Chapter 25, Problem 8QP

Duration Blue Stool Community Bank has the following market value balance shoot:

Asset or Liability	Market value (in $ million)	Duration (in year)
Federal funds deposits	$ 31	0
Accounts receivable	540	.20
Short-term loans	320	.65
Long-term loans	98	5.25
Mortgages	435	12.85
Checking and savings deposits	615	0
Certificates of deposit	390	1.60
Long-term financing	285	9.80
Equity	134	N/A

1. a. What is the duration of the assets?
2. b. What is the duration of the liabilities?
3. c. Is the bank immune to interest rate risk?

Summary Introduction

To calculate: Duration of assets.

Duration:

It is an object used in bond valuation to determine the sensitivity of the bond. High duration comes with high probability of interest rate risk while in lower duration there is low risk of interest rate fluctuation.

Explanation of Solution

Calculation of duration of assets

Assets	Amount ($)	Duration (in years)
Federal Fund deposits	31	0
Account receivables	540	0.20
Short term loans	320	0.65
Long term loan	98	5.25
Mortgage	435	12.85
Total	1,424

Table (1)

Given,

Amount of federal fund deposit is $31.

Duration is 0 years.

Total amount of assets is $1,424.

Formula to calculate duration,

$\text{Duration}=\text{Number}\text{of}\text{year}\times \frac{\text{Amount}\text{of}\text{asset}}{\text{Total}\text{amount}\text{of}\text{assets}}$

Substitute $31 for amount of federal fund deposit, 0 for number of years and $1,424 for total amount of assets.

$\begin{array}{c}\text{Duration}=0\times \frac{\$31}{\$1,424}\\ =0\text{year}\end{array}$

So, duration for federal fund deposit is 0 year.

Given,

Amount of account receivables is $540.

Duration is 0.20 years.

Total amount of assets is $1,424.

Formula to calculate duration,

$\text{Duration}=\text{Number}\text{of}\text{year}\times \frac{\text{Amount}\text{of}\text{asset}}{\text{Total}\text{amount}\text{of}\text{assets}}$

Substitute $540 for amount of account receivables, 0.20 years for number of years and $1,424 for total amount of assets.

$\begin{array}{c}\text{Duration}=0.20\times \frac{\$540}{\$1,424}\\ =0.07584\text{year}\end{array}$

So, duration for account receivables is 0.07584 year.

Given,

Amount of short term loans is $320.

Duration is 0.65 years.

Total amount of assets is $1,424.

Formula to calculate duration is

$\text{Duration}=\text{Number}\text{of}\text{year}\times \frac{\text{Amount}\text{of}\text{asset}}{\text{Total}\text{amount}\text{of}\text{assets}}$

Substitute $320 for amount of short term loans, 0.65 years for number of years and $1,424 for total amount of assets.

$\begin{array}{c}\text{Duration}=0.65\times \frac{\$320}{\$1,424}\\ =0.14606\text{year}\end{array}$

So duration for Short term loans is 0.14606 year.

Given,

Amount of long term loan is $98.

Duration is 5.25 years.

Total amount of assets is $1,424.

Formula to calculate duration,

$\text{Duration}=\text{Number}\text{of}\text{year}\times \frac{\text{Amount}\text{of}\text{asset}}{\text{Total}\text{amount}\text{of}\text{assets}}$

Substitute $98 for amount of long term loan, 5.25 years for number of years and $1,424 for total amount of assets.

$\begin{array}{c}\text{Duration}=5.25\times \frac{\$98}{\$1,424}\\ =0.36130\text{year}\end{array}$

So, duration for long term loans is $0.36130$ year.

Given,

Amount of mortgage is $435.

Duration is 12.85 years.

Total amount of assets is $1,424.

Formula to calculate duration,

$\text{Duration}=\text{Number}\text{of}\text{year}\times \frac{\text{Amount}\text{of}\text{asset}}{\text{Total}\text{amount}\text{of}\text{assets}}$

Substitute $435 for amount of mortgage, 12.85 years for number of years and $1,424 for total amount of assets.

$\begin{array}{c}\text{Duration}=12.85\times \frac{\$435}{\$1,424}\\ =3.9253\text{years}\end{array}$

So, duration for mortgage is 3.9253 years.

Calculation of total duration of assets

Calculated values,

Duration of federal fund deposits 0 year.

Duration of account receivables is 0.07584 year.

