DURATION MATCHING with a SWAP TO IMMUNIZE INTEREST RATE RISK Consider a bank with the following balance sheet: First the asset side: Assets Value Duration

of the Asset

Cash $200 0 3yr loan @7% $1000 2.20 7yr loan @8% $800 4 Total Value of Assets, as you can see, is $200+$1000+$800=$2000

• Duration of a financial security is the extent of its exposure to interest rate risk. For example, the duration of the 3yr loan @7% in this example is 2.20. This means that if interest rates go up by 1%, the value of this loan, which is $1000 now, will go down by 2.20 %. Of course, if interest rates go down by 1%, then the value of this loan will go up by 1%.

• Duration of the total assets is just the weighted average of the

component durations on the asset side.

• Duration of Assets = DA = (200/2000)*0 + (1000/2000)* 2.20 +

(800/2000)*4 = 2.7 • This means that the total value of the assets will go DOWN by 2.7% if

interest rates go UP by 1%. Remember again that duration is a measure of interest rate risk. It tells us the sensitivity of the value of an asset or portfolio to changes in interest rates. Also note that cash always has zero duration.

Now let’s look at the liability side of the bank’s balance sheet: Liabilities Value Duration of

the Liability5yr loan @ Libor +0.1%

$750 1.5

5yr loan @6% $150 3 7yr loan @8% $300 6

• Total Value of Liabilities, as you can see, is $750+$150+$300=$1200 • Duration of the total liabilities is again just the weighted average of

the component durations on the liability side. • Duration of Total Liabilities= DL = (750/1200)*1.5 + (150/1200)* 3 +

(300/1200)*6 = 2.8125 • This means that the total value of the liabilities will go DOWN by

2.8125% if interest rates go UP by 1%.

• Now, what is the COMPLETE situation here in terms of interest rate risk? Recall that DA = 2.7, therefore the total value of the assets will go DOWN by 2.7 % if interest rates go down by 1%. But DL = 2.8125, therefore the total value of the liabilities will DOWN by 2.8125% if interest rates go UP by 1%.

• In other words, the bank’s net worth (assets minus liabilities) may be

exposed to a loss in case interest rates go up, because if that happens, value of assets may decrease more than the value of liabilities. To check whether this is the case in our example, we quantify this interest rate risk exposure as follows:

Duration Gap = DA – {Total Liabilities/Total Assets}* DL

= 2.7– (1200/2000)* 2.8125 =1.0125 • Since duration gap is positive, this means the bank’s net worth (assets

minus liabilities) will suffer if interest rates go up. In our case, duration gap is 1.0125. This says that if interest rates go up by 1%, then the net worth of the bank’s balance sheet (which is now $2000-$1200=$800) will go down by 1.0125%.

• Now, we will discuss how we can immunize this interest rate risk. This practice of interest rate risk management is called duration matching. The purpose of duration matching, which is a hedging method, is to decrease the duration gap to zero, because if the duration gap is zero, then this means that the net worth of the balance sheet is immune to changes in interest rate, i.e. the net worth (value of assets minus the value of liabilities) do not change if interest rate changes. (It becomes immune). To see how one can achieve this, consider the duration gap equation again: Duration Gap = DA – {Total Liabilities/Total Assets}* DL At the moment, this is positive, it is equal to 1.0125 and we want to decrease it to zero. We can concentrate on the liability side and increase the duration of liability side just enough to make Duration gap zero.

• Now, look at the floating rate loan on the liability side, that’s a 5yr

loan @ Libor +0.1%. The face value of this loan is $750. Suppose you swap part of this loan in exchange of paying a fixed interest rate of 7.25%. Since this is a liability, you are at the moment paying Libor +0.1% on a value of $750 over the course of next 5 years. Say you swap $x out of this $750, i.e. the other party pays you Libor +0.1% over the notional principal of $x (i.e. she assumes your liability) and you pay her a fixed rate of 7.25% over the next 5 years. After this swap, your liability side will look as follows:

Liabilities Value Duration of

the Liability5yr loan @ Libor +0.1%

$750-x 1.5

5yr loan @7.25%

$x 4.3

7yr loan @8% $300 6 5yr loan @6% $150 3 All we have to decide now is the value of x such that Duration Gap = 0 (see the solution in class).

Practice Questions on Duration Matching: Question 1) Reducing Duration Gap To Zero Consider a bank with the following balance sheet: Assets Value Duration of

the Asset Cash $1500 0 3yr loan @6% $3000 3 5yr loan @8% $3000 4 Liabilities Value Duration of

the Liability5yr loan @ Libor $2000 2 4yr loan @6% $1000 3 5yr loan @6% $2000 4.9

a) Find the duration of the asset side. If interest rate goes up by 1%, what will happen to the total value of the assets? Find the duration of the liability side. If interest rate goes up by 1%, what will happen to the total value of the liabilities?

Answer: Duration of Assets = 2.8 Duration of Liabilities = 3.36

b) Find the duration gap. If interest rate goes up by 1%, what will happen to the net worth of the bank’s balance sheet?

Answer: Duration Gap= 0.56

c) Suppose the portfolio manager wants to reduce the duration gap to zero and

thus immunize all interest rate risk. For that purpose he wants to swap $x of the 5yr loan @ Libor with a 5yr loan at 8% fixed rate. The duration of the fixed rate 5yr loan at 8% is 4.5

What is the size $x of the swap such that duration gap becomes zero? Answer : x = $1680

Question 2 (Reducing Duration Gap To Zero) Consider a bank with the following balance sheet

Assets Value Duration of

the Asset Cash $3000 0 6yr loan @5% $1500 5.3 5yr loan @6% $1500 4.2 Liabilities Value Duration of

the Liability4yr loan @ Libor $1000 1 4yr loan @5% $1000 3 5yr loan @6% $2000 4

a) Find the duration gap. If interest rate goes up by 1%, what will happen to the net worth of the bank’s balance sheet?

b) Suppose the portfolio manager wants to reduce the duration gap to zero and

thus immunize all interest rate risk. For that purpose he wants to swap $x of the 4yr loan @ Libor with a 4yr loan at 7% fixed rate. The duration of the fixed rate 4yr loan at 7% is 3.5. What is the size of the swap $x so that duration gap becomes 0?