Mgcr 341 Final Exam
April 2014
Final Examination
Finance 1
MGCR 341
April 24, 2:00PM-5:00PM
FINAL EXAM: Solutions
Examiner:
Vadim di Pietro
Student Name:
McGill ID:
INSTRUCTIONS:
a) This is a CLOSED BOOK and CLOSED NOTES examination.
b) The exam is 180 minutes in length.
c)
SHOW YOUR WORK: In order to receive credit for your answers, you must show your work.
Correct answers with no work shown will not receive any credit. Incorrect answers with partial correct work may receive partial credit.
d) Answer all questions DIRECTLY ON THE EXAM
EXAM.
e) This exam has a total of 17 PAGES including the cover sheet and formula sheet. You may
PAGES,
detach the formula sheet if you like.
f)
There are a total of 11 questions worth a total …show more content…
of 100 points points. g)
STANDARD CALCULATOR or FINANCIAL CALCULATOR permitted ONLY.
h) This examination paper MUST BE RETURNED
RETURNED.
MGCR 341
Page 1 of 17
1) (4 points, 5 minutes) A bank is offering a 12% APR with monthly compounding on deposits. You deposit $300 in the bank at t = 0, and then make another deposit of $X at month t = 8. You then let the money stay in the bank until month t = 35. At month t = 35 you make a $1,000 withdrawal and after that withdrawal you have $20,000 left in the bank at month t = 35. Write down one equation where the only unknown is X. You do not need to isolate or solve for X.
Monthly rate = 12%/12 = 1%
300(1.01)35 + X(1.01)35-8 = 1,000 + 20,000 or PV(deposits) = PV(withdrawals) + PV(final balance)
300 + X/1.018 = 1,000/1.0135 + 20,000/1.0135
2) (4 points, 10 minutes) Today is t = 0 and the yield curve is flat at 5%. Consider a “2-year, 8% real coupon bond”. Specifically, the bond payment at t = 1 will be enough to buy 8 hamburgers at t = 1, and the total bond payment at t = 2 will be enough to buy 108 hamburgers at t = 2. If a hamburger costs $1 at t = 0 and the inflation rate is 2%, what is the fair price of the bond?
Real approach: rr = 1.05/1.02 – 1 = 2.94%
P0 = 8/1.0294 + 108/1.02942 = 109.69
Nominal approach:
P0 = [8(1.02)]/1.05 + [108(1.022)]/1.052 = 109.69
MGCR 341
Page 2 of 17
3) (8 points, 15 minutes) Today is month t = 0. A bank is offering you the following financial product:
You make monthly payments to the bank of $400 per month from month t = 15 to month t = 254. In exchange, the bank will give you monthly payments of X from month t = 255 to month t = 494.
The annual interest rate (APR) is 12%, with monthly compounding.
The inflation rate is 0.2% per month.
The contract is priced in such a way that the bank makes a $5,000 profit in PV terms.
a) Write down an equation where the only unknown is X (4 points).
Monthly rate = 12%/12 = 1%
ࢄ
൬ −
൰
=
൬ −
൰
+ ,
(ିା) .
(ૢିା) .
.
.
.
.
You could also use the real approach, but that would take longer.
MGCR 341
Page 3 of 17
(Question 3 continued)
Suppose that at t = 254, the bank makes you the following offer: Instead of giving you monthly payments of X from t = 255 to t = 494, the bank will pay you a lump sum of
$380,000 at t = 254, and another lump sum of $350,000 at t = 354.
Assume that after some analysis you decide to accept this alternative lump sum offer.
b) Suppose your goal in retirement is to consume a certain amount of hamburgers at t = 255, and each month thereafter you want to consume 5% fewer hamburgers until t = 300, and then each month after that you want to consume 15% fewer hamburgers until t = 494. If a hamburger costs $1 at t = 0, how many hamburgers would you be consuming at t = 255, based only on the alternative lump sum offer you accepted from the bank? (4 points) Nominal approach:
First nominal growth rate: g1 = (1.002)(1-0.05) – 1
Second nominal growth rate: g2 = (1.002)(1-0.15) – 1
Let C be the amount you are spending on hamburgers at t = 255.
۱
( + ࢍ )ିା
ቆ −
ቇ
. − ࢍ
. ିା
.
ି
۱( + ࢍ )
( + ࢍ )
( + ࢍ )ૢିା
+
ቆ −
ቇ
.
− ࢍ
. ૢିା
.
ૡ, ,
=
+
.
.
C = 30,958.51
(Note: You could have also gotten the same result by doing the analysis in terms of value as of 254 instead of PV. You could have also split up the cash flows differently. For example, instead of 255 to 300 and 301 to 494 you could have done 255 to 299 and 300 to 494.)
Now that we know how much is spent at t = 255, we can convert that to hamburgers at t = 255:
, ૢૡ.
= ૡ, ૢૢ. ૢ
.
MGCR 341
Page 4 of 17
Real approach (for left hand side):
You could also have solved this using the real approach (for all terms, or more simply for the terms on the left as shown below) where rr = (1.01/1.002)-1 = 0.7984%:
Let C be the number of hamburgers purchased at t = 255.
۱
( − . )ିା
ቆ −
ቇ
࢘࢘ + .
( + ࢘࢘ )ିା ( + ࢘࢘ )
۱( − . )ି ( − . )
( − . )ૢିା
+
ቆ −
ቇ
ૢିା
࢘࢘ + .
( + ࢘࢘ )
( + ࢘࢘ )
ૡ, ,
=
+
.
.
C = ૡ, ૢૢ. ૢ
where C is already in terms of hamburgers since we are using the real approach on the left hand side.
(Note: as long as you set up the equation correctly, no points were deducted for
calculation errors.) MGCR 341
Page 5 of 17
4) (9 points, 15 minutes) You are given information about the following zero coupon bonds. Bonds A and B are denominated in Dollars ($). Bonds C and D are denominated in Euros (€).
i. Bond A: Maturity = 1 year, Face Value = $2,000, Price = $1,900 ii. Bond B: Maturity = 2 years, Face Value = $3,000, Price = $2,800 iii. Bond C: Maturity = 1 year, Face Value = €1,000, Price = €950 iv. Bond D: Maturity = 2 years, Face Value = €1,000, Price = €900
At t = 0, the exchange rate is 1 Dollar equals 1 Euro. Do not, however, assume that the exchange rate will remain constant over time.
Bond E is a 2-year double currency bond with the following payments: $100 and €200 at t = 1, and $1,100 and €2,200 Euros at t = 2.
a) What is the fair price of bond E in Dollars? (Hint: use the discount factor logic.) (4 points)
For a given currency, the ratio of a t-year zero coupon bond price to face value is the cost
(in that currency) of receiving one unit of currency at time t. And since at t = 0 one Euro equals one dollar it follows that the fair price of the bond is:
=
, ૢ
ૢ
, ૡ
ૢ
+
( ) + ,
+ ,
( ) = , ૢ. ૠ
,
,
,
,
Longer approach: You could have also backed out the 1-year and 2-year risk free rates in both currencies and discounted the cash flows. Note that for a given maturity there is no reason that the risk free rate in one currency should equal to the risk free rate in another currency. For further explanation on why that is the case you may want to look up the concept of “covered interest rate parity”.
b) Suppose bond E were trading at a price that was $10 greater than your answer to part a.
