Question

Professor Wendy Smith has been offered the following​ opportunity: A law firm would like to retain her for an upfront payment of $ 49000. In​ return, for the next year the firm would have access to eight hours of her time every month. As an alternative payment​ arrangement, the firm would pay Professor​ Smith's hourly rate for the eight hours each month. ​ Smith's rate is $ 545 per hour and her opportunity cost of capital is 15 % per year. What does the IRR rule advise regarding the payment​ arrangement? (Hint: Find the monthly rate that will yield an effective annual rate of 15 %​.) What about the NPV​ rule?

Answer 1

Effective Interest Rate or EAR = [{1+(APR/n)}^n]-1

Where, APR = Annual Interest Rate or Nominal Rate, n = Number of times compounded in a year

Therefore, 0.15 = [{1+(APR/12)}^12]-1

1.15^(1/12) = 1+(APR/12)

1.0117149-1 = APR/12

Therefore, APR = 0.0117149*12 = 0.140579 = 14.0579%

Monthly APR = 0.0117149(as above)

Opportunity Cost per month = 8*545 = $4360

		1.1715%	1%	1.10%
Period	Cash Flow	Discountig Factor [1/(1.0117149^period)]	PV of cash flows (cash flow*discounting factor)	Discountig Factor [1/(1.01^period)]	PV of cash flows (cash flow*discounting factor)	Discountig Factor [1/(1.011^period)]	PV of cash flows (cash flow*discounting factor)
0	49000	1	49000	1	49000	1	49000
1	-4360	0.9884207	-4309.51447	0.990099	-4316.83168	0.9891197	-4312.56182
2	-4360	0.9769756	-4259.61352	0.980296	-4274.09078	0.9783577	-4265.63978
3	-4360	0.9656629	-4210.29039	0.9705901	-4231.77304	0.9677129	-4219.22827
4	-4360	0.9544813	-4161.53839	0.9609803	-4189.8743	0.9571839	-4173.32173
5	-4360	0.9434291	-4113.35089	0.9514657	-4148.3904	0.9467694	-4127.91467
6	-4360	0.9325049	-4065.72137	0.9420452	-4107.31723	0.9364683	-4083.00165
7	-4360	0.9217072	-4018.64337	0.9327181	-4066.65072	0.9262792	-4038.5773
8	-4360	0.9110345	-3972.11049	0.9234832	-4026.38685	0.916201	-3994.6363
9	-4360	0.9004854	-3926.11643	0.9143398	-3986.52163	0.9062324	-3951.1734
10	-4360	0.8900585	-3880.65494	0.905287	-3947.05112	0.8963723	-3908.18338
11	-4360	0.8797523	-3835.71987	0.8963237	-3907.97141	0.8866195	-3865.66111
12	-4360	0.8695654	-3791.30511	0.8874492	-3869.27862	0.8769728	-3823.60149
		NPV =	455.420763	NPV =	-72.1377844	NPV =	236.4990938

IRR is the rate of return at which NPV=0

Here, NPV@1.1% is positive and @1% is negative.

Therefore, IRR is between 1.1% and 1%

IRR = Rate at which positive NPV - [Positive NPV/(Positive NPV-Negative NPV)]

= 1.1% - [236.5/(236.5-(-72.1378)]

= 1.1% - [236.5/308.6378]

= 1.1% - 0.0766% = 1.0234%

(Explanation & Logic of the method: NPV @1.1% is 236.5 and NPV@1% is -72.1378. i.e. 0.1% decrease in required rate of return reduces NPV by 236.5+72.1378 =308.6378. We want NPV=0. Therefore, Proportionate decrease in required rate of return to reduce NPV by 236.5 is calculated)

1.0234% is Monthly IRR.

Therefore, Annual IRR(APR) = 1.0234%*12 = 12.2808% and Annual IRR(EAR) = [(1+0.010234)^12]-1 = [1.010234^12]-1 = 12.996%

IRR Rule: If IRR<Required Rate of Return, then ACCEPT.

Therefore, ACCEPT.

NPV = $455.42

NPV Rule: If NPV>0, then ACCEPT.

Therefore, ACCEPT