An engineering firm can pay for its liability insurance on an annual or quarterly basis. If paid quarterly, the insurance costs $10,000. If paid annually, the insurance costs $35,000. What is the rate of return for paying on annual basis? As shown in the cash flow diagrams, insurance payments are

Question

An engineering firm can pay for its liability insurance on an annual or quarterly basis. If paid quarterly, the insurance costs $10,000. If paid annually, the insurance costs $35,000. What is the rate of return for paying on annual basis? As shown in the cash flow diagrams, insurance payments are made in advance at the beginning of the period covered.

Accepted Answer

TABLE SOLUTION

This is most easily solved by using quarterly periods. As shown in the cash flow diagram and table, insurance must be paid at the start of each quarter. The cash flow table adds a column for the incremental difference to the information in the cash flow diagram.

Quarter Annual Payment Quarterly Payments Incremental Difference

–$25,000	–$10,000	–$35,000	0
–10,000	–10,000		1
–10,000	–10,000		2
–10,000	–10,000		3

Setting the two PWs equal to each other we have

[latex]-\$35,000=-\$10,000(1+(P/A,i_{qtr},3))[/latex]

[latex](P/A,i_{qtr},3)=-\$25,000/-\$10,000=2.5[/latex]

[latex](P/A,9\%,3)=2.531[/latex]

[latex](P/A,10\%,3)=2.487[/latex]

[latex]i_{qtr}=0.09+(0.10-0.09)(2.531-2.5)/(2.531-2.487)[/latex]

[latex]=0.09+0.01(0.7045)=9.705\%[/latex]

On an annual basis the effective interest rate is [latex](1.09705^{4} − 1) = 44.8[/latex]%.

Unless the firm is planning to go out of business, it is clearly better to buy the insurance a year at a time.

5-BUTTON SOLUTION

The incremental cash flows were identified as −$25,000 at time 0 and $10,000 at the end of the first three quarters. The resulting interest rate is shown in the first calculation of the 5-BUTTON SOLUTION. It is 9.70% for 3 months, which is a high rate.

The second calculation shows four quarterly payments and an initial investment of −$35,000. These annuity due payments are identified as beginning-of-period cash flows by using BEGIN on a financial calculator or Type = 1 in a spreadsheet.

Compounding the quarterly interest rate for 4 periods with a PV = −1 leads to an FV that equals 1 + effective interest rate. Subtracting 1 from the FV value of 1.448 shows the annual rate is 44.8%, which means the firm should buy its insurance a year at a time.

H	G	F	E	D	C	B	A
Answer	Solve for	FV	PV	PMT	n	i	Problem	1
9.70%	[latex]i_{qtr}[/latex]	0	-25,000	10,000	3		Exp. 7-9	2
9.70%	[latex]i_{qtr}[/latex]	0	-35,000	10,000	4	using Type=1 or Begin		3
1.448	FV		-1	0	4	9.70%		4
44.8%	effective annual rate							5

Question 7.9: An engineering firm can pay for its liability insurance on a...

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