Describe how you would construct an arbitrage. Specifically, you will short 1 unit of bond E.
You must specify how many units of each of bonds A, B, C, and D you will be trading, and make sure to specify whether you are buying or shorting each bond. (5 points)
Short 1 unit of bond E, and buy A, B, C, and D units of bonds A, B, C, and D, respectively.
For t = 1 and t = 2 you need your $ inflow to equal your $ outflow, and you also need your
Euro inflow to equal your Euro outflow. So there are 4 conditions in total:
MGCR 341
Page 6 of 17
INFLOW = OUTFLOW
2000A = 100
($ t = 1 condition)
3000B = 1,100
($ t = 2 condition)
1000C = 200
(€ t = 1 condition)
1000D = 2,200
(€ t = 2 condition)
A = 0.05
B = 0.37
C = 0.2
D = 2.2
Thus, the arbitrage is
Short 1 unit of E
Buy 0.05 units of A
Buy 0.3667 units of B
Buy 0.2 units of C
Buy 2.2 units of D
Although not asked to compute it, you can verify that the magnitude of the arbitrage profit is $10.
MGCR 341
Page 7 of 17
5) (8 points, 16 minutes) At t = 0 the yield curve is flat at 5%. Imagine the following very simplified pension fund: At t = 0 the pension fund takes in a $10,000 contribution from an employee. The pension fund is a defined benefit plan and guarantees that it will pay the employee $5,000(1.05)15 at t = 15 and $5,000(1.05)25 at t = 25. Suppose the fund manager running the pension plan decides to invest the $10,000 in a 30-year zero coupon bond. Now, suppose that at t = 0 the yield curve shifts up to 5.3% (parallel shift of the entire yield curve).
a) Calculate the exact amount by which the pension plan is underfunded at t = 0 after the yield curve has shifted up to 5.3%. Answer this question by directly computing the new values of the assets and liabilities, and not by using the Modified Duration approach. (2 points)
The face value of the 30-year zero coupon bond is 10,000(1.05)30. After the yield curve shift, the amount of underfunding is
, (. ) , (. ) , (. )
+
−
= .
.
.
.
b) Answer part a again, but this time using the Modified Duration methodology. (2 points)
∆ࡼࢀ࢚ࢇ = ∆ࡼ − ∆ࡼ − ∆ࡼ
∆ࡼࢀ࢚ࢇ = ࡼ, (−ࡹࡰ )(. ) − ࡼ, (−ࡹࡰ )(. ) − ࡼ, (−ࡹࡰ )(. )
൰ (. ) − , ൬−
൰ (. )
.
.
൰ (. ) = −ૡ. ૠ
− , ൬−
.
∆ࡼࢀ࢚ࢇ = , ൬−
The (approximate) amount of underfunding is 285.71.
MGCR 341
Page 8 of 17
(Question 5, continued)
c) Suppose that after the yield curve shift the pension fund decides it no longer wants to be exposed to parallel shifts in the yield curve. It decides to liquidate the position in the 30-year bond, and invest those proceeds, plus your answer to part a in the following 2 bonds: $5,000 worth invested in a 10-year zero coupon bond, and the rest invested in a T-year zero coupon bond. What would T have to be in order for the fund to no longer be exposed to parallel shifts in the yield curve? You do not need to isolate or solve numerically for T, just set up an equation where the only unknown is T, but make sure there are no other variables in the equation. (4 points) ࡼ,ࢀ
, (. )
=
+ . − , = , .
.
∆ࡼࢀ࢚ࢇ = ∆ࡼࢀ − ∆ࡼ − ∆ࡼ
∆ࡼࢀ࢚ࢇ = ࡼ,ࢀ (−ࡹࡰࢀ )∆࢟ࢀ + ࡼ, (−ࡹࡰ )∆࢟ − ࡼ, (−ࡹࡰ )∆࢟
− ࡼ, (−ࡹࡰ )∆࢟
ࢀ
൰ ∆࢟ࢀ + , (−
)∆࢟
.
.
, (. )
, (. )
−
(−
)∆࢟ −
(−
)∆࢟
.
.
.
.
∆ࡼࢀ࢚ࢇ = , . ൬−
Note that the yield curve has already shifted to 5.3%. So our new P initials in the 15 and 25 year bonds are no longer 5,000.
Mathematically, a parallel shift in the yield curve means that all delta y’s are the same (call this common change simply delta y). When this happens, we want ∆ࡼࢀ࢚ࢇ to equal 0.
ࢀ
, (. )
= , . ൬−
൰ ∆࢟ + , (−
)∆࢟ −
(−
)∆࢟
.
.
.
.
, (. )
−
(−
)∆࢟
.
.
We can divide both sides of the equation by delta y.
ࢀ
, (. )
൰ + , (−
)−
(−
)
.
.
.
.
, (. )
−
(−
)
.
.
= , . ൬−
MGCR 341
Page 9 of 17
6) (5 points, 5 minutes) Given the following information:
•
Portfolio X has a Sharpe ratio of 0.2
•
Portfolio Y has an expected return of 10%
•
Expected return on the market portfolio = 13%
•
Standard deviation of the market portfolio = 20%
•
Risk free rate = 3%
•
The CAPM holds and you can trade the risk free asset
a) Construct an optimal portfolio that has a standard deviation of 40%. (2.5 points)
Since the CAPM holds, optimal portfolios are combinations of the risk free asset and the market portfolio.
40% = wm(20%) wm = 2 wf = -1
b) Construct an optimal portfolio that has an expected return of 50%. (2.5 points)
Since the CAPM holds, optimal portfolios are combinations of the risk free asset and the market portfolio.
50% = wm(13%) + (1-wm)(3%) wm = 4.7 wf = -3.7
MGCR 341
Page 10 of 17
7) (9 points, 15 minutes) The risk free rate is 5%. Lobee has mean-variance preferences. You are given the following information:
• Portfolio A has an expected return of 15% and a standard deviation of 20%
• Portfolio B has a Sharpe ratio of 0.6
• Portfolio C has a standard deviation of 5% and an expected return of X
• Stock D has a standard deviation of 45%, a beta of 1.3 and an expected return of Y
• The market portfolio has an expected return of 15%
• The CAPM may or may not hold
a) If Lobee has to invest 100% of his wealth in either portfolio A or B, can you say which he would prefer? If yes, state which one, and explain why. If no, explain why not. (3 points)
You do not have enough information to conclude which portfolio is preferred.
b) Suppose Lobee is allowed to mix the risk free asset with one of either A, B, or C. You observe that he has decided to invest 20% in the risk free asset and 80% in portfolio C. Based on his actions, what can you say about the value of X, the expected return of portfolio C? (3 points)
Lobee will want to mix the risk free asset with the portfolio that has the highest Sharpe ratio.
