Financial-Institutions Management
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FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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Chapter 14: FX VAR 24.
An FI has $100,000 of net positions outstanding in British pounds (£) and -$30,000 in Swiss francs (SF). The standard deviation of the net positions as a result of exchange rate changes is 1 percent for the SF and 1.3 percent for the £. The correlation coefficient between the changes in exchange rates of the £ and the SF is 0.80.
Since the FI has a positive £ position, an appreciation of the £ will increase the value of its £- denominated assets more than its liabilities, providing a net gain. The opposite will occur if the £ depreciates.
Since the FI has a negative net position in SFs, the value of its Swiss-denominated assets will increase in value, but not as much as the value of its liabilities. Hence, an appreciation of the SF will lead to a net loss. The opposite will occur if the currency depreciates.
Use the DEAR formula for a portfolio: ) . )( )(- )( . )( . ( + ) . ( ) + (- ) . ( ) ( = DEAR p 8 0 30 100
013 0 01 0 2 01 0 30 013 0 100
2 2 2 2 = $1,075.20
have been estimated at $100,000 - $30,000 = $70,000.
FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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Chapter 22: Micro-Hedge 8.
In each of the following cases, indicate whether it would be appropriate for an FI to buy or sell a forward contract to hedge the appropriate risk.
The bank should sell a forward contract to protect against an increase in interest rates.
b. An insurance company plans to buy bonds in two months.
The insurance company should buy a forward contract to protect against a decrease in interest rates.
c. A savings bank is going to sell Treasury securities it holds in its investment portfolio next month.
The savings bank should sell a forward contract to protect against an increase in interest rates.
d. A U.S. bank lends to a French company; the loan is payable in euros.
The bank should sell francs forward to protect against a decrease in the value of the euro, or an increase in the value of the dollar.
e. A finance company has assets with a duration of six years and liabilities with a duration of 13 years.
The finance company should buy a forward contract to protect against decreasing interest rates that would cause the value of liabilities to increase more than the value of assets, thus causing a decrease in equity value. FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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9. The duration of a 20-year, 8 percent coupon Treasury bond selling at par is 10.292 years. The bond’s interest is paid semiannually, and the bond qualifies for delivery against the Treasury bond futures contract.
a. What is the modified duration of this bond?
The modified duration is 10.292/1.04 = 9.896 years.
b. What is the impact on the Treasury bond price if market interest rates increase 50 basis points?
∆ P = -MD( ∆ R)$100,000 = -9.896 x 0.005 x $100,000 = -$4,948.08.
c. If you sold a Treasury bond futures contract at 95 and interest rates rose 50 basis points, what would be the change in the value of your futures position?
67 . 700 , 4 $ 000 , 95 $ 6 8 ) (
= (0.005) 9 9. - = P R MD - = P ∆ ∆
d. If you purchased the bond at par and sold the futures contract, what would be the net value of your hedge after the increase in interest rates?
-$4,948.08
Gain in value from the sale of futures contract $4,700.67
Net gain or loss from hedge -$247.41
10. What are the differences between a microhedge and a macrohedge for a FI? Why is it generally more efficient for FIs to employ a macrohedge than a series of microhedges? FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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A microhedge uses a derivative contract such as a forward or futures contract to hedge the risk exposure of a specific transaction, while a macrohedge is a hedge of the duration gap of the entire balance sheet. FIs that attempt to manage their risk exposure by hedging each balance sheet position will find that hedging is excessively costly, because the use of a series of microhedges ignores the FI’s internal hedges that are already on the balance sheet. That is, if a long-term fixed-rate asset position is exposed to interest rate increases, there may be a matching long-term fixed-rate liability position that also is exposed to interest rate decreases. Putting on two microhedges to reduce the risk exposures of each of these positions fails to recognize that the FI has already hedged much of its risk by taking matched balance sheet positions. The efficiency of the macrohedge is that it focuses only on those mismatched positions that are candidates for off-balance-sheet hedging activities.
FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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Chapter 22: Macro-Hedge 12.
Hedge Row Bank has the following balance sheet (in millions):
Assets $150
Liabilities $135
Equity 15 Total
$150 Total
$150
The duration of the assets is six years and the duration of the liabilities is four years. The bank is expecting interest rates to fall from 10 percent to 9 percent over the next year.
