Question

Consider the following balance sheet (in millions) for an FI:

Assets Liabilities

Duration = 10 years $950 Duration = 2 years $860

Equity     90

a)What is the FI's duration gap?

b)What is the FI's interest rate risk exposure?

c)How can the FI use futures and forward contracts to put on a macrohedge?

d) What is the impact on the FI's equity value if the relative change in interest rates is an increase of 1 percent? That is, DR/(1+R) = 0.01.

e)Suppose that the FI in part (c) macrohedges using Treasury bond futures that are currently priced at 96. What is the impact on the FI's futures position if the relative change in all interest rates is an increase of 1 percent? That is, DR/(1+R) = 0.01. Assume that the deliverable Treasury bond has a duration of nine years.

f)If the FI wants a perfect macrohedge, how many government bond futures contracts does it need?

g)How does consideration of basis risk change your answers?

Answer 1

a) FI's Duration gap= duration of assets - duration of liabilities*(liabilities / assets)  

= 10 - 2 * (860/950) = 8.19 years

b) The FI directly exposed to interest rate increases. The market value of equity will decrease if interest rate increases.

c) The FI can hedge its interest rate risk by selling future or forward contracts.

d) It is given that, (DR) /(1+R) = 0.01

change in equity(E) = - duration gap * assets * interest rate

= -8.19 * (950,000) * (0.01) = - $77,805

e) Change in Equity (DE)= 9 * (96,000) * (.01) = -$8,640 per future contracts. since the macrohedge is a short hedge,

    this will be a profit of $8,640 per contract.

f) To perfectly hedge, the treasury bonds futures position should yield a profit equal to the loss in equity value (for any given increase in interest rates). Thus the number of future contracts must be sufficient to offset the $77,805 loss in equity value. This will necessitate the sale of $77,805/8,640=9.005 contracts. Rounding down, to construct a perfect macrohedge

requires the FI to sell 9 Treasury bonds futures contracts.

e) we assumed that basis rate did not exist. That allowed us to assert that the percentage change in interest rates (DR/(1+R)) would be the same for both the futures and underlying cash positions. if there is basis risk, then (DR/(1+R)) is not necessarily equal to (DR_f/(1+R_f)). If the FI wants to fully hedge its interest rate risk exposure in an environment with basis risk, the required number of futures contracts must reflect the disparity in volatilities between the futures and cash markets.