SA = (15% – 5%)/20% = 0.5
SB = 0.6
Thus portfolio C must have a Sharpe ratio that is greater than or equal to 0.6.
SC = (X – 5%)/5% ≥ 0.6
X ≥ 8%
(Note: I also accepted X > 8%)
c) Suppose now that Lobee is allowed to mix the risk free asset with the market portfolio and/or with Stock D. You observe that he has selected the following portfolio: 100% in the market portfolio,
-1% in the risk free asset and 1% in Stock D. Based on this information, what can you say about the value of Y, the expected return on stock D? (3 points)
If the CAPM holds, optimal portfolios are mixtures of the market portfolio and the risk free asset.
Since Lobee’s optimal portfolio also has some extra Stock D in it, the CAPM must not hold. Indeed,
Lobee’s portfolio must have a higher Sharpe ratio than the market (otherwise he would have just mixed the risk free with the market). Based on the proof of the CAPM formula, this situation occurs because stock D’s expected return is not given by the CAPM formula. Since there is a positive weight invested in D, it must be that E[rD] > 5% + 1.3(15% – 5%) = 18%.
Based on the above we can conclude that Y > 18%.
(Note: I also accepted Y ≠ 18%.)
MGCR 341
Page 11 of 17
8) (10 points, 15 minutes) Today is t = 0. The market believes Morgan Stanley will pay a $2 dividend at t = 1. But the market also believes that Morgan Stanley’s t = 2 dividend will be cut to $1 due to real estate related losses. The market believes that thereafter dividends are expected to grow at a 3% rate until t = 7, and then grow at a 6% rate until t = 12, and then grow at a rate of g in perpetuity.
The market also believes that at t = 9 the stock price will be trading based on a P9/Div10 ratio of 8.
The discount rate is 10%.
a) Write down an equation where the only unknown is P0, the price of Morgan Stanley at t = 0? You do not need to solve for P0. (2.5 points)
Since we do not know g, but we do know something about the sale price at t = 9, we will imagine a
9-year investment horizon.
ࡼ =
.
(. )(. )
.
+
+
ቆ −
ቇ
ቆ −
ቇ
. . − .
. .
. − .
. . ૠ
ૡ[()൫. ൯൫. ൯]
+
. ૢ
The second term represents the PV of Div2 to Div7, the third term represents the PV of Div8 to Div9, and the fourth term represents the PV of P9, where is P9 = 8[Div10]
There are other ways of solving this. For example the dividends can be broken up slightly differently. Or you could figure out what g is (as is shown in part b) and then write out P0 based on the PV of all dividends assuming an infinite investment horizon.
b) Write down an equation where the only unknown is g? You do not need to isolate or solve for g.
(2.5 points)
ૡൣ()൫. ൯൫. ൯൧ =
(. )(. )
.
(. )(. )
ቆ −
ቇ+
. − .
.
. − ࢍ
.
The left hand side is the expected price at t = 9 based on P9 = 8[Div10].
The right hand side represents Div10 to Div11 discounted back to t = 9 and also Div12 to Divinfinity discounted back to t = 9.
MGCR 341
Page 12 of 17
Alternatively:
ૡൣ()൫. ൯൫. ൯൧ =
(. )(. )
.
(. )(. )( + ࢍ)
ቆ −
ቇ+
. − .
.
. − ࢍ
.
The left hand side is the expected price at t = 9 based on P9 = 8[Div10].
The right hand side represents Div10 to Div12 discounted back to t = 9 and also Div13 to Divinfinity discounted back to t = 9.
Alternatively
ࡼ =
.
(. )(. )
.
+
+
ቆ −
ቇ
ቆ −
ቇ
. . − .
. .
. − .
. . ૠ
(. )(. )
+
. − ࢍ
.
Where P0 is based on the expression in part a.
Alternatively
ࡼ =
.
(. )(. )
.
+
+
ቆ −
ቇ
ቆ −
ቇ
. . − .
. .
. − .
. . ૠ
(. )(. )( + ࢍ)
+
. − ࢍ
.
Where P0 is based on the expression in part a.
There are other variations as well.
c) What is the expected return (including the dividend) on the stock from t = 0 to t = 1? (2.5 points)
Expected return equals required rate of return = 10%
You could also try doing this the long way by first finding P1.
MGCR 341
Page 13 of 17
(Question 8, continued)
d) Suppose you believe the following: Morgan Stanley will pay a $3 dividend at t = 1, a $2 dividend at t = 2, a $4 dividend at t = 3 and that dividends will remain constant thereafter. You also believe that at t = 2 the market will revise its views on Morgan Stanley. Specifically, you believe that at t = 2 the market will believe that Div3 will equal $5, Div4 will equal $6, Div5 will equal $7 and that the
P4/Div5 ratio will equal 10. Assuming you have a 2 year investment horizon, write down an equation where the only unknown is H0, the most you would be willing to pay for Morgan Stanley at t = 0?
(2.5 points)
Use your views for dividends up to t = 2, then use market views to figure out the sale price at t = 2.
where
where
ࡴ =
ࡼ
+
+
. .
.
ࡼ =
ࡼ
+
+
. .
.
ࡼ = (ૠ) = ૠ
So,
ૠ
+
+
. . .
ࡴ =
+
+
. .
.
MGCR 341
Page 14 of 17
9) (10 points, 15 minutes) Companies A and B are expected to pay dividends in the form of a growing perpetuity. All earnings are paid out as dividends, so earnings = dividends. For each of the below, circle the best answer and justify briefly.
a) A has a higher dividend growth rate than B, and A has a lower required rate of return than B. Which company should have a higher PE ratio? Justify briefly. (2.5 points)
i) A ii) B iii) Can’t tell iv) They should have the same PE ratio
Higher growth and lower required rate of return implies investors are willing to pay more per dollar of expected dividend, where dividends are equal to earnings in this question.
b) A and B have the same dividend growth rate and the same required rate of return, but A has a lower profit margin than B. (Profit margin is equal to earnings as a fraction of sales. Assume that the profit margin will remain constant.) Which company should have a higher PE ratio? Justify briefly. (2.5 points)
i) A ii) B iii) Can’t tell iv) They should have the same PE ratio
Given that the dividend growth rate and required rate of return are the same, it does not matter what the profit margin is. Also, here earnings equal dividends.
c) A and B have the same growth rate and the same required rate of return, but company A has a lower profit margin than company B. (Profit margin is equal to earnings as a fraction of sales. Assume that the profit margin will remain constant.) Which company should be trading at a higher multiple of sales?
Justify briefly. (2.5 points)
i) A ii) B iii) Can’t tell iv) They should be trading at the same multiple of sales
That A has a lower profit margin implies
ࡼ
ࡼ
ࡱ
ࡿ
<
ࡱ
ࡿ
From part b, we know ࡱ = ࡱ
Multiplying both sides of the top expression by the respective side of the second expression:
MGCR 341
ࡱ ࡼ ࡱ ࡼ
ࡼ
< →
ࡿ ࡱ ࡿ ࡱ
ࡿ
<
ࡼ
ࡿ
Page 15 of 17
Thus, B is trading at a higher multiple of sales. Intuitively, investors are willing to pay more per dollar of sales for B because B is able to generate more earnings per $ of sales.
