Question

Professor Wendy Smith has been offered the following​ deal: A law firm would like to retain her for an upfront payment of $60,000. In​ return, for the next​ year, the firm would have access to eight hours of her time every month.​ Smith's rate is $636 per​ hour, and her opportunity cost of capital is 16% ​(equivalent annual​ rate, EAR). What is the IRR​ (annual)? What does the IRR rule advise regarding this​ opportunity? What is the​ NPV? What does the NPV rule say about this​ opportunity?

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, 0.16 = [{1+(APR/12)}^12]-1

1.16^(1/12) = 1+(APR/12)

1.012445-1 = APR/12

Therefore, APR = 0.012445*12 = 0.14934 = 14.934%

Monthly APR = 0.012445(as above)

Opportunity Cost per month = 8*636 = $5088

		1.2445%	1%	0.50%	0.30%	0.20%
Period	Cash Flow	Discountig Factor [1/(1.012445^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/(1.005^period)]	PV of cash flows (cash flow*discounting factor)	Discountig Factor [1/(1.003^period)]	PV of cash flows (cash flow*discounting factor)	Discountig Factor [1/(1.002^period)]	PV of cash flows (cash flow*discounting factor)
0	60000	1	60000	1	60000	1	60000	1	60000	1	60000
1	-5088	0.987708	-5025.45817	0.990099	-5037.62376	0.9950249	-5062.68657	0.997009	-5072.782	0.998004	-5077.844
2	-5088	0.975567	-4963.68511	0.980296	-4987.7463	0.9900745	-5037.49907	0.9940269	-5057.609	0.996012	-5067.709
3	-5088	0.9635753	-4902.67137	0.9705901	-4938.36267	0.9851488	-5012.43689	0.9910537	-5042.481	0.9940239	-5057.594
4	-5088	0.9517311	-4842.4076	0.9609803	-4889.46799	0.9802475	-4987.49939	0.9880895	-5027.399	0.9920398	-5047.499
5	-5088	0.9400324	-4782.88461	0.9514657	-4841.05742	0.9753707	-4962.68596	0.9851341	-5012.362	0.9900597	-5037.424
6	-5088	0.9284774	-4724.09326	0.9420452	-4793.12616	0.9705181	-4937.99598	0.9821875	-4997.37	0.9880836	-5027.369
7	-5088	0.9170646	-4666.02459	0.9327181	-4745.66946	0.9656896	-4913.42884	0.9792497	-4982.423	0.9861113	-5017.334
8	-5088	0.905792	-4608.66969	0.9234832	-4698.68264	0.9608852	-4888.98392	0.9763208	-4967.52	0.984143	-5007.32
9	-5088	0.894658	-4552.01981	0.9143398	-4652.16103	0.9561047	-4864.66061	0.9734006	-4952.662	0.9821787	-4997.325
10	-5088	0.8836608	-4496.06626	0.905287	-4606.10003	0.9513479	-4840.45832	0.9704891	-4937.849	0.9802183	-4987.35
11	-5088	0.8727988	-4440.8005	0.8963237	-4560.49507	0.9466149	-4816.37644	0.9675864	-4923.079	0.9782617	-4977.396
12	-5088	0.8620704	-4386.21407	0.8874492	-4515.34166	0.9419053	-4792.41437	0.9646923	-4908.354	0.9763091	-4967.461
		NPV =	3609.00495	NPV =	2734.165815	NPV =	882.873644	NPV =	118.10945	NPV =	-269.6249

IRR is the rate of return at which NPV=0

Here, [email protected]% is positive and @0.2% is negative.

Therefore, IRR is between 0.3% and 0.2%

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

= 0.3% - [118.109/(118.109-(-269.62)]

= 0.3% - [118.109/388.029]

= 0.3% - 0.03% = 0.27%

(Explanation & Logic of the method: NPV @0.3% is 118.109 and [email protected]% is -269.92. i.e. 1% decrease in required rate of return reduces NPV by 118.109+269.62 =388.029. We want NPV=0. Therefore, Proportionate decrease in required rate of return to reduce NPV by 118.109 is calculated)

0.27% is Monthly IRR.

Therefore, Annual IRR(APR) = 0.27%*12 = 3.24% and Annual IRR(EAR) = [(1+0.0027)^12]-1 = [1.0027^12]-1 = 3.29%

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

Therefore, ACCEPT.

NPV = $3609

NPV Rule: If NPV>0, then ACCEPT.

Therefore, ACCEPT