# Rsm333 Week 4

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UNIVERSITY OF TORONTO
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
: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
(d) An approximate 95% con…dence interval for the number of defaults for the portfop p p(1 p)=n, or equivalently np 1:96 np(1 p). lio is given by: np 1:96 n
Plugging in p = :119, and n = 50, we get a con…dence interval of 5:95 4:49, or
[1:46; 10:44]. This suggests we are approximately 95% con…dent that we will see between 2 and 10 defaults in our portfolio.
(e) The Binomial distribution is great for independent trials like ‡ ipping a fair coin.
However, here the probability that one bond will default is likely not independent of the probability that other bonds may default, as they are all likely exposed to the same negative (systematic) economic factors, which could lead to hardly any bonds defaulting, or very many bonds defaulting.
(f) The non-zero co-variances between the bonds would cause the portfolio variance to be larger, and hence, the con…dence interval would be wider. That is, a 95% con…dence interval might run from 0 to 14 defaults, for instance.
6. (a) We start by considering the case without corporate taxes:
i. VLF = VU = \$2m=0:1 =

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