Question

You have been hired as a risk manager for Acorn savings and loan. Currently, Acorn's balance sheet is as follows (in millions of dollars).

Assets. Liabilities

Cash Reserves 51.8 Checking and savings 81.3

Auto Loan 103.7 Certificates of deposit 98.3

Mortgages 147.1 Long term Financing 102.3

Total assests= 302.6 Total Liabilities= 281.9

Owners equity=20.7

Total liabilities and equity 302.6

When you analyze the duration of loans you find that the duration of the auto loans is 2.2 years. While the mortgages have a duration of 7.1 years. Both the cash reserves and the checking and savings accounts have zero duration. The CD's have a duration of 2.2 years, and the long term financing has a 9.5-year duration.

A. What is the duration of Acorn's equity?

B. Suppose Acorn experience a rash of mortgage pre-payments, reducing the size of the mortgage portfolio from $147.1 million to 98.1, million, and increasing cash reserves to 100.8 million. What is the duration of Acorn's equity now? If interest rates are currently 4% and were to fall to 3%. Estimate the approximate change in the value of Acorn's equity (Assume interest rates are APR's based on monthly compounding).

C. Suppose that after the pre-payments in part (b) but before a change in interest rates. Acorn considers managing its risk by selling mortgages and a buying 10-year treasury strips(zero coupon bonds). How many should the firm buy or sell to eliminate its current interest rate risk?

Assets

Cash reserves 51.8

Auto Loans 103.7

Mortgages 147.1

Total Assets 302.6

Liabilities

Checking and savings 81.3

Certificates of Deposit 98.3

Long term financing 102.3

Total Liabilities 281.9

Owners Liabilities 20.7

Total Liabilities and equity 302.6

Answer 1

Assets	$ value	Portfolio weight	Duration	Weighted duration = duration * portfolio weight
Cash reserves	51.8	17.12%	0	-
Auto Loans	103.7	34.27%	2.2	0.7539
Mortgages	147.1	48.61%	7.1	3.4515
Total Assets	302.6	100.00%	Total	4.2054
Liabilities	$ value	Portfolio weight	Duration	Weighted duration = duration * portfolio weight
Checking and savings	81.3	28.84%	0	-
Certificates of Deposit	98.3	34.87%	2.2	0.7672
Long term financing	102.3	36.29%	9.5	3.4475
Total Liabilities	281.9	100.00%	Total	4.2147
Owners Liabilities	20.7
Total Liabilities and equity	302.6
Duration gap	(0.01)
=asset duration- liability duration
Duration of equity=	(0.01)

B
New balance sheet
Assets	$ value	Portfolio weight	Duration	Weighted duration = duration * portfolio weight
Cash reserves	100.8	33.31%	0	-
Auto Loans	103.7	34.27%	2.2	0.7539
Mortgages	98.1	32.42%	7.1	2.3018
Total Assets	302.6	100.00%	Total	3.0557
Liabilities	$ value	Portfolio weight	Duration	Weighted duration = duration * portfolio weight
Checking and savings	81.3	28.84%	0	-
Certificates of Deposit	98.3	34.87%	2.2	0.7672
Long term financing	102.3	36.29%	9.5	3.4475
Total Liabilities	281.9	100.00%	Total	4.2147
Owners Liabilities	20.7
Total Liabilities and equity	302.6
Duration gap	(1.16)
=asset duration- liability duration
=3.0557-4.2147
Duration of equity=	(1.16)

As the assets shift from higher duration mortgages to lower duration cash, the duration of assets decreases. Hence the duration gap increases

Interest rates change from 4% to 3%

% change in price= -Duration * change in interest rate/ (1+ original interest rate=
i.e.
% change P= -dur * -1%/(1+4%)

% change in asset value=
=-3.0557*-1%/(1+4%)	2.94%
$ change in asset value=
= % change in asset * asset $ value
=2.94%*302.6	8.8909
New asset value	311.4909
% change in liability value=
=-4.2147*-1%/(1+4%)	4.05%
$ change in liability value=
= % change in liability * liability $ value
=4.05%*281.9	11.4243
New liability value	293.3243
New equity
=total assets- total liability	18.1666
Change in equity
=new equity- old equity	-2.5334

Since the duration of liabilities is more than that of assets, an equal change in interest rates increased the value of liabilities more than that of assets. Hence, the equity after change in interest rate has gone down.

C
New balance sheet
Assets	$ value	Portfolio weight	Duration	Weighted duration = duration * portfolio weight
Cash reserves	100.8	33.31%	0	-
Auto Loans	103.7	34.27%	2.2	0.7539
Mortgages	98.1	32.42%	7.1	2.3018
Total Assets	302.6	100.00%	Total	3.0557
Liabilities	$ value	Portfolio weight	Duration	Weighted duration = duration * portfolio weight
Checking and savings	81.3	28.84%	0	-
Certificates of Deposit	98.3	34.87%	2.2	0.7672
Long term financing	102.3	36.29%	9.5	3.4475
Total Liabilities	281.9	100.00%	Total	4.2147
Owners Liabilities	20.7
Total Liabilities and equity	302.6
Duration gap	(1.16)
=asset duration- liability duration
=3.0557-4.2147
Duration of equity=	(1.16)

The duration of 10-year treasury strips can be considered as 10 since they are zero coupon bonds

Value of ZCBs to be transacted can be computed by following equation-
Duration of equity* value of equity= duration of ZCB* value of ZCB - duration of mortgage* value of ZCB

-1.16*20.7=10*Z - 7.1*(98.1-Z)
Z= -1.16*20.7/(10-7.1)-98.1*7.1

121

Assets	$ value	Portfolio weight	Duration	Weighted duration = duration * portfolio weight
Cash reserves	100.8	33.31%	0	-
Auto Loans	103.7	34.27%	2.2	0.7539
ZCB	121	39.99%	10	3.9987
Mortgages	-22.9	-7.57%	7.1	(0.5373)
Total Assets	302.6	100.00%	Total	4.2153
Liabilities	$ value	Portfolio weight	Duration	Weighted duration = duration * portfolio weight
Checking and savings	81.3	28.84%	0	-
Certificates of Deposit	98.3	34.87%	2.2	0.7672
Long term financing	102.3	36.29%	9.5	3.4475
Total Liabilities	281.9	100.00%	Total	4.2147
Owners Liabilities	20.7
Total Liabilities and equity	302.6
Duration gap	0.00
=asset duration- liability duration
Duration of equity=	0.00

Hence, the firm should have a total $121M in ZCB on the balance sheet and sell of $98.1M mortgage + $22.9M additional mortgage to match the asset and liability duration so that the new equity duration is zero. This will eliminate the interest rate risk