# RSM333 assignment 1

**Topics:**Stock, Cost of capital, Modigliani-Miller theorem

**Pages:**6 (2051 words)

**Published:**September 23, 2014

Joseph L. Rotman School of Management

RSM333

PROBLEM SET #1

SOLUTIONS

1. Let’ denote the payment to the manufacturer by x. The following cash ‡ s

ows are

created by the project:

Selling the existing machine: its book value is $45; 000 5 $3; 000 = $30; 000. So selling the machine will produce a capital gain equal to $35; 000 $30; 000 = $5; 000, and the …rm will pay taxes on the capital gain so that the net cash in‡ ow

is CF (selling existing machine) = $30; 000 + $5; 000(1 :35) = $33; 250: Cash ‡ increments over years 1-10, given by the changes in operation costs and ow

in tax savings due to depreciation:

OC

(1

t) + t

depreciation = ($12; 000)(:65) + :35

x

8000

10

$3; 000

= :035x + $6; 470

Selling the asset in the …nal period: we need to compare the CF we would obtain by selling either the existing machine or the new machine. CF (salvage existing machine) = 0, CF (salvage new machine) = $8; 000, so incremental CF = $8; 000.

We can now compute the NPV of the project and solve for the price of the machine that sets it equal to zero:

x + $33; 250 +

10

P :035x + $6; 470

$8; 000

+

= 0 =) x = $82; 124:7

t

(1:15)

(1:15)10

t=1

2. (a) Share price A = $50m=100; 000 = $500, Share price B = ($50m $20m)=100; 000 = $300:

(b) Let’ de…ne Company A’ periodic cash ‡

s

s

ows as X, which depends purely on

Company A’ assets. If we purchase a share of Company A we are entitled to s

X/100,000 each period. In order to replicate this periodic cash ‡ we have to ow

construct a portfolio that consists of n shares of Company B and an investment of D in the risk-free asset:

X=100; 000 = (n=100; 000) (X $20m r) + D r

X=100; 000 = n X=100; 000 n 200 r + D r

1

This last equation is satis…ed if: X=100; 000 = n X=100; 000 and n 200 r = D r. Solving the two equations (after simplifying for r the second one), we have n = 1 and D = 200, which means that the replicating portfolio is long on 1 stock of Company B and lends 200 dollars at the risk-free rate.

(c) Currently, Company B’ debt to equity ratio is D=E = $20m=($50m $20m) = s

2=3. To achieve the target debt-to-equity ratio of 0.5, Company B has to buy back debt and pay for it with additional equity (since in the question it is stated that a buy-back of equity or debt is targeted).

:5

= ($20m X)=($30m + X) =) $15m + :5X = $20m

=) $5m = 1:5X =) X = $5m=1:5 = $3:33m

X

So Dnew = $20m $3:33m = $16:67m. The new value of equity becomes Snew = $30m + $3:33m = $33:33m. As the value of the company stays the same after the change in the capital structure, the share price is still $300. So the company will issue n = $3:33m=$300 = 11; 100 new shares.

(d) Let’ consider the value of the unlevered …rm with M&M Proposition 2, VU = s

VL T D = $50m :35 $20m = $43m: A target debt-to-equity ratio of 0.5 implies the following leverage ratio: D=E = :5, D=(D + E) = :5=(1 + :5) = 1=3. The value of the company under the proposed debt buy-back is: VL = VU + T

Using that VL = 3

3

Dnew = $43 + :35

Dnew

Dnew , we obtain:

Dnew = $43 + :35 Dnew =) Dnew = $16:23m

VL = Dnew 3 = $16:23m 3 = $48:68m

Snew = VL Dnew = $48:68m $16:23m = $32:45m

So the …rm buys back: D = Dold Dnew = $20m $16:23m = $3:77m. As the investors that buy the new equity will only be willing to pay the post-restructuring value of each share, if we issue n new shares, the post-restructuring stock price, Pnew , will be:

$32:45m

Pnew =

100; 000 + n

Moreover the proceeds obtained by issuing the new shares should be equal to the amount of debt we want to buy back:

Pnew

n = $3:77m

Solving these two equations for Pnew and n gives: Pnew = $286:8 and n = 13; 145 shares.

2

(e) The di¤erence in the values of debt and equity in parts (3) and (4) can be explained with M&M Proposition II. Due to the tax shields of debt decrease in the level of debt decreases the value of the levered company.

3. First step is to estimate the WACC of the project. The...

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