Question

Question 1
The following balance sheet information is available (amounts in $ thousands and duration in years) for a financial institution
	Amount	Duration
T-bills	90	0.5
T-notes	55	0.9
T-bonds	176	To be calculated
Loans	2724	7
Deposits	2092	1
Federal Funds	238	0.01
Equity	715
Treasury bonds are 5-year maturities paying 6% semiannually and selling at par
What is the duration of the T-Bond portfolio? What is the average duration of all the assets?				What is the average duration of all the liabilities?	What is the leverage-adjusted duration gap?   What is the interest rate risk exposure? What is the forecasted impact on the market value of equity caused by a relative upward shift in the entire yield curve of .5%? What is the market-determined risk premium for the corresponding probability of default for each bond? If the yield curve shifts downward by .25% what is the forecasted impact on the market value of equity? If the rate on 1-year T-bills currently is 6%, what is the repayment probability for each of the following 2 securities? Assume that is the loan is defaulted no payments are expected.   Yield 1-year AA rated bond 9.50% 1-year BB rated bond 13.50%

Answer 1

Since, the question has multiple sub-parts, I have answered the first 7 (part a to part g).

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Part a)

The duration of T-Bond portfolio is determined as below:

Time [A]	Cash Flow [B]	PVIF (at 6%/2 = 3%) [C]	Cash Flow*PVIF*Time [A*B*C]
0.5	5.28	0.9709	2.56
1	5.28	0.9426	4.98
1.5	5.28	0.9151	7.25
2	5.28	0.8885	9.38
2.5	5.28	0.8626	11.39
3	5.28	0.8375	13.27
3.5	5.28	0.8131	15.03
4	5.28	0.7894	16.67
4.5	5.28	0.7664	18.21
5	181.28	0.7441	674.45
Total			$773.18

Duration of T-Bond Portfolio = Total of Cash Flow*PVIF*Time/Current Value of Treasury Bonds = 773.18/176 = 4.3931 or 4.39 years

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Notes:

1) PVIF indicates Present Value Interest Factor for $1. It can be dervied from the Present Value Tables.

2) When the bond is selling at par, the coupon rate and YTM is same. Therefore, coupon rate and YTM will be same at 6%. We will take 3% (6%/2) as the rate (YTM) because the bond is semi-annual.

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Part b)

The value of average duration of all assets is determined as below:

Average Duration of All Assets = (Duration of T-Bills*Amount of T-Bills + Duration of T-Notes*Amount of T-Notes + Duration of T-Notes*Amount of T-Notes + Duration of Loans*Amount of Loans)/Total Value of Assets

Substituting values in the above formula, we get,

Average Duration of All Assets = (.50*90 + .90*55 + 4.3931*176 + 7*2,724)/(90 + 55 + 176 + 2,724) = 6.5470 or 6.55 Years

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Part c)

The value of average duration of all liabilities is calculated as follows:

Average Duration of All Liabilities = (Duration of Deposits*Amount of Deposits + Duration of Federal Funds*Amount of Federal Funds)/Total Value of Liabilities

Substituting values in the above formula, we get,

Average Duration of All Liabilities = (1*2,092 + .01*238)/(2,092 + 238) = .8989 or .90 Years

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Part d)

The leverage adjusted duration gap is calculated as below:

Leverage Adjusted Duration Gap = Average Duration of All Assets - (Total Value of Liabilities/Total Value of Assets)*Average Duration of Liabilities = 6.5470 - (2,092 + 238)/(90 + 55 + 176 + 2,724)*.8989 = 5.8592 or 5.96Years

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Part e) - Answer to What is Interest Rate Risk Exposure?

There is an inverse relationship between interest rate and value of equity. Since, duration gap is positive, any increase in interest rate will result in a decline in the market value of equity.

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Part f)

The forecasted impact on the market value of equity is determined as below:

Change in the Market Value of Equity = -Leverage Adjusted Duration Gap*(Total Value of All Assets)*?R/(1+R)

Substituting Values in the above formula, we get,

Change in Market Value of Equity = -5.8592*(90 + 55 + 176 + 2,724)*.0050 = -$89.207

The total market value of equity will decrease from $715,000 to $625,793 (715,000 - 89.207*1,000).

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Part g)

The forecasted impact on the market value of equity is determined as below:

Change in the Market Value of Equity = -Leverage Adjusted Duration Gap*(Total Value of All Assets)*?R/(1+R)

Substituting Values in the above formula, we get,

Change in Market Value of Equity = -5.8592*(90 + 55 + 176 + 2,724)*-.0025 = $44.603

The total market value of equity will increase from $715,000 to $759,603 (715,000 + 44.603*1,000).