Question

You are looking at replacing your Widget producing machine for $8,500,000. This will last for 9 year at which point it could be salvaged for $3,200,000. The old machine could be sold today for $3,000,000 and could be salvaged for $750,000 in 9 years. The quantity of widgets produced will be 10,000 units per year. The variable costs for producing each widget will be $49 and the annual fixed costs for production will be $300,000. The project will require you to incur an increase in net working capital of $150,000 which you can reclaim at the end. The machine is in the 25% CCA bracket, and the tax rate is 35%. Debt A is in the form of 15 year bonds with annual compounding. The bonds have a 7% coupon rate and are priced at $955.86 with a face value of $1,000. Debt B is in the form of 10 year zero coupon bonds with annual compounding and a face value of $1,000. These bonds are currently selling for $452.61. Your company is financed by two kinds of equity and two kinds of debt. The debt to equity ratio is 1.3. The value of equity A is twice that of equity B. The value of debt A is 80% of that of debt B. The correlation between the returns on Equity A and the market is 0.75 and the variance of the market returns is 0.001348 while the variance of the returns on equity A is 0.004054. The market risk premium is 8% and the risk free rate is 2.5%. Equity B is expected to pay a dividend of $3.33 one year from now. The growth rate in dividends is 2.5% and the stock is priced at $37. The floatation costs on class A equity are 6%, the floatation costs on class B equity are 5.5%, the floatation costs on class A debt are 4% and the floatation costs on class B debt are 3.5%.
What is the minimum price you should sell the widgets

Answer 1

First we will find the beta for Widget's Equity A stock, by using the formula,

Beta=Correlation between stock's & market returns/(std. devn. Of stock's returns/std. devn. Of market's returns)

As we have variances for the denominator-part of the above formula, we need to use Square roots of both

ie.0.75/(0.004054/0.001348)^(1/2)=

0.43

so, now the cost of Class A Equity can be found as per CAPM

ie.ke=RFR+(Beta*Market return risk premium)

ie.2.5%+(0.43*8%)=

5.94%

Adding the Floatation costs on class A equity 6%,

we get the total issue cost of Class A equity as

5.94%+6%=

11.94%

Cost of Class B equity

can be found by Gordon's formula for constant-dividend growth model

ie.ke=(D1/Net proceeds)+g

where D1=dividend one year from now, ie given as $ 3.33

net proceeds= current market price-Flotation costs, ie.37*(1-5.5%)= $ 34.965

& g= growth rate of dividends=2.5%

Plugging in the values,we get the cost of Class B equity as

ie.ke=(3.33/34.965)+2.5%=

12.02%

After-tax Cost of Debt A

Net proceeds of debt=PV of its future coupons +Pv of face value to be recd. At maturity

ie.Selling price-Flotation costs=$ Semi-annual coupon *(1-(1+r)^-n)/r)+(1000/(1+r)^n)

where,

Selling price = $ 955.86

Net proceeds after Flotation cost=955.86*(1-4%)=917.63 /bond

$ Annual coupon= 1000*7%= $ 70

r= The annual yield /cost of debt--- to be found--??

n= no.of compounding periods--ie. 15

Now, plugging these values, in the formula,

917.63=70*(1-(1+r)^-15)/r)+(1000/(1+r)^15)

we have the annual before-tax cost as

7.9600%

Now the after tax annual cost =

BT cost*(1-Tax rate)

ie. 7.96%*(1-35%)=

5.17%

After-tax Cost of Debt B--zero coupon bonds

Net proceeds=Face value/(1+r)^n

where net proceeds= Selling price-Flotation costs, ie.452.61*(1-3.5%)=436.77

Face value= $ 10000

r= the annual   yield --- to be found out??

n=no.of years to maturity

Now, plugging these values, in the formula,

436.77=1000/(1+r)^10

solving for r, we get the annual yield as,

8.64%

As, no periodic interest is paid on zero-coupon bonds, there is no need to find the after-tax cost

Now,for finding WACC, we find the weights of diff. components as follows:
Debt=1/4= 0.25		Equity=3/4= 0.75
Debt A	Debt B	Eq. A	Eq. B
4/9*0.25=	5/9*0.25=	2/3*0.75=	1/3*0.75=
0.1111	0.1389	0.5	0.25

So, the WACC=(Wt.D ebt A*k Debt A)+(wt.Debt B*k Debt B)+(Wt. Eq.A*keA)+(Wt. Eq.B*keB)

ie.(0.1111*5.17%)+(0.1389*8.64%)+(0.5*11.94%)+(0.25*12.02%)=

10.75%

Year	Depn.	Book value(Prev. BV-depn.)	DTS=Depn.*35%	PV F at 10.75%	PV of DTS at 10.75%
0		8500000
1	2125000	6375000	743750	0.90293	671557.56
2	1593750	4781250	557812.5	0.81529	454779.39
3	1195312.5	3585937.5	418359.4	0.73615	307977.01
4	896484	2689453.125	313769.5	0.66470	208562.31
5	672363	2017089.844	235327.1	0.60018	141238.58
6	504272	1512817.383	176495.4	0.54192	95646.90
7	378204	1134613.037	132371.5	0.48932	64772.17
8	283653	850960	99278.64	0.44182	43863.77
9	212740	638220	74458.98	0.39894	29704.58
	salvage	3200000		5.59127	2018102.27
	Gain on salvge	2561780	Salvage-BV at end yr. 9
	Tax on gain at 35%	896623	Gain *35%
	ATCF on salvage	2303377	$ salvage-$ tax on gain

Initial cost	-8500000
Yr.0--Increase in NWC	-150000
After-tax sale value of old m/c(3000000*(1-35%))	1950000
PV of After-tax Salvage value of new m/c at end yr.9(2303377/1.1075^9)	918906.69
PV of After-tax sale value of old m/c lost(750000*(1-35%)/1.1075^9)	-194482.71
PV of NWC recovered at end yr. 9(150000/1.1075^9)	59840.84
Operating cash flows:
PV of after-tax sale value                                        (10000*x*(1-35%)*5.59127)
PV of After-tax variable costs (10000* $ 49*(1-35%)*5.59127	-1780819.50
PV of After-tax fixed costs (300000*(1-35%)*5.59127	-1090297.65
PV of depn. Tax shields(as per Table)	2018102.27
Total PV (without PV of sale values)	-6768750.07

With the above tabulated values,

The minimum price you should sell the widgets, is where the Net present value of the project's cash flows excatly EQUALS ZERO

Supposing " $ x" to be that per unit sale price,

adding all other rows , except the sales value row,& equating the NPV to 0,

(10000*x*(1-35%)*5.59127)-6768750.07=0

Solving for x, we get, the sale price /unit as,

$186.25 (ANSWER)

NOTE: P/A,i=10.75%;n=9----5.59127