91Ó°ÊÓ

Dr. Harold Wolf of Medical Research Corporation (MRC) was thrilled with the response he had received from drug companies for his latest discovery, a unique electronic stimulator that reduces the pain from arthritis. The process had yet to pass rigorousFederal Drug Administration (FDA) testing and was still in the early stages ofdevelopment, but the interest was intense. He received the three offers described in the following paragraph. (A 10 percent interest rate should be used throughout this analysisunless otherwise specified.)

Offer I\(1,000,000 now plus \)200,000 from year 6 through 15. Also if the productdid over \(100 million in cumulative sales by the end of year 15, he wouldreceive an additional \)3,000,000. Dr. Wolf thought there was a 70 percentprobability this would happen.

Offer IIThirty percent of the buyer’s gross profit on the product for the next fouryears. The buyer in this case was Zbay Pharmaceutical. Zbay’s gross profitmargin was 60 percent. Sales in year one were projected to be \(2 millionand then expected to grow by 40 percent per year.

Offer IIIA trust fund would be set up for the next eight years. At the end of thatperiod, Dr. Wolf would receive the proceeds (and discount them back tothe present at 10 percent). The trust fund called for semiannual paymentsfor the next eight years of \)200,000 (a total of $400,000 per year).

The payments would start immediately. Since the payments are coming at thebeginning of each period instead of the end, this is an annuity due. Assumethe annual interest rate on this annuity is 10 percent annually (5 percent semiannually).Determine the present value of the trust fund’s final value. Hint:See the section on Annuities Due.

Required: Find the present value of each of the three offers and indicatewhich one has the highest present value.

Step by step solution

01

Computation of present value of offer 1

From year 6 to year 15, No. of years = 10

Annual Amount = $200,000

Annual interest rate = 10%

$\begin{array}{rcl}\mathbf{Present}\mathbf{}\mathbf{value}\mathbf{}\mathbf{of}\mathbf{}\mathbf{Annuity}\mathbf{}\mathbf{at}\mathbf{}\mathbf{yesr}\mathbf{}\mathbf{end}\mathbf{}\mathbf{6}& \mathbf{=}& \mathbf{Annual}\mathbf{}\mathbf{amount}\mathbf{×}\left[\frac{1-{\displaystyle \frac{1}{{\left(1+r\right)}^n}}}{r}\right]\\ & \mathbf{=}& \mathbf{\$}\mathbf{200}\mathbf{,}\mathbf{000}\mathbf{×}\left[\frac{1-{\displaystyle \frac{1}{{\left(1+0.1\right)}^{10}}}}{0.1}\right]\\ & \mathbf{=}& \mathbf{\$}\mathbf{200}\mathbf{,}\mathbf{000}\mathbf{×}\mathbf{6}\mathbf{.}\mathbf{145}\\ & \mathbf{=}& \mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{229}\mathbf{,}\mathbf{000}\end{array}$

$\begin{array}{rcl}\mathbf{Present}\mathbf{}\mathbf{value}\mathbf{}\mathbf{of}\mathbf{}\mathbf{Annuity}\left(t=0\right)& \mathbf{=}& \frac{\mathbf{Present}\mathbf{}\mathbf{value}\mathbf{}\mathbf{of}\mathbf{}\mathbf{Annuity}}{{\left(1+i\right)}^{\mathbf{n}}}\\ & \mathbf{=}& \frac{\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{229}\mathbf{,}\mathbf{000}}{{\left(1+0.1\right)}^{\mathbf{6}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{693}\mathbf{,}\mathbf{738}\mathbf{.}\mathbf{46}\end{array}$

$\begin{array}{rcl}\mathbf{Expected}\mathbf{}\mathbf{additional}\mathbf{}\mathbf{amount}\mathbf{}\mathbf{to}\mathbf{}\mathbf{be}\mathbf{}\mathbf{received}& \mathbf{=}& \mathbf{Expected}\mathbf{}\mathbf{amount}\mathbf{×}\mathbf{Probability}\\ & \mathbf{=}& \mathbf{\$}\mathbf{3}\mathbf{,}\mathbf{000}\mathbf{,}\mathbf{000}\mathbf{×}\mathbf{0}\mathbf{.}\mathbf{7}\\ & \mathbf{=}& \mathbf{}\mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{100}\mathbf{,}\mathbf{000}\end{array}$