DGAP = D A – k D
L = 6 – (0.9)(4) = 6 – 3.6 = 2.4 years
Expected ∆ E = -DGAP[ ∆ R/(1 + R)]A = -2.4(-0.01/1.10)$150m = $3.272 million
c. What will be the effect on net worth if interest rates increase 100 basis points?
Expected ∆ E = -DGAP[ ∆ R/(1 + R)]A = -2.4(0.01/1.10)$150 = -$3.272.
d. If the existing interest rate on the liabilities is 6 percent, what will be the effect on net worth of a 1 percent increase in interest rates?
FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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Solving for the impact on the change in equity under this assumption involves finding the impact of the change in interest rates on each side of the balance sheet, and then determining the difference in these values. The analysis is based on the equation:
∆ E =
∆ A -
∆ L
∆ A = -D A [ ∆ R A /(1 + R A )]A = -6[0.01/1.10]$150m = -$8.1818 million
and ∆ L = -D
L [ ∆ R L /(1 + R L )]L = -4[0.01/1.06]$135m = -$5.0943 million
Therefore, ∆ E =
∆ A -
∆ L = -$8.1818m – (-$5.0943m) = - $3.0875 million
16. Tree Row Bank has assets of $150 million, liabilities of $135 million, and equity of $15 million. The asset duration is six years and the duration of the liabilities is four years. Market interest rates are 10 percent. Tree Row Bank wishes to hedge the balance sheet with Treasury bond futures contracts, which currently have a price quote of $95 per $100 face value for the benchmark 20-year, 8 percent coupon bond underlying the contract.
Calculation of Duration for Problem 16
$1,000 bond, 8% coupon, R = 8.5295% and R = 8.2052%, n = 20 years Cash
Price = $95
Time Flow PV of CF PV of CF x t 1 80 73.71268 73.71268 2 80
67.91950 135.83900 3 80
62.58161 187.74482 4 80
57.66323 230.65291 5 80
53.13139 265.65695 6 80
48.95572 293.73430 7 80
45.10822 315.75751 8 80
41.56301 332.50477 9 80
38.29659 344.66933 10 80
35.28681 352.86807 FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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11
80 32.51357 357.64923 12
80 29.95828 359.49933
13 80 27.60381 358.84957 14
80 25.43439 356.08145 15
80 23.43546 351.53196 16
80 21.59364 345.49819 17
80 19.89656 338.24155 18
80 18.33286 329.99152 19
80 16.89206 320.94906 20
1080 210.12054 4202.41084
Total950.00000 9853.84304
Duration = 10.3725
a. Should the bank go short or long on the futures contracts to establish the correct macrohedge?
value of the equity and the futures contracts to decrease. But the bank could buy back the futures contracts to realize a gain to offset the decreased value of the equity.
If the market value of the underlying 20-year, 8 percent benchmark bond is $95 per $100, the market rate is 8.5295 percent (using a calculator) and the duration is 10.3725 as shown on the last page of this chapter solutions. The number of contracts to hedge the bank is:
FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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contracts x m P x D A kD D N F F L A F 365
000 , 95 $ 3725
. 10 150 )$ 4 ) 9 . 0 ( 6 ( ) ( − = − = − − =
c. Verify that the change in the futures position will offset the change in the cash balance sheet position for a change in market interest rates of plus 100 basis points and minus 50 basis points.
For an increase in rates of 100 basis points, the change in the cash balance sheet position is: Expected ∆ E = -DGAP[ ∆ R/(1 + R)]A = -2.4(0.01/1.10)$150m = -$3,272,727.27. The change in bond value = -10.3725(0.01/1.085295)$95,000 = -$9,079.41, and the change in 365 contracts is - $9,079.41 x -365 = $3,313,986.25. Since the futures contracts were sold, they could be repurchased for a gain of $3,313,986.25. The sum of the two values is a net gain of $41,258.98.
For a decrease in rates of 50 basis points, the change in the cash balance sheet position is:
Expected ∆ E = -DGAP[ ∆ R/(1 + R)]A = -2.4(-0.005/1.10)$150m = $1,636,363.64. The change in each bond value = -10.37255(-0.005/1.085295)$95,000 = $4,539.71 and the change in 365 contracts is $4,539.71 x -365 = -$1,656,993.13. Since the futures contracts were sold, they could be repurchased for a loss of $1,656,993.13. The sum of the two values is a loss of $20,629.49.
