Rsm333 Week 4

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
+ $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 weights of debt and equity are given at 35% and 65%, respectively. The after tax cost of debt is kd = 14% (1 0:36) =
8:96% and the cost of equity is ke = 5% + :8 1:25 8:5% = 13:5%. We can compute the WACC: kwacc = :35 8:96% + :65 13:5% = 11:91%
The NPV of the project can be decomposed in di¤erent parts:
NPV = PV(Cash ‡ ows) + PV(CCA Tax Shields) + PV(Change in NWC, end of the project) –Initial Investment - Change in NWC (beginning of the project)
P V (Cash F lows) = $1:2m

Annuity(6; kwacc ) = $4; 946; 290

P V (CCA T ax Shields) = [$4; 200; 000 :3 :36=(:3+:1191)] [(1+:5 :1191)=(1:1191)] =
$1; 024; 726
P V (Change in N W C; end of the project) = $200; 000=(1:1191)6 = $101; 816
N P V = $4; 946; 290 + $1; 024; 726 + $101; 816

$4; 200; 000

$200; 000 = $1; 672; 832

Since NPV is positive, we would recommend for the project (notice that using the risk-free rate for discounting CCA would have been considered correct as well).
4. As we have seen in class, the value of the …rm will decrease, in a world with taxes, because now the government is taking a portion of the expected pro…ts each year. Since the debt of the …rm remains unchanged, the value of the equity will decrease; we can new write ks = k0 +(D=S new )(1 T )(k0 kd ), where S new is the value of the equity, for the new new
…rm, in a world with taxes. Now it is also true that D + S new = VL = VU + T D new and VU = VU (1 T ) = (D + S)(1 T ), so D + S new = (D + S)(1 T ) + T D.
This implies:
S new = (D + S)(1 T )
= D(1 T ) + S(1
= S(1 T )

D+T D
T ) D(1 T )

Therefore: new ks
= k0 +
= k0 +

= k0 +
= ks

D
(1 T )(k0
S new
D
(1 T )(k0
S(1 T )
D
(k0 kd )
S

3

kd ) kd )

5. (a) The expected number of defaults is 50p = 50(:119) = 5:95
(b) The variance is 50p(1 p) = 50(:119)(1 deviation is 2:2895 ' 2:29.

:119) = 5:242 and so the standard

(c) The probability that the number of defaults will be 4 or less is obtained as follows:
Prob of exactly 0 defaults: 50 (0:119)0 (1 :119)50 = 0:0018
0
Prob of exactly 1 defaults:
Prob of exactly 2 defaults:
Prob of exactly 3 defaults:

50
1
50
2
50
3
50
4

(0:119)1 (1

:119)49 = 0:0120

(0:119)2 (1

:119)48 = 0:0396

(0:119)3 (1

:119)47 = 0:0857

Prob of exactly 4 defaults:
(0:119)4 (1 :119)46 = 0:1359
So the probability of 4 or less defaults is 0:0018+0:0120+0:0396+0:0857+0:1359 =
0:2750.
(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 =
$20m
VInd = VU + T D = $20m + 0 = $20m
LF
Ind ii. ks = k0 = 10%, ks = k0 + (k0 kd )(1 T )(D=E) = 10% + (10% 5%)(1
0)(10=10) = 15%:
Ind
iii. SInd = [(EBIT Dkd )(1 T )]=ks = [($2m $0:5m)(1 0)]=0:15] = $10m
VInd = SInd + DInd = $10m + $10m = $20m iv. W ACCLF = 10% 1 = 10%
1
W ACCInd = 15% 2 + 5% 1 = 10%
2
(b) We now introduce taxes:
i. VLF = VU = $2m 0:6=0:1 = $12m
VIndebted = VU + T D = $12m + 0:4
4

$10m = $16m

LF
Ind
ii. ks = k0 = 10%, ks = k0 + (k0 kd )(1 T )(D=E) = 10% + (10% 5%)(1
0:4)(10=6) = 15%:
Ind
iii. SInd = [(EBIT Dkd )(1 T )]=ks = [($2m $0:5m)(1 0:4)]=0:15] = $6m
VInd = SInd + DInd = $6m + $10m = $16m iv. W ACCLF = 10% 1 = 10%
6
W ACCInd = 15% 16 + 5%(1 0:4) 10 = 7:5%
16

7. (a) P aybackX = 2 + 500=3; 000 = 2:17 years
P aybackY = 2 + 3; 000=3; 500 = 2:86 years
N P VX = 10; 000+6; 500=(1:12)+3; 000=(1:12)2 +3; 000=(1:12)3 +1; 000=(1:12)4 =
$966:01
N P VY = 10; 000+3; 500=(1:12)+3; 500=(1:12)2 +3; 500=(1:12)3 +3; 500=(1:12)4 =
$630:72
IRRX = 18:0%
IRRY = 15:0%
P IX = PV of bene…ts / PV Costs = 10; 966:01=10; 000 = 1:10
P IY = PV of bene…ts / PV Costs = 10; 630:72=10; 000 = 1:06
(b) Both projects have a positive NPV and are acceptable based on all measures. The
…rm should therefore accept both projects assuming these are independent.
(c) Project X ranks above Project Y based on all measures used. Therefore if the projects are mutually exclusive and only 1 can be selected, the …rm should go for
Project X.
(d) To determine the e¤ects of a change in the cost of capital on the NPV, we must calculate the NPV of the two projects while changing the cost of capital.
Cost of Capital
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%

NPVX
$3,500
$3,007
$2,545
$2,113
$1,707
$1,325
$966
$627
$307
$5
($282)
5

NPVY
$4,000
$3,327
$2,705
$2,128
$1,592
$1,095
$631
$198
($206)
($585)
($939)

1. (a) Based on the NPV table, we can tell that the crossover occurs between 6% and 8%.
We therefore create a new project, Project , that is the di¤erence in cash‡ ows between Projects X and Y.
Year
0
1
2
3
4

Project

Cash‡ ows $$3,000
($500)
($500)
($2,500)

1. (a) We calculate the IRR as of the cash‡ ows to arrive at the crossover rate: IRRcross =
6:2187%. According to the IRR, Project X is always preferred to Project Y. When we use NPV to compare the projects, however, their ranking will depend on the cost of capital, k. When k > 6:2187%, then N P VX > N P VY and there is no con‡ ict. However, when k < 6:2187%, N P VX < N P VY and there is a con‡ ict, as according to NPV we should select Project Y. Indeed IRR is not taking into account the di¤erent timing of cash ‡ ows: when the cost of capital is low, project
Y that produces higher cash ‡ ows, even if later in time, is preferable.

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