(And because dividend growth and required rate of return are the same.)
d) Now assume that dividends are not necessarily in the form of a growing perpetuity but that companies A and B have the same required rate of return and are both expected to have a 0% sales growth rate in perpetuity. Company B’s profit margin is constant. For t = 0 to t = 5 company A has a lower profit margin than company B. But for t = 6 to infinity, company A’s profit margin will equal that of company B. Which company should be trading at a higher PE ratio at t = 0? Justify briefly. (2.5 points)
i) A ii) B iii) Can’t tell iv) They should have the same PE ratio
Imagine that the profit margin for A were constant. In this case, both companies would have a zero dividend growth rate in perpetuity. This, together with the fact that they have the same required rate of return implies they should trade at the same PE ratio (since dividends are the same as earnings in this question).
But company A’s profit margin will increase in the future. Hence earnings will at some point increase.
Thus, A should be trading at a higher price than if its profit margin were constant. Thus at t = 0 company A should be trading at a higher PE ratio than company B.
MGCR 341
Page 16 of 17
10) (15 points, 30 minutes) Today is t = 0. Your steel company has an existing blast furnace and you are evaluating a proposal to replace it with a new furnace. You previously paid a consulting company
$3M for a feasibility study to determine the impact of replacing the furnace. The conclusions are outlined below. As part of your agreement with the consulting company, you have to pay the consulting company an extra $0.3M in cash at t = 0 if you decide to replace the furnace (assume this is a tax deductible “operating” expense at t = 0, and assume that at t = 0 you would have positive pretax income regardless of whether you decide to replace the furnace or not).
Ten years ago, at t = -10, you paid $50M for the old furnace, and for depreciation purposes you assumed a 20 year asset life, and a $0 salvage value.
The new furnace costs $105M at t = 0, and for depreciation purposes you will assume a 10 year asset life, and a $90M salvage value.
Regardless of whether you replace the furnace or not at t = 0, at t = 5 you will sell whichever furnace you have left. At t = 5, you believe you could sell the old furnace for $30M (before taxes), if you still have it, or the new furnace for $97.5M (before taxes). Note, however, that if you replace the furnace at t = 0, then you will sell the old furnace at t = 0 for $40M (before taxes).
The new furnace is eco-friendly, and if you replace the old furnace with the new furnace the government will allow you to pay $0.1M fewer taxes per year for t = 1 to t = 5.
Replacing the furnace will result in $13M greater revenue per year and will save $25M per year in operating expenses, from t = 1 to t = 5. The current sales and costs are $690M and $425M per year, respectively. Assume that for your company’s annual revenue, clients pay you 10% in cash at the time of sale, and the remaining 90% is paid in cash one year later. Assume that for your annual operating expenses, you pay 30% in cash at the time the expenses are incurred and the remaining 70% is paid in cash one year later. (Aside from these two points, all transactions are 100% in cash. For example, purchase/sale of capital assets (old/new furnace) is 100% in cash at the time of purchase/sale.)
The company’s tax rate is 40%. Assume that if the company does not replace the furnace, it will continue operations with the existing furnace until t = 5, and then sell the old furnace at t = 5. Fill in the table below indicating all the incremental values associated with replacing the furnace as appropriate. Use the space on the following page to show relevant calculations.
MGCR 341
Page 17 of 17
Incremental
Value
t=0
Operating
Expenses
Depreciation
13M
13M
25M
25M
25M
1M
(300,000)
t = 2 to 4
13M
Revenue
t=1
t=5
1M
t=6
1M
Operating profit
(300,000)
39M
39M
39M
Taxes
120,000
(40%*39M)
+0.1M
(40%*39M)
+0.1M
(40%*39M)
+0.1M
Net Income
(180,000)
23.5M
23.5M
23.5M
Depreciation
-1M
-1M
-1M
Accounts
Payable
-70%*25M
-70%*25M
70%*25M
-70%*25M
70%*25M
70%*25M
Accounts
Receiv.
-90%*13M
-90%*13M
90%*13M
-90%*13M
90%*13M
90%*13M
(excluding profit from sale of equipment) Adjustments:
Purchase/Sale
at t = 0
-105M
+34M
Purchase/Sale at t = 5
Net cash flow
MGCR 341
97.5M-23M
-71.18M
-6.7M
22.5M
97M
29.2M
Page 18 of 17
Use this page to show your work for Q10.
Extra 300,000 at t = 0 to consultants:
If you implement the project, you have to pay an extra 300,000 at t = 0. This, however, is a taxdeductible expense, which saves you from paying 40%(300,000) = 120,000 in taxes. (We assume the company has enough revenue to make the tax savings matter). Also, the $3M you already paid is a sunk cost and should be ignored.
Incremental depreciation for t = 1 to 5:
Previous depreciation expenses were (50 – 0)/20 = 2.5
New depreciation expense = (105 – 90)/10 = 1.5
So incremental depreciation expense is 1.5 – 2.5 = -1
(This is 1 less depreciation, which implies a positive contribution to operating profit. But since this is a non-cash contribution, you have to subtract it afterwards in the adjustments.)
Purchase of new machine and sale of old machine at t = 0:
If you implement project, it will cost you 105 at t = 0 to buy the new machine. You will sell the old furnace for 40, on which you have to pay taxes. The taxes equals the tax rate times the profit, where the profit equals the sale price minus the book value at time of sale, and where the book value at the time of sale = purchase price minus accumulated depreciation.
Net cash flow from sale at t = 0
= 40 – 40%(40 – (50 – 10(2.5))) = 34
Sale of new machine at t = 5
If you implement the project, then at t = 5, you sell the new machine for 97.5, on which you don’t have to pay taxes since the BV at t = 5 is also 97.5.
Opportunity Cost at t = 5
Note, if you implement the project, there is an opportunity cost of not being able to sell the old machine at t = 5, since you will have sold it at t = 0.
Opportunity cost net cash flow
= 30 – 40%(30 - (50 – 15(2.5))) = 23
Account Payable (AP) adjustments:
For t = 1 to 5 there is a savings of 25M in expenses. But not all those savings are cash savings since some of your expenses are paid on credit. So, for t = 1 to 5 you have to subtract 70% of that 25M. (25M less expenses implies 70% of 25M fewer AP. A decrease in AP needs to be subtracted from net income.)
But, the decreased 25M in expenses from t = 1 to 5 also implies that there will be 70% of 25M less cash being paid out from t = 2 to t = 6. This needs to be added back for t = 2 to 6.
Account Receivable (AR) adjustments:
For t = 1 to 5 there is 13M in extra revenue, but not all that increase in revenue is a cash increase since some of your sales are made on credit. So, for t = 1 to 5 you have to subtract off 90% of the 13M.
The increase of 13M in revenue from t = 1 to 5 also implies that there will be 90% of 13M more cash being received from clients for t = 2 to t = 6. This needs to be added for t = 2 to 6.