$100 of face value, how many futures contracts would have been necessary to hedge fully the balance sheet?
the face value of the contract is $1,000,000, and the number of contracts necessary to hedge the bank is:
469
, 1 000 , 245
$ 000
, 000
, 360
$ 000
, 980
$ 25 . 0 150
)$ 4 ) 9 . 0 ( 6 ( ) ( − = − = − − = − − =
e. What additional issues should be considered by the bank in choosing between T-bond or T-bill futures contracts? FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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In cases where a large number of Treasury bonds are necessary to hedge the balance sheet with a macrohedge, the FI may need to consider whether a sufficient number of deliverable Treasury bonds are available. The number of Treasury bill contracts necessary to hedge the balance sheet is greater than the number of Treasury bonds, the bill market is much deeper and the availability of sufficient deliverable securities should be less of a problem.
17. Reconsider Tree Row Bank in problem 16 but assume that the cost rate on the liabilities is 6 percent.
a. How many contracts are necessary to fully hedge the bank?
In this case, the bank faces different average interest rates on both sides of the balance sheet. Further, the yield on the bonds underlying the futures contracts is a third interest rate. Thus, the hedge also has the effects of basis risk. Determining the number of futures contracts necessary to hedge this balance sheet must consider separately the effect of a change in rates on each side of the balance sheet, and then consider the combined effect on equity. Estimating the number of contracts can be determined with the modified general equation shown on the next page.
b. Verify that the change in the futures position will offset the change in the cash balance sheet position for a change in market interest rates of plus 100 basis points and minus 50 basis points.
For an increase in rates of 100 basis points, ∆ E = 0.01[(4/1.06)$135 m – (6/1.10)$150 m] = - $3,087,478.56. The change in the bond value is –10.3725(.01/1.085295)$95,000 = -$9,079.41, and the change for -340 contracts = $3,087,000.89. The sum of the two values is a net gain of $477.67.
For a decrease in rates of 50 basis points, ∆ E = -0.005[(4/1.06)$135 m – (6/1.10)$150 m] = $1,543,739.28. The change in the bond value is –10.3725(-.005/1.085295)$95,000 = $4,539.71, and the change for -340 contracts = -$1,543,500.45. The sum of the two values is a net loss of $238.83.
Modified Equation Model for part (b): FIN 683 Financial-Institutions Management Professor Robert Hauswald Kogod School of Business, AU
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{ } { } } {
} { } { } { } { } { } { } ( ) ( ) contracts MD P L MD A MD MD P A MD L MD N A MD L MD MD P N A MD L MD R P N D A R R D L R R D R R P N D L R R D A R R D R R P N D L A F E F F F L A F F A L F A L F F F A L F F F F A A L L F F F F L L A A F F F F 340
38 . 944 , 907
$ 92 . 855 , 747 , 308
$ 38 . 944 , 907 $ 26 . 962 , 433 , 509
$ 18 . 818 , 181 , 818
$ 085295
. 1 3725 . 10 * 000 , 95 000 , 000 , 135
* 06 . 1 4 000 , 000
, 150
* 10 . 1 6 * * * * * * 0 ) * * ( * 1 * ) * * ( * ) 1 ( ) * ( * * ) 1 ( * * ) 1 ( ) 1 ( * ) * ( * ) 1 ( * * ) 1 ( * ) 1 ( * ) * ( 0 0 − = − = − − = − − = − − = + − − = = − − + = − + + − = ∆ + − ∆ + + + + ∆ − = + ∆ − − + ∆ − = + ∆ − = ∆ − ∆ + ∆ = ∆ + ∆
c. If the bank had hedged with Treasury bill futures contracts that had a market value of $98 per $100 of face value (implying a discount rate of 8 percent), how many futures contracts would have been necessary to fully hedge the balance sheet?
underlying T-bills. Therefore, the equation developed above in part (a) to determine the number of contracts necessary to hedge the bank can be adjusted as follows for the use of T-bill contracts:
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