Question

Julie is a portfolio manager at Know Better plc. She wants to estimate the interest rate risk of assets of the company consisting of 1 million sharesof 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 by 12.77 pounds. The convexity of Bond C is 300 b) What is the duration and convexity of the bond portfolio?

Answer 1

The change in the price of a bond can be summarized as follow:

Change in price= Duration effect + Convexity effect

≈(-Duration×ΔYield)+(1/2×Convexity×(ΔYield)^2)

Given duration of Bond A( D_A )= 5.59

When interest rates decline by 30bp = -0.30%

Given Value of Bond A will increase to 83.5

i.e. Price of Bond A + Change in price due to Duration effect + Convexity effect  When interest rates decline by 30bp should take the bond price to 83.5

or, Price of bond (1 - 5.59 * (-0.30%) + 1/2 * Convexity * (-0.30%)^2) = 83.5

or, Price Of Bond (1 + 5.59 * 0.30% + 1/2 * Convexity * 0.30%^2) = 83.5

or, Price Of Bond (1.0167 + 0.0000045Convexity) = 83.5 -----(1)

When interest rates increase by +30 bps = +0.30%

i.e. Price of Bond A + Change in price due to Duration effect + Convexity effect  When interest rates rises by 30bp should take the bond price to 80.75

Price Of Bond (1 - 5.59 * 0.30% + 1/2 * Convexity * 0.30%^2) = 80.75

Or, Price Of Bond (0.98323 + 0.0000045Convexity) = 80.75 ---------------(2)

Now let us divide equation (1) by equation (2)

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

On solving,

Convexity = 361.41

From equation(1)

Price Of Bond (1.0167 + 0.0000045Convexity) = 83.5 -----(1)

Or, Price Of Bond (1.0167 + + 0.0000045 * 361.41) = 83.5

Or, Price Of Bond = 81.9972

Current price of Bond B is 100 pounds.

Convexity of bond C= 300

when interest rates decline by 1%

Price Of Bond (- Duration Of Bond C * (-1%) + 1/2 * 300 * (-1%)²) = 8.46

Or, Price Of Bond (0.01 * Duration Of Bond C + 0.015)= 8.46 -----(3)

When interest rates increase by 3%

Price Of Bond (- Duration Of Bond C * 3% + 1/2 * 300 * 3%^2) = 12.77

Or, Price Of Bond (-0.03 *Duration Of Bond C + 0.135) = 12.77 --------(4)

On dividing equation (4) by (3),

(0.135 - 0.03) / (0.015 + 0.01 * Duration Of Bond C) = 12.77 / 8.46 = 1.51

On Solving,

Duration Of Bond C = 2.49

From equation (3)

Price Of Bond (0.01 * Duration Of Bond C + 0.015) = 8.46 -----(3)

Putting Duration Of Bond C = 2.49 back in (3) : Price Of Bond C = 211.94

Current value of the bond portfolio = Price Of Bond A x Number of Bond A + Price Of Bond B x Number of Bond B + Price Of Bond C x Number of Bond C = 82 x 1 million + 100 x 2 million + 211.94 x 2 million = 705.88 million pounds

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