MGCR 341
Page 19 of 17
11) (18 points, 20 minutes) True or False: write True or False directly to the left of the statements below.
No explanation is required.
a) It is possible for markets to be efficient even if a stock has a statistically significant positive alpha relative to the CAPM. True. The CAPM could be the wrong model of risk control.
b) The value effect, also known as the book-to-market effect, is a reference to the fact that historically stocks with high book-to-market ratios have, on average, outperformed stocks with low book-to-market ratios, even after controlling for beta risk. True
c) If markets are weak form efficient, then strategies based on momentum should be able to generate superior risk-adjusted returns. False
d) According to the single index model, firm-specific shocks are correlated to the market return.
False
e) If two stocks have a positive correlation, then there are no benefits of diversification.
False. There are benefits as long as the correlation is less than 1.
f)
Stock A has an abnormally low PE ratio at the moment because earnings are high, but are expected to drop significantly in the future. This suggests that stock A’s stock price is expected to drop going forward. False. The price at t = 0 already incorporates the fact that earnings will drop. MGCR 341
Page 20 of 17
MGCR 341: Formula Sheet
HPR =
annualized HPR = (1 + HPR )1 / t − 1
ending price
−1
beginning price
ending price
=
beginning price
S
E[r ] = p1r1 + .... + pS rS = ∑ pi ri
T
cov[rA , rB ]
y realized
σ Aσ B
FW
=
beginning price
E ( X + Y ) = E ( X ) + E (Y )
E (aX + bY ) = aE ( X ) + bE (Y )
−1
Var (aX ) = a 2Var ( X )
Cov (aX , bY ) = abCov ( X , Y )
Var (aX + bY ) = a 2Var ( X ) + b 2Var (Y ) + 2abCov ( X , Y )
Var ( X ) = 0 if X is a constant
E (rp ) − r f
Cov ( X , X ) = Var ( X )
Cov ( X , Y ) = 0 if either X or Y is a constant
σp
σ 2 = ∑ w i2 ⋅ σ i2 + 2 ∑ w i w j cov(ri , r j ) p n
E(rp ) = ∑i =1 w i ⋅ E(ri ) n i =1
E[ri ] = rf + β i (E[rm ] − rf ) βi =
1/ t
Var (a + X ) = Var ( X )
E (a + X ) = a + E ( X )
E (aX ) = aE ( X )
S=
−1
PBond (1 + y realized ) = FW
i =1
ρ A, B =
1/ t
cov(rm , ri ) σ i,m σ = 2 = i ρ i, m var(rm ) σm σm
i≠ j
σ 2 = w 2 ⋅ σ 2 + w 2 ⋅ σ 2 + 2w A w B cov(rA , rB ) p A
A
B
B
β P = w 1 × β1 + ... + w N × β N
( F + C n ) / (1 + y )
C1 / (1 + y )
C / (1 + y )
+ 2× 2
+ ... + n ×
P
P
P
2
D = 1×
MD = D / (1 + YTM)
∆P
≈ − MD × ∆y
P
PV (perpetuity) = C/r
PV (annuity ) =
PV (growing perpetuity) = C/(r - g)
PV ( growing _ annuity ) =
n
C
1
1 −
(1 + r ) n r
C (1 + g )
1 − r - g (1 + r ) n
n
APR
1 + EAR = 1 +
k
MGCR 341
k
1 + rr =
1+ r
1+ i
CFreal,t =
CFnominal,t
(1 + i) t
Page 21 of 17
Final Examination
Finance 1
MGCR 341
April 24, 2:00PM-5:00PM
FINAL EXAM: Solutions
Examiner:
Vadim di Pietro
Student Name:
McGill ID:
INSTRUCTIONS:
a) This is a CLOSED BOOK and CLOSED NOTES examination.
b) The exam is 180 minutes in length.
c)
SHOW YOUR WORK: In order to receive credit for your answers, you must show your work.
Correct answers with no work shown will not receive any credit. Incorrect answers with partial correct work may receive partial credit.
d) Answer all questions DIRECTLY ON THE EXAM
EXAM.
e) This exam has a total of 17 PAGES including the cover sheet and formula sheet. You may
PAGES,
detach the formula sheet if you like.
f)
There are a total of 11 questions worth a total …show more content…
of 100 points points. g)
STANDARD CALCULATOR or FINANCIAL CALCULATOR permitted ONLY.
h) This examination paper MUST BE RETURNED
RETURNED.
MGCR 341
Page 1 of 17
1) (4 points, 5 minutes) A bank is offering a 12% APR with monthly compounding on deposits. You deposit $300 in the bank at t = 0, and then make another deposit of $X at month t = 8. You then let the money stay in the bank until month t = 35. At month t = 35 you make a $1,000 withdrawal and after that withdrawal you have $20,000 left in the bank at month t = 35. Write down one equation where the only unknown is X. You do not need to isolate or solve for X.
Monthly rate = 12%/12 = 1%
300(1.01)35 + X(1.01)35-8 = 1,000 + 20,000 or PV(deposits) = PV(withdrawals) + PV(final balance)
300 + X/1.018 = 1,000/1.0135 + 20,000/1.0135
2) (4 points, 10 minutes) Today is t = 0 and the yield curve is flat at 5%. Consider a “2-year, 8% real coupon bond”. Specifically, the bond payment at t = 1 will be enough to buy 8 hamburgers at t = 1, and the total bond payment at t = 2 will be enough to buy 108 hamburgers at t = 2. If a hamburger costs $1 at t = 0 and the inflation rate is 2%, what is the fair price of the bond?
Real approach: rr = 1.05/1.02 – 1 = 2.94%
P0 = 8/1.0294 + 108/1.02942 = 109.69
Nominal approach:
P0 = [8(1.02)]/1.05 + [108(1.022)]/1.052 = 109.69
MGCR 341
Page 2 of 17
3) (8 points, 15 minutes) Today is month t = 0. A bank is offering you the following financial product:
You make monthly payments to the bank of $400 per month from month t = 15 to month t = 254. In exchange, the bank will give you monthly payments of X from month t = 255 to month t = 494.
The annual interest rate (APR) is 12%, with monthly compounding.
The inflation rate is 0.2% per month.
The contract is priced in such a way that the bank makes a $5,000 profit in PV terms.
a) Write down an equation where the only unknown is X (4 points).
Monthly rate = 12%/12 = 1%
ࢄ
൬ −
൰
=
൬ −
൰
+ ,
(ିା) .
(ૢିା) .
.
.
.
.
You could also use the real approach, but that would take longer.
MGCR 341
Page 3 of 17
(Question 3 continued)
Suppose that at t = 254, the bank makes you the following offer: Instead of giving you monthly payments of X from t = 255 to t = 494, the bank will pay you a lump sum of
$380,000 at t = 254, and another lump sum of $350,000 at t = 354.