Duration of short term loans is 0.14606 year.

Duration of long term loan is $0.36130$ year.

Duration of mortgage is 3.9253 years.

Formula to calculate to total duration,

$\text{Total duration}=\left(\begin{array}{l}\text{Duration}\text{of}\text{asset}1+\text{Duration}\text{of}\text{asset}2\\ +\text{Duration}\text{of}\text{asset}3+\text{Duration}\text{of}\text{asset}4\end{array}\right)$

Substitute 0 year for asset 1, 0.07584 year for asset 2, 0.14606 year for asset 3, 0.36130 year for asset 4, and 3.9253year for asset 5.

$\begin{array}{c}\text{Total}\text{Duration}=0+0.07584+0.14606+0.36130+3.9253\\ =\text{4}\text{.5085}\text{years}\end{array}$

Conclusion

So, total duration of assets is 4.5085 years.

Summary Introduction

To calculate: Duration of liabilities.

Explanation of Solution

Calculation of duration of liabilities

Liabilities	Amount ($)	Duration (in years)
Checking and saving deposit	615	0
Certificates of deposit	390	1.60
Long term financing	285	9.80
Equity	134	N/A
Total	1,424

Table (2)

Given,

Amount of checking and saving deposit is $615.

Duration is 0 years.

Total amount of liabilities is $1,424.

Formula to calculate duration,

$\text{Duration}=\text{Number}\text{of}\text{year}\times \frac{\text{Amount}\text{of}\text{liabilities}}{\text{Total}\text{amount}\text{of}\text{liabilities}}$

Substitute $615 for amount of checking and saving deposit, 0 for number of years and $1,424 for total amount of liabilities.

$\begin{array}{c}\text{Duration}=0\times \frac{\$615}{\$1,424}\\ =0\text{year}\end{array}$

So, duration for checking and saving deposit is 0 year.

Given,

Amount of certificates of deposit is $390.

Duration is 1.60 years.

Total amount of liabilities is $1,424.

Formula to calculate duration,

$\text{Duration}=\text{Number}\text{of}\text{year}\times \frac{\text{Amount}\text{of}\text{liabilities}}{\text{Total}\text{amount}\text{of}\text{liabilities}}$

Substitute $390 for amount of certificates of deposit, 1.60 years for number of years and $1,424 for total amount of liabilities.

$\begin{array}{c}\text{Duration}=1.60\times \frac{\$390}{\$1,424}\\ =0.43820\text{year}\end{array}$

So, duration for certificates of deposit is 0.43820 year.

Given,

Amount of long term financing is $285.

Duration is 9.80 years.

Total amount of liabilities is $1,424.

Formula to calculate duration,

$\text{Duration}=\text{Number}\text{of}\text{year}\times \frac{\text{Amount}\text{of}\text{liabilities}}{\text{Total}\text{amount}\text{of}\text{liabilities}}$

Substitute $285 for amount of long term financing, 9.80 years for number of years and $1,424 for total amount of liabilities.

$\begin{array}{c}\text{Duration}=9.80\times \frac{\$285}{\$1,424}\\ =1.96\text{year}\end{array}$

So, duration for long term financing is 1.96 year.

Calculation of total duration of liabilities

Given,

Duration of checking and saving deposit is 0 year.

Duration of certificates of deposit is 0.43820 year.

Duration of long term financing is 1.96 year.

Formula to calculate to total duration,

$\text{Total duration}=\left(\begin{array}{l}\text{Duration}\text{of}\text{liability}1+\text{Duration}\text{of}\text{liability}2\\ +\text{Duration}\text{of}\text{liability}3\end{array}\right)$

Substitute 0 year for liability 1, 0.43820 year for liability 2, and 1.96 year for liability 3.

$\begin{array}{c}\text{Total}\text{Duration}=0+0.43820+1.96\\ =\text{2}\text{.3982}\text{years}\end{array}$

Conclusion

So total duration of liabilities is 2.3982 years

Summary Introduction

To explain: Whether the bank is immune to interest rate risk or not.

Answer to Problem 8QP

No, bank is not immune to interest rate risk.

Explanation of Solution

Since, the duration of assets and duration of liabilities is not the same, bank is not immune to the interest rate risk. It is mandating that the duration of assets and duration of liability to match for immune the interest rate of risk.

Conclusion

Therefore, bank is not immune to the interest rate of risk due the different duration of assets and liabilities.