the loan today at 5%, 6%, and 7% yield, assuming loan termination term changes with interest rate (60 months at 5%; 120 months at 6%, and extends to 120 months @ 7% ).

b. calculate numerical duration and convexity at 6% market interest rate based on pricing from 4a

Answer 1

3a). s1 = yield on one year zero = 5%

s2 calculation: Let the par value of the 2 year coupon bond be 1,000. Then annual coupon = 6%*1,000 = 60

Price of 2 year coupon bond = year 1 coupon/(1+s1) + (year 2 coupon + par value)/(1+s2)^2

1,000 = 60/(1+5%) + (60+1,000)/(1+s2)^2

Solving for s2, we get s2 = 6.03%

s3 calculation: Let the par value of the 3 year coupon bond be 1,000. Then annual coupon = 7%*1,000 = 70

Price of 3 year coupon bond = year 1 coupon/(1+s1) + year 2 coupon/(1+s2)^2 + (year 3 coupon + par value)/(1+s3)^3

1,000 = 70/(1+5%) + 70/(1+6.03%)^2 + (70+1,000)/(1+s3)^3

871.07 = 1,070/(1+s3)^3

Solving for s3, we s3 = 7.10%

3b). f12 calculation:

(1+s2)^2 = (1+s1)*(1+f12)

f12 = [(1+6.03%)^2/(1+5%)] -1 = 7.07%

f23 calculation:

(1+s3)^3 = (1+s2)^2*(1+f23)

f23 = [(1+7.10%)^3/(1+6.03%)^2] -1 = 9.26%

3c). Price of the bond can be calculated using the spot rates. Current price of a 3 year 10% coupon bond (annual coupon = 10%*1,000 = 100) will be

price = 100/(1+5%) + 100/(1+6.03%)^2 + (1,000+100)/(1+7.10%)^3

= 1,079.68

3d). FV = 1,000; PV = -1,079.68; PMT = 100; N = 3, solve for RATE.

Annual yield = 6.97%