Assume that after some analysis you decide to accept this alternative lump sum offer.
b) Suppose your goal in retirement is to consume a certain amount of hamburgers at t = 255, and each month thereafter you want to consume 5% fewer hamburgers until t = 300, and then each month after that you want to consume 15% fewer hamburgers until t = 494. If a hamburger costs $1 at t = 0, how many hamburgers would you be consuming at t = 255, based only on the alternative lump sum offer you accepted from the bank? (4 points) Nominal approach:
First nominal growth rate: g1 = (1.002)(1-0.05) – 1
Second nominal growth rate: g2 = (1.002)(1-0.15) – 1
Let C be the amount you are spending on hamburgers at t = 255.
۱
( + ࢍ )ିା
ቆ −
ቇ
. − ࢍ
. ିା
.
ି
۱( + ࢍ )
( + ࢍ )
( + ࢍ )ૢିା
+
ቆ −
ቇ
.
− ࢍ
. ૢିା
.
ૡ, ,
=
+
.
.
C = 30,958.51
(Note: You could have also gotten the same result by doing the analysis in terms of value as of 254 instead of PV. You could have also split up the cash flows differently. For example, instead of 255 to 300 and 301 to 494 you could have done 255 to 299 and 300 to 494.)
Now that we know how much is spent at t = 255, we can convert that to hamburgers at t = 255:
, ૢૡ.
= ૡ, ૢૢ. ૢ
.
MGCR 341
Page 4 of 17
Real approach (for left hand side):
You could also have solved this using the real approach (for all terms, or more simply for the terms on the left as shown below) where rr = (1.01/1.002)-1 = 0.7984%:
Let C be the number of hamburgers purchased at t = 255.
۱
( − . )ିା
ቆ −
ቇ
࢘࢘ + .
( + ࢘࢘ )ିା ( + ࢘࢘ )
۱( − . )ି ( − . )
( − . )ૢିା
+
ቆ −
ቇ
ૢିା
࢘࢘ + .
( + ࢘࢘ )
( + ࢘࢘ )
ૡ, ,
=
+
.
.
C = ૡ, ૢૢ. ૢ
where C is already in terms of hamburgers since we are using the real approach on the left hand side.
(Note: as long as you set up the equation correctly, no points were deducted for
calculation errors.) MGCR 341
Page 5 of 17
4) (9 points, 15 minutes) You are given information about the following zero coupon bonds. Bonds A and B are denominated in Dollars ($). Bonds C and D are denominated in Euros (€).
i. Bond A: Maturity = 1 year, Face Value = $2,000, Price = $1,900 ii. Bond B: Maturity = 2 years, Face Value = $3,000, Price = $2,800 iii. Bond C: Maturity = 1 year, Face Value = €1,000, Price = €950 iv. Bond D: Maturity = 2 years, Face Value = €1,000, Price = €900
At t = 0, the exchange rate is 1 Dollar equals 1 Euro. Do not, however, assume that the exchange rate will remain constant over time.
Bond E is a 2-year double currency bond with the following payments: $100 and €200 at t = 1, and $1,100 and €2,200 Euros at t = 2.
a) What is the fair price of bond E in Dollars? (Hint: use the discount factor logic.) (4 points)
For a given currency, the ratio of a t-year zero coupon bond price to face value is the cost
(in that currency) of receiving one unit of currency at time t. And since at t = 0 one Euro equals one dollar it follows that the fair price of the bond is:
=
, ૢ
ૢ
, ૡ
ૢ
+
( ) + ,
+ ,
( ) = , ૢ. ૠ
,
,
,
,
Longer approach: You could have also backed out the 1-year and 2-year risk free rates in both currencies and discounted the cash flows. Note that for a given maturity there is no reason that the risk free rate in one currency should equal to the risk free rate in another currency. For further explanation on why that is the case you may want to look up the concept of “covered interest rate parity”.
b) Suppose bond E were trading at a price that was $10 greater than your answer to part a.
Describe how you would construct an arbitrage. Specifically, you will short 1 unit of bond E.
You must specify how many units of each of bonds A, B, C, and D you will be trading, and make sure to specify whether you are buying or shorting each bond. (5 points)
Short 1 unit of bond E, and buy A, B, C, and D units of bonds A, B, C, and D, respectively.
For t = 1 and t = 2 you need your $ inflow to equal your $ outflow, and you also need your
Euro inflow to equal your Euro outflow. So there are 4 conditions in total:
MGCR 341
Page 6 of 17
INFLOW = OUTFLOW
2000A = 100
($ t = 1 condition)
3000B = 1,100
($ t = 2 condition)
1000C = 200
(€ t = 1 condition)
1000D = 2,200
(€ t = 2 condition)
A = 0.05
B = 0.37
C = 0.2
D = 2.2
Thus, the arbitrage is
Short 1 unit of E
Buy 0.05 units of A
Buy 0.3667 units of B
Buy 0.2 units of C
Buy 2.2 units of D
Although not asked to compute it, you can verify that the magnitude of the arbitrage profit is $10.
MGCR 341
Page 7 of 17
5) (8 points, 16 minutes) At t = 0 the yield curve is flat at 5%. Imagine the following very simplified pension fund: At t = 0 the pension fund takes in a $10,000 contribution from an employee. The pension fund is a defined benefit plan and guarantees that it will pay the employee $5,000(1.05)15 at t = 15 and $5,000(1.05)25 at t = 25. Suppose the fund manager running the pension plan decides to invest the $10,000 in a 30-year zero coupon bond. Now, suppose that at t = 0 the yield curve shifts up to 5.3% (parallel shift of the entire yield curve).
a) Calculate the exact amount by which the pension plan is underfunded at t = 0 after the yield curve has shifted up to 5.3%. Answer this question by directly computing the new values of the assets and liabilities, and not by using the Modified Duration approach. (2 points)
The face value of the 30-year zero coupon bond is 10,000(1.05)30. After the yield curve shift, the amount of underfunding is
, (. ) , (. ) , (. )
+
−
= .
.
.
.
b) Answer part a again, but this time using the Modified Duration methodology. (2 points)
∆ࡼࢀ࢚ࢇ = ∆ࡼ − ∆ࡼ − ∆ࡼ
∆ࡼࢀ࢚ࢇ = ࡼ, (−ࡹࡰ )(. ) − ࡼ, (−ࡹࡰ )(. ) − ࡼ, (−ࡹࡰ )(. )
൰ (. ) − , ൬−
൰ (. )
.
.
൰ (. ) = −ૡ. ૠ
− , ൬−
.
∆ࡼࢀ࢚ࢇ = , ൬−
The (approximate) amount of underfunding is 285.71.
MGCR 341
Page 8 of 17
(Question 5, continued)
c) Suppose that after the yield curve shift the pension fund decides it no longer wants to be exposed to parallel shifts in the yield curve. It decides to liquidate the position in the 30-year bond, and invest those proceeds, plus your answer to part a in the following 2 bonds: $5,000 worth invested in a 10-year zero coupon bond, and the rest invested in a T-year zero coupon bond. What would T have to be in order for the fund to no longer be exposed to parallel shifts in the yield curve? You do not need to isolate or solve numerically for T, just set up an equation where the only unknown is T, but make sure there are no other variables in the equation. (4 points) ࡼ,ࢀ
, (. )
=
+ . − , = , .
