Question

a) A portfolio manager wants to estimate the interest rate risk of a bond using duration. The current price of the bond is 98. A valuation model found that if interest rates decline by 35 basis points, the price will increase to 101 and if interest rates increase by 35 basis points, the price will decline to 96. What is the duration of this bond?

b) A portfolio manager purchased a bond portfolio with a market value of $75 million. The portfolio’s duration is 12. Estimate the change in the market value of the bond portfolio for a parallel shift in interest rates of +250 basis points. Comment on this duration based estimate of the market value change.

Answer 1

Part (a)

I assume that the duration being referred to in the question is modified duration.

current price of the bond is 98

%change in price = - (Modified) Duration x % change in yield + 0.5 x Convexity x (% change in yield)²

Case 1:  if interest rates decline by 35 basis points (bps), the price will increase to 101

100 basis points = 1%

% change in yield = - 35 bps = - 0.35%

% change in price = 101 / 98 - 1 = + 3.06%

Hence, 3.06% = - Duration x (- 0.35%) + 0.5 x Convexity x (-0.35%)²

Or 3.06% = Duration x 0.35% + 0.5 x Convexity x 0.35%² -------Equation (1)

Case 2:  if interest rates increase by 35 basis points, the price will decline to 96

100 basis points = 1%

% change in yield = 35 bps = 0.35%

% change in price = 96 / 98 - 1 = - 2.04%

Hence, - 2.04% = - Duration x 0.35% + 0.5 x Convexity x 0.35%² -------Equation (2)

Hence, equation (1) - equation (2) gives:

3.06% + 2.04% = 5.10% = 2 x Duration x 0.35%

Hence, Duration of this bond = 5.10% / (2 x 0.35%) =  7.2886

Part (b)

Parallel shift by + 250 bps

(Assumption: Ignore the impact of convexity)

Hence, %change in price = - (Modified) Duration x % change in yield = - 12 x 250 basis points = -12 x 2.5% = - 30%

Hence, the change in the market value of the bond portfolio = -30% x Current value = - 30% x $ 75 mn = - $ 22.50 mn

Hence, the market value will decline by 30% or an amount of $ 22.50 mn

250 bps change in yield is quite large and hence duration based price change approximation may not give the correct estimate. The effect of convexity can't be ignored in such a case. The missing link is provided by the concept of convexity. The %age change in price will have impact due to duration and convexity.