Question

This is another exam type question, covering numerouse subtopics. Consider the following balance sheet (in million) for a bank: Liabilities Assets Duration- 10 years $950 Duration-2 years $860 $90 Equity- The bank's leverage adjusted duration gap is about R) 0.01, equity would change by (calculate to two decimals). With this duration gap the bank should worry rising/falling) interest rates. In fact, if the relative change in interest rates is an increase of 1 per cent, that is AR/(1 million dollars (use +/ -to indicate increases/ falls). The bank could (buy/sell) bond futures to create a macrohedge. Suppose that T-bond futures (contract size 1 million are currently priced at 960.000. The deliverable T-bond has a duration of nine years. Calculate how many contracts the bank should trade to hedge its risk. The rounded number of contracts is

Answer 1

Interest risk is risk of change in yield-interest rate. Assets consist future cash inflows and Liabilities consist future cash flows.

Duration is measure of price sensitivity with change in yield-interest rate. In other words, duration measures how price of assets or liability if interest rate change by 1%.

Interest rate and value of assets and liabilities inversely related.

So, Duration of Assets is more than duration of liabilities(Positive Duration Gap) then, if interest rate rise- value of assets reduces more than liabilities and thus value of equity would reduce and vice versa.

Duration Gap is difference between Duration of Assets and Duration of Liabilities. When Duration Gap is equal to Zero then interest risk immunized means no interest risk. Leverage Adjusted Duration Gap (LADG) can be calculated with following formula

$LADG = Duration of Assets - Duration of Liabilities * \frac{Liability}{Assets}$

Following information provided about the Bank -

Duration of Assets	10 years
Assets	$ 950 m
Liability	$ 860 m
Equity	$ 90 m
Duration of Liability	2 years

1. Bank's LADG = 10 - 2*950/860

= 8.90 years

2. Bank's LADG is positive i.e 8.90 which means when interest rate rise the value of assets reduce more than value liability and consequently Value of Equity fall, thus Bank should worry more about rise in interest rate.

3. Duration of Assets is 10 years which means 1 % rise in interest rate would reduce 10% value of Assets and Duration of liability is 2 years which means 1 % rise in interest rate would reduce 2% value of Liabilities.

Equity = Assets - Liability

when 1% rise in interest rate -

Assets = 950*90% = $ 855 m

Liability = 860*98% = $ 842.8 m

Thus, Equity = $ 855 m - $ 842.8 m

= $ 12.2 m

Before interest rate rise, Equity value was $ 90 m

Therefore, Change in Equity = $ 90 - $ 12.2

= $ 77.8 millions

If interest rate rise by 1 % then Equity reduce by $ 77.8 millions i.e 86.44%

4.

The Bank could Sell Bond future to create macro hedge.In above calculation we have observed that the leverage adjusted duration gap (LADG) of Bank is positive which means if interest rate rise then value of Bank or Equity would reduce and vice verse. Therefore, if bank sell Bond future and interest rate rise then Bond future price fall and Profit from Bond future offset the loss of bank equity value.

5.

T-Bond future having contract size $ 1 million currently price at $ 960,000 and duration of bond is 9 years.

We have earlier calculated, a 1% rise in interest rate could reduce Equity value by $ 77.8 millions and vice versa.Thus, to completely hedge the interest rate risk we need to sell Bond future so that 1 % change interest rate could change T-Bond future value of total contract by $ 77.8 millions.

Thus, Total contract value of T-bond future -

77.8 = Contract value * Bond duration(%)

77.8 = Contract value*0.09

Contract value = 77.8/0.09

Contract value = $ 864.44 millions.

No. T-bond future contract required to sell = 864.44/0.96*

= 900.46 contracts. (approx)

Thus, Rounded number of Contract is 900 .

* $ 960,000 = $ 0.96 millions (T-bond future contract price today)