Question

Adirondack Savings Bank (ASB) has $1 million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are 5% for home loans, 10% for personal loans, and 7% for automobile loans. The bank’s planning committee has decided that at least 40% of the new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans.

(a)	Formulate a linear programming model that can be used to determine the amount of funds ASB should allocate to each type of loan to maximize the total annual return for the new funds. If the constant is "1" it must be entered in the box. If your answer is zero enter “0”.
	Let H = amount allocated to home loans P = amount allocated to personal loans A = amount allocated to automobile loans
	Max H + P + A s.t. H + P + A ≥ Minimum Home Loans H + P + A ≤ Personal Loan Requirement H + P + A = Amount of New Funds
(b)	How much should be allocated to each type of loan?
	Loan type Allocation Home $ Personal $ Automobile $
	What is the total annual return?
	If required, round your answer to nearest whole dollar amount.
	$
	What is the annual percentage return?
	If required, round your answer to two decimal places.
	%
(c)	If the interest rate on home loans increases to 9%, would the amount allocated to each type of loan change?
	- Select your answer -YesNoItem 21
	Explain.
	The input in the box below will not be graded, but may be reviewed and considered by your instructor.
(d)	Suppose the total amount of new funds available is increased by $10,000. What effect would this have on the total annual return? Explain.
	If required, round your answer to nearest whole dollar amount.
	An increase of $10,000 to the total amount of funds available would increase the total annual return by $ .
(e)	Assume that ASB has the original $1 million in new funds available and that the planning committee has agreed to relax the requirement that at least 40% of the new funds must be allocated to home loans by 1%. How much would the annual return change?
	If required, round your answer to nearest whole dollar amount.
	$
	How much would the annual percentage return change?
	If required, round your answer to two decimal places.
	%

Answer 1

ANSWER:

Part (a)

The annual rates of return for the three types of loans are 5% for home loans, 10% for personal loans, and 7% for automobile loans.

Hence, objective function should be:

Max 0.05H + 0.10P + 0.07A

Subject to:

Total loan amount is $ 1 mn: H + P + A ≤ 1

at least 40% of the new funds must be allocated to home loans: H / (H + P + A) ≥ 40% = 0.4

Hence, H ≥ 0.4(H + P + A)

Hence, 0.6H - 0.4P - 0.4A ≥ 0

Or, 3H - 2P - 2A ≥ 0

And finally, the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans

Hence, P ≤ 0.6A

Or, P - 0.6A ≤ 0

Hence, your final answers should be:

Max	0.05H	+	0.10P	+	0.07A
s.t.
	3H	-	2P	-	2A	≥		0
			P	-	0.60A	≤		0
	H	+	P	+	A	=		1

Part (b)

We need to solve it using solver in excel.

Solver Solution:

Objective Cell (Max)
	Cell	Name	Original Value	Final Value
	$V$38	Max z = Allocation	0	0.06875
Variable Cells
	Cell	Name	Original Value	Final Value	Integer
	$V$34	Home, H Allocation	0	0.4000	Contin
	$V$35	Personal, P Allocation	0	0.2250	Contin
	$V$36	Automobile, A Allocation	0	0.3750	Contin
Constraints
	Cell	Name	Cell Value	Formula	Status	Slack
	$V$39		0	$V$39>=$Y$39	Binding	0
	$V$40		8.9407E-09	$V$40<=0	Binding	0
	$V$41		1	$V$41=$Y$41	Binding	0

Hence,  

Loan type	Allocation ($)
Home, H	400,000
Personal, P	225,000
Automobile, A	375,000

What is the total annual return?

Total annual return = $ 0.06875 mn = $ 68,750

What is the annual percentage return?

Annual percentage return = Total annual return / Total loan = 0.06875 / 1 = 6.88%

Part (c)

If the interest rate on home loans increases to 9%, would the amount allocated to each type of loan change?

Yes, the amount allocated will change. The new allocation will be:

H = $ 1 mn; P = 0; A = 0

Part (d)

Suppose the total amount of new funds available is increased by $10,000. What effect would this have on the total annual return? Explain.

An increase of $10,000 to the total amount of funds available would increase the total annual return by $  0.0694 mn - 0.06875 mn = $ 688

Part (e)

Total annual return now =  69,753

Hence, annual return increases by  69,753 - 68,750 = $ 1,003

Annual %age return = 69,753 / 1,010,000 = 6.91%

Annual %age return increases by 6.91% - 6.875% = 0.03%

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