$\begin{array}{rcl}\mathbf{Present}\mathbf{}\mathbf{value}\mathbf{}\mathbf{of}\mathbf{}\mathbf{expected}\mathbf{}\mathbf{amount}\mathbf{}\mathbf{to}\mathbf{}\mathbf{be}\mathbf{}\mathbf{received}& \mathbf{=}& \mathbf{Expected}\mathbf{}\mathbf{amount}\mathbf{×}\frac{\mathbf{1}}{{\left(1+r\right)}^{\mathbf{n}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{100}\mathbf{,}\mathbf{000}\mathbf{×}\frac{\mathbf{1}}{{\left(1+0.1\right)}^{\mathbf{10}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{100}\mathbf{,}\mathbf{000}\mathbf{×}\mathbf{0}\mathbf{.}\mathbf{386}\\ & \mathbf{=}& \mathbf{\$}\mathbf{810}\mathbf{,}\mathbf{600}\end{array}$

02

Computation of present value of offer 2

Buyer’s Gross profit and its share’s computation

Year	Buyer’s incremental gross profit (40% per year)	Percent of share	Share value
1	$1,200,000	30%	$360,000
2	$1,680,000	30%	$504,000
3	$2,352,000	30%	$705,600
4	$3,292,800	30%	$987,840

Computation of present value of cash flows

Year	Gross profit’s Share value	Present value factor at 10%	Present value
1	$360,000	0.9091	$ 327,276
2	$504,000	0.8264	$ 416,506
3	$705,600	0.7513	$ 530,117
4	$987,840	0.6830	$ 674,695
Total			$1,948,594

Present value of offer II = $1,948,594

03

Computation of present value of offer 3

$\begin{array}{rcl}\mathbf{Semiannual}\mathbf{}\mathbf{interest}\mathbf{}\mathbf{rate}& \mathbf{=}& \frac{\mathbf{Annual}\mathbf{}\mathbf{rate}}{\mathbf{2}}\\ & \mathbf{=}& \frac{\mathbf{10}}{\mathbf{2}}\\ & \mathbf{=}& \mathbf{5}\mathbf{\%}\end{array}$

$\begin{array}{rcl}\mathbf{No}\mathbf{.}\mathbf{}\mathbf{of}\mathbf{}\mathbf{periods}& \mathbf{=}& \mathbf{No}\mathbf{.}\mathbf{}\mathbf{of}\mathbf{}\mathbf{years}\mathbf{×}\mathbf{2}\\ & \mathbf{=}& \mathbf{8}\mathbf{×}\mathbf{2}\\ & \mathbf{=}& \mathbf{16}\end{array}$

$\begin{array}{rcl}\mathbf{Future}\mathbf{}\mathbf{value}\mathbf{}\mathbf{of}\mathbf{}\mathbf{the}\mathbf{}\mathbf{interest}& \mathbf{=}& \mathbf{Annuity}\mathbf{}\mathbf{due}\mathbf{×}\left[\frac{{\left(1+r\right)}^{n+1}-1}{r}-1\right]\\ & \mathbf{=}& \mathbf{\$}\mathbf{200}\mathbf{,}\mathbf{000}\mathbf{×}\left[\frac{{\left(1+0.05\right)}^{16+1}-1}{0.05}-1\right]\\ & \mathbf{=}& \mathbf{\$}\mathbf{200}\mathbf{,}\mathbf{000}\mathbf{×}\mathbf{24}\mathbf{.}\mathbf{84}\\ & \mathbf{=}& \mathbf{\$}\mathbf{4}\mathbf{,}\mathbf{968}\mathbf{,}\mathbf{000}\end{array}$

$\begin{array}{rcl}\mathbf{Present}\mathbf{}\mathbf{value}\mathbf{}\mathbf{b}\mathbf{}\mathbf{of}\mathbf{}\mathbf{trut}\mathbf{}\mathbf{find}\mathbf{\text{'}}\mathbf{s}\mathbf{}\mathbf{final}\mathbf{}\mathbf{value}& \mathbf{=}& \mathbf{Fund}\mathbf{\text{'}}\mathbf{s}\mathbf{}\mathbf{final}\mathbf{}\mathbf{value}\mathbf{×}\frac{\mathbf{1}}{{\left(1+r\right)}^{\mathbf{n}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{4}\mathbf{,}\mathbf{968}\mathbf{,}\mathbf{000}\mathbf{×}\frac{\mathbf{1}}{{\left(1+0.1\right)}^{\mathbf{8}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{4}\mathbf{,}\mathbf{968}\mathbf{,}\mathbf{000}\mathbf{×}\mathbf{0}\mathbf{.}\mathbf{4665}\\ & \mathbf{=}& \mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{317}\mathbf{,}\mathbf{572}\end{array}$

Present value of offer III = $2,317,572

04

Recommended offer

Since present value of offer I is the highest, it should be given priority.

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