Question

Problem 2

Maturity	Spot Rate
1	5.0700
2	5.1248
3	5.1873
4	5.2557
5	5.3281
6	5.4026
7	5.4773

a) Compute the Forward Rate Curve for the adjacent Term Structure

b) Price 5 year, 8% annual coupon bond. Assume these are risk adjusted rates.

c) Show that you will arrive at the same price by using the forward rates

Problem 3

What is the YTM of the above bond? What should its Par yield be?

Answer 1

a) Forward Rate f (t-1, 1) = [(1 + s(t))^t / (1 + s(t-1))^t-1 ] – 1

where,

s(t):t-period Spot Rate

s(t)-1:t-1-period Spot Rate

f(t-1, 1): forward Rate applicable for the period (t-1,1)

Maturity (t)	Spot Rate (S)	Forward Rate(t-1,1) F(t-1,1)
1	5.0700%	5.0700%
2	5.1248%	5.1796%
3	5.1873%	5.3124%
4	5.2557%	5.4612%
5	5.3281%	5.6182%
6	5.4026%	5.7759%
7	5.4773%	5.9266%

Excel formula for Forward Rate at t=2 is,

F(1,1) = ((1+S2)^t2)/((1+S1)^t1)-1

F(1,1) = ((1+5.1248%)^2)/((1+5.0700%)^1)-1 = 5.1796%

Similarly, we can compute F(t-1,1) for all maturities as in above excel and draw the Forward rate curve as below:

b) Price of 5 year, 8% annual coupon bond

Using Spot rate, Price of bond = C/(1+S1) + C/(1+S2)² + C/(1+S3)³ + C/(1+S4)⁴ + C/(1+S5)⁵ + FV/(1+S5)⁵

Where, C is coupon payment

FV is par value

S1, S2, S3, S4, S5 are spot rates for respective maturity

Considering FV i,e, par value = $1000

Maturity (t)	Spot Rate (S)	Cash flows (CF)	Discounting factor using Spot rates D = 1/(1+S)t	PV CF*D
1	5.0700%	80	0.951746455	76.13972
2	5.1248%	80	0.904877178	72.39017
3	5.1873%	80	0.859231277	68.7385
4	5.2557%	80	0.814737122	65.17897
5	5.3281%	1080	0.771398428	833.1103

Price of Bond = sum of PV = $1115.56

C) Using Forward rate, Price of bond

P = C/(1+F(0,1))

+C/{(1+F(0,1))* (1+F(1,1))}

+ C/{(1+F(0,1))* (1+F(1,1))* (1+F(2,1)) }

+ C/{(1+F(0,1))* (1+F(1,1))* (1+F(2,1))* (1+F(3,1))}

+ C/{(1+F(0,1))* (1+F(1,1))* (1+F(2,1))* (1+F(3,1))* (1+F(4,1))}

+ FV/{(1+F(0,1))* (1+F(1,1))* (1+F(2,1))* (1+F(3,1))* (1+F(4,1))}

Where, P is price of bond

C is coupon payment

FV is par value

F(0,1), F(1,1) F(2,1), F(3,1), F(4,1) are forward rates for respective maturity

Considering FV i,e, par value = $1000

Maturity (t)	Forward Rate(t-1,1)	Cash flows CF	Discounting factor using Spot rates D = 1/{(1+F(0,1))* (1+F(1,1))…. (1+F(t-1,1))}	PV (CF*D)
1	5.0700%	80	0.951746455	76.13972
2	5.1796%	80	0.904877178	72.39017
3	5.3124%	80	0.859231277	68.7385
4	5.4612%	80	0.814737122	65.17897
5	5.6182%	1080	0.771398428	833.1103

Price of Bond = sum of PV = $1115.56

Hence, we can see that the price of bond is same by using either spot rates or forward rates.

3) YTM of Bond

Price of bond = C/(1+YTM) + C/(1+YTM)² + C/(1+YTM)³ + C/(1+YTM)⁴ + C/(1+YTM)⁵ + FV/(1+YTM)⁵

Where, C is coupon payment

FV is par value

YTM is Yield to maturity

For above bond,

1115.56 = 80/(1+YTM) + 80/(1+YTM)² + 80/(1+YTM)³ + 80/(1+YTM)⁴ + 90/(1+YTM)⁵ + 1000/(1+YTM)⁵

Solve for YTM,

YTM = 5.3081%