Question

Julie is the portfolio manager at know better plc. She wants to estimate the interest rate risk of assets of the company consisting of 1 million shares of Bond A, 2 million shares of Bond B, and 2 million shares of Bond C. The duration of Bond A is 5.59, a valuation model found that if interest rates decline by 30 basis points, the value of Bond A will increase to 83.5 pounds, and if interest rates increase by 30 basis points, the value of Bond to A will decline to 80.75 pounds. The same valuation model also found that if interest rates decreases by 50 basis points, the value of Bond B increases to 104.6 pounds, and if interest rates increases by 50 basis points, the value of Bond B decreases to 96.4 pounds, and the current value of Bond B is 100 pounds. Kirstin also knows from the valuation model that, by using the duration and convexity rule, if interest rates decline by 1%, the price of bond C increases approximately by 8.46 pounds, and if interest rates increase by 3%, the price of Bond C decreases approximately by12.77 pounds. The convexity of Bond C is 300. a ) What is current value of the bond portfolio?

Answer 1

Let P_i, D_i and C_i denote the price, duration and convexity respectively of bond i.

%change in the price of the bond = - D x Δi + 1/2 x C x (Δi)²

Case of Bond A: D_A = 5.59

When Δi = -30 bps = -0.30%

P_A[1 - 5.59 x (-0.30%) + 1/2 x C_A x (-0.30%)²] = 83.5

Or, P_A[1 + 5.59 x 0.30% + 1/2 x C_A x 0.30%²] = 83.5

Or, P_A(1.01677 + 0.0000045C) = 83.5 --------------(1)

When Δi = +30 bps = +0.30%

P_A[1 - 5.59 x 0.30% + 1/2 x C_A x 0.30%²] = 80.75

Or, P_A(0.98323 + 0.0000045C) = 80.75 ---------------(2)

Divide (1) by (2) to get:

(1.01677 + 0.0000045C) / (0.98323 + 0.0000045C) = 83.5 / 80.75 = 1.0341

Solve for C = 361.41

Put it back in (1) to get, P_A = 81.99 = say 82 pounds

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current value of Bond B is 100 pounds. Hence, P_B = 100 pounds

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Case of Bond C: C_C = 300

When Δi = -1%

P_C[- D_C x (-1%) + 1/2 x 300 x (-1%)²] = 8.46

Or, P_C[0.01D_C + 0.015] = 8.46 --------------(3)

When Δi = +3%

P_C[- D_C x 3% + 1/2 x 300 x 3%²] = 12.77

Or, P_C[-0.03D_C + 0.135] = 12.77 --------------(4)

Divide (4) by (3) to get,

(0.135 - 0.03D_C) / (0.015 + 0.01D_C) = 12.77 / 8.46 = 1.51

Solve for D_C = 2.49

Put it back in (3) to get, P_C = 211.94

Hence, the current value of the bond portfolio = P_A x N_A + P_B x N_B + P_C x N_C = 82 x 1 + 100 x 2 + 211.94 x 2 = 705.88 million pounds = 705,880,541 pounds