.
∆ࡼࢀ࢚ࢇ = ∆ࡼࢀ − ∆ࡼ − ∆ࡼ
∆ࡼࢀ࢚ࢇ = ࡼ,ࢀ (−ࡹࡰࢀ )∆࢟ࢀ + ࡼ, (−ࡹࡰ )∆࢟ − ࡼ, (−ࡹࡰ )∆࢟
− ࡼ, (−ࡹࡰ )∆࢟
ࢀ
൰ ∆࢟ࢀ + , (−
)∆࢟
.
.
, (. )
, (. )
−
(−
)∆࢟ −
(−
)∆࢟
.
.
.
.
∆ࡼࢀ࢚ࢇ = , . ൬−
Note that the yield curve has already shifted to 5.3%. So our new P initials in the 15 and 25 year bonds are no longer 5,000.
Mathematically, a parallel shift in the yield curve means that all delta y’s are the same (call this common change simply delta y). When this happens, we want ∆ࡼࢀ࢚ࢇ to equal 0.
ࢀ
, (. )
= , . ൬−
൰ ∆࢟ + , (−
)∆࢟ −
(−
)∆࢟
.
.
.
.
, (. )
−
(−
)∆࢟
.
.
We can divide both sides of the equation by delta y.
ࢀ
, (. )
൰ + , (−
)−
(−
)
.
.
.
.
, (. )
−
(−
)
.
.
= , . ൬−
MGCR 341
Page 9 of 17
6) (5 points, 5 minutes) Given the following information:
•
Portfolio X has a Sharpe ratio of 0.2
•
Portfolio Y has an expected return of 10%
•
Expected return on the market portfolio = 13%
•
Standard deviation of the market portfolio = 20%
•
Risk free rate = 3%
•
The CAPM holds and you can trade the risk free asset
a) Construct an optimal portfolio that has a standard deviation of 40%. (2.5 points)
Since the CAPM holds, optimal portfolios are combinations of the risk free asset and the market portfolio.
40% = wm(20%) wm = 2 wf = -1
b) Construct an optimal portfolio that has an expected return of 50%. (2.5 points)
Since the CAPM holds, optimal portfolios are combinations of the risk free asset and the market portfolio.
50% = wm(13%) + (1-wm)(3%) wm = 4.7 wf = -3.7
MGCR 341
Page 10 of 17
7) (9 points, 15 minutes) The risk free rate is 5%. Lobee has mean-variance preferences. You are given the following information:
• Portfolio A has an expected return of 15% and a standard deviation of 20%
• Portfolio B has a Sharpe ratio of 0.6
• Portfolio C has a standard deviation of 5% and an expected return of X
• Stock D has a standard deviation of 45%, a beta of 1.3 and an expected return of Y
• The market portfolio has an expected return of 15%
• The CAPM may or may not hold
a) If Lobee has to invest 100% of his wealth in either portfolio A or B, can you say which he would prefer? If yes, state which one, and explain why. If no, explain why not. (3 points)
You do not have enough information to conclude which portfolio is preferred.
b) Suppose Lobee is allowed to mix the risk free asset with one of either A, B, or C. You observe that he has decided to invest 20% in the risk free asset and 80% in portfolio C. Based on his actions, what can you say about the value of X, the expected return of portfolio C? (3 points)
Lobee will want to mix the risk free asset with the portfolio that has the highest Sharpe ratio.
SA = (15% – 5%)/20% = 0.5
SB = 0.6
Thus portfolio C must have a Sharpe ratio that is greater than or equal to 0.6.
SC = (X – 5%)/5% ≥ 0.6
X ≥ 8%
(Note: I also accepted X > 8%)
c) Suppose now that Lobee is allowed to mix the risk free asset with the market portfolio and/or with Stock D. You observe that he has selected the following portfolio: 100% in the market portfolio,
-1% in the risk free asset and 1% in Stock D. Based on this information, what can you say about the value of Y, the expected return on stock D? (3 points)
If the CAPM holds, optimal portfolios are mixtures of the market portfolio and the risk free asset.
Since Lobee’s optimal portfolio also has some extra Stock D in it, the CAPM must not hold. Indeed,
Lobee’s portfolio must have a higher Sharpe ratio than the market (otherwise he would have just mixed the risk free with the market). Based on the proof of the CAPM formula, this situation occurs because stock D’s expected return is not given by the CAPM formula. Since there is a positive weight invested in D, it must be that E[rD] > 5% + 1.3(15% – 5%) = 18%.
Based on the above we can conclude that Y > 18%.
(Note: I also accepted Y ≠ 18%.)
MGCR 341
Page 11 of 17
8) (10 points, 15 minutes) Today is t = 0. The market believes Morgan Stanley will pay a $2 dividend at t = 1. But the market also believes that Morgan Stanley’s t = 2 dividend will be cut to $1 due to real estate related losses. The market believes that thereafter dividends are expected to grow at a 3% rate until t = 7, and then grow at a 6% rate until t = 12, and then grow at a rate of g in perpetuity.
The market also believes that at t = 9 the stock price will be trading based on a P9/Div10 ratio of 8.
The discount rate is 10%.
a) Write down an equation where the only unknown is P0, the price of Morgan Stanley at t = 0? You do not need to solve for P0. (2.5 points)
Since we do not know g, but we do know something about the sale price at t = 9, we will imagine a
9-year investment horizon.
ࡼ =
.
(. )(. )
.
+
+
ቆ −
ቇ
ቆ −
ቇ
. . − .
. .
. − .
. . ૠ
ૡ[()൫. ൯൫. ൯]
+
. ૢ
The second term represents the PV of Div2 to Div7, the third term represents the PV of Div8 to Div9, and the fourth term represents the PV of P9, where is P9 = 8[Div10]
There are other ways of solving this. For example the dividends can be broken up slightly differently. Or you could figure out what g is (as is shown in part b) and then write out P0 based on the PV of all dividends assuming an infinite investment horizon.
b) Write down an equation where the only unknown is g? You do not need to isolate or solve for g.
(2.5 points)
ૡൣ()൫. ൯൫. ൯൧ =
(. )(. )
.
(. )(. )
ቆ −
ቇ+
. − .
.
. − ࢍ
.
The left hand side is the expected price at t = 9 based on P9 = 8[Div10].
The right hand side represents Div10 to Div11 discounted back to t = 9 and also Div12 to Divinfinity discounted back to t = 9.
MGCR 341
Page 12 of 17
Alternatively:
ૡൣ()൫. ൯൫. ൯൧ =
(. )(. )
.
(. )(. )( + ࢍ)
ቆ −
ቇ+
. − .
.
. − ࢍ
.
The left hand side is the expected price at t = 9 based on P9 = 8[Div10].
The right hand side represents Div10 to Div12 discounted back to t = 9 and also Div13 to Divinfinity discounted back to t = 9.
Alternatively
ࡼ =
.
(. )(. )
.
