a. Forward rates

Spot rates from bonds.

Bond A (1 year): 105 = 10 + 100 / (1+r1), 95 = 100 / (1+r1), r1 = 5.26%

Bond B (2 year): 96 = 4 / (1+r2) + 104 / (1+r2)^2

Using approx, r2 = (100/96)^0.5 -1 ≈ 2.08% wait, calculate properly.

To find forward f1,2 = (1+r2)^2 / (1+r1) -1

First, r1 = (100+10)/105 -1 = 9.52%

No, price 105 for coupon 10, redemption 100, so PV = coupon + redemption / (1+r1) = 10 + 100 / (1+r1) = 105

110 / (1+r1) = 105, 1+r1 = 110/105 = 1.0476, r1 = 4.76%

Bond B: 4 + 100 / (1+r2)^2 + 4 / (1+r2) = 96

Let (1+r2) = y, 4/y + 104/y^2 = 96

Multiply y^2: 4y + 104 = 96 y^2

96 y^2 – 4y – 104 = 0

y^2 – 0.0417y – 1.083 = 0

y ≈ 1.05, r2 ≈ 5%

More precise calculation needed, but assume r1 = 4.76%, r2 = 5.5% approx.

For forward, the two-year forward at end year 1 is the rate from year 1 to 2, f1,2 = (1+r2)^2 / (1+r1) -1

From bond C for r3.

This requires solving for spot rates.

Assume spot r1 = 4.76%

For bond B: solve for r2.

Using code, but since no, approximate.

The question is to estimate the two-year forward rate at end year 1, which is f0,1,2 the forward for year 2 from year 1.

And one-year forward at end year 2, f2,3

First, find spot rates r1, r2, r3.

r1 = (10 + 100)/105 -1 = 4.76%

For r2, solve 4/(1+r2) + 100/(1+r2)^2 +4/(1+r2) wait, coupon 4, so 4 +4/(1+r2) +100/(1+r2)^2 = 96

8 +100/(1+r2)^2 = 96, 100/(1+r2)^2 = 88, (1+r2)^2 = 1.1364, r2 = 6.55%

For bond C: 6/(1+r3) +6/(1+r3)^2 +100/(1+r3)^3 = 98

This is harder, approximate r3 = 6.8%

Then f1,2 = (1.0655)^2 / 1.0476 -1 = 1.135 / 1.0476 -1 = 8.3%

f2,3 = (1.068)^3 / (1.0655)^2 -1 = 1.22 / 1.135 -1 = 7.5%

(5 Marks)

b. i. Price and YTM of 9% bond

Assume the spot rates from above, but the question is for a new bond, but to calculate price, but the bond is 3 year 9% , face 1000, but to find price, but it says calculate the price, but using what? Perhaps using the spot rates from a.

Using r3 = 6.8%, price = 90 /1.0476 + 90 /1.0655^2 + 1090 /1.068^3 ≈ 900.9546 +900.881 +1090*0.864 ≈ 85.9 +79.3 +942 = 1,107.2

YTM solve for r where 90/(1+r) +90/(1+r)^2 +1090/(1+r)^3 = 1107.2, r ≈ 6.8%

Why not realise: If sold before maturity, price changes with interest rates; reinvestment risk.

(6 Marks)

ii. One year later, no change in rates, price will increase (pull to par, since premium bond).

(2 Marks)

iii. Modified duration = Macaulay duration / (1+y), Macaulay ≈ 2.8 years, mod ≈ 2.6

Interpretation: % change in price ≈ – mod * Δy

Limitations: Assumes parallel shift in yield curve; linear approximation, not for large changes; ignores convexity.

(7 Marks)