+
+
ቆ −
ቇ
ቆ −
ቇ
. . − .
. .
. − .
. . ૠ
(. )(. )
+
. − ࢍ
.
Where P0 is based on the expression in part a.
Alternatively
ࡼ =
.
(. )(. )
.
+
+
ቆ −
ቇ
ቆ −
ቇ
. . − .
. .
. − .
. . ૠ
(. )(. )( + ࢍ)
+
. − ࢍ
.
Where P0 is based on the expression in part a.
There are other variations as well.
c) What is the expected return (including the dividend) on the stock from t = 0 to t = 1? (2.5 points)
Expected return equals required rate of return = 10%
You could also try doing this the long way by first finding P1.
MGCR 341
Page 13 of 17
(Question 8, continued)
d) Suppose you believe the following: Morgan Stanley will pay a $3 dividend at t = 1, a $2 dividend at t = 2, a $4 dividend at t = 3 and that dividends will remain constant thereafter. You also believe that at t = 2 the market will revise its views on Morgan Stanley. Specifically, you believe that at t = 2 the market will believe that Div3 will equal $5, Div4 will equal $6, Div5 will equal $7 and that the
P4/Div5 ratio will equal 10. Assuming you have a 2 year investment horizon, write down an equation where the only unknown is H0, the most you would be willing to pay for Morgan Stanley at t = 0?
(2.5 points)
Use your views for dividends up to t = 2, then use market views to figure out the sale price at t = 2.
where
where
ࡴ =
ࡼ
+
+
. .
.
ࡼ =
ࡼ
+
+
. .
.
ࡼ = (ૠ) = ૠ
So,
ૠ
+
+
. . .
ࡴ =
+
+
. .
.
MGCR 341
Page 14 of 17
9) (10 points, 15 minutes) Companies A and B are expected to pay dividends in the form of a growing perpetuity. All earnings are paid out as dividends, so earnings = dividends. For each of the below, circle the best answer and justify briefly.
a) A has a higher dividend growth rate than B, and A has a lower required rate of return than B. Which company should have a higher PE ratio? Justify briefly. (2.5 points)
i) A ii) B iii) Can’t tell iv) They should have the same PE ratio
Higher growth and lower required rate of return implies investors are willing to pay more per dollar of expected dividend, where dividends are equal to earnings in this question.
b) A and B have the same dividend growth rate and the same required rate of return, but A has a lower profit margin than B. (Profit margin is equal to earnings as a fraction of sales. Assume that the profit margin will remain constant.) Which company should have a higher PE ratio? Justify briefly. (2.5 points)
i) A ii) B iii) Can’t tell iv) They should have the same PE ratio
Given that the dividend growth rate and required rate of return are the same, it does not matter what the profit margin is. Also, here earnings equal dividends.
c) A and B have the same growth rate and the same required rate of return, but company A has a lower profit margin than company B. (Profit margin is equal to earnings as a fraction of sales. Assume that the profit margin will remain constant.) Which company should be trading at a higher multiple of sales?
Justify briefly. (2.5 points)
i) A ii) B iii) Can’t tell iv) They should be trading at the same multiple of sales
That A has a lower profit margin implies
ࡼ
ࡼ
ࡱ
ࡿ
<
ࡱ
ࡿ
From part b, we know ࡱ = ࡱ
Multiplying both sides of the top expression by the respective side of the second expression:
MGCR 341
ࡱ ࡼ ࡱ ࡼ
ࡼ
< →
ࡿ ࡱ ࡿ ࡱ
ࡿ
<
ࡼ
ࡿ
Page 15 of 17
Thus, B is trading at a higher multiple of sales. Intuitively, investors are willing to pay more per dollar of sales for B because B is able to generate more earnings per $ of sales.
(And because dividend growth and required rate of return are the same.)
d) Now assume that dividends are not necessarily in the form of a growing perpetuity but that companies A and B have the same required rate of return and are both expected to have a 0% sales growth rate in perpetuity. Company B’s profit margin is constant. For t = 0 to t = 5 company A has a lower profit margin than company B. But for t = 6 to infinity, company A’s profit margin will equal that of company B. Which company should be trading at a higher PE ratio at t = 0? Justify briefly. (2.5 points)
i) A ii) B iii) Can’t tell iv) They should have the same PE ratio
Imagine that the profit margin for A were constant. In this case, both companies would have a zero dividend growth rate in perpetuity. This, together with the fact that they have the same required rate of return implies they should trade at the same PE ratio (since dividends are the same as earnings in this question).
But company A’s profit margin will increase in the future. Hence earnings will at some point increase.
Thus, A should be trading at a higher price than if its profit margin were constant. Thus at t = 0 company A should be trading at a higher PE ratio than company B.
MGCR 341
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10) (15 points, 30 minutes) Today is t = 0. Your steel company has an existing blast furnace and you are evaluating a proposal to replace it with a new furnace. You previously paid a consulting company
$3M for a feasibility study to determine the impact of replacing the furnace. The conclusions are outlined below. As part of your agreement with the consulting company, you have to pay the consulting company an extra $0.3M in cash at t = 0 if you decide to replace the furnace (assume this is a tax deductible “operating” expense at t = 0, and assume that at t = 0 you would have positive pretax income regardless of whether you decide to replace the furnace or not).
Ten years ago, at t = -10, you paid $50M for the old furnace, and for depreciation purposes you assumed a 20 year asset life, and a $0 salvage value.
The new furnace costs $105M at t = 0, and for depreciation purposes you will assume a 10 year asset life, and a $90M salvage value.
Regardless of whether you replace the furnace or not at t = 0, at t = 5 you will sell whichever furnace you have left. At t = 5, you believe you could sell the old furnace for $30M (before taxes), if you still have it, or the new furnace for $97.5M (before taxes). Note, however, that if you replace the furnace at t = 0, then you will sell the old furnace at t = 0 for $40M (before taxes).
The new furnace is eco-friendly, and if you replace the old furnace with the new furnace the government will allow you to pay $0.1M fewer taxes per year for t = 1 to t = 5.
Replacing the furnace will result in $13M greater revenue per year and will save $25M per year in operating expenses, from t = 1 to t = 5. The current sales and costs are $690M and $425M per year, respectively. Assume that for your company’s annual revenue, clients pay you 10% in cash at the time of sale, and the remaining 90% is paid in cash one year later. Assume that for your annual operating expenses, you pay 30% in cash at the time the expenses are incurred and the remaining 70% is paid in cash one year later. (Aside from these two points, all transactions are 100% in cash. For example, purchase/sale of capital assets (old/new furnace) is 100% in cash at the time of purchase/sale.)
The company’s tax rate is 40%. Assume that if the company does not replace the furnace, it will continue operations with the existing furnace until t = 5, and then sell the old furnace at t = 5. Fill in the table below indicating all the incremental values associated with replacing the furnace as appropriate. Use the space on the following page to show relevant calculations.
MGCR 341
Page 17 of 17
Incremental
Value
t=0
Operating
Expenses
Depreciation
13M
13M
25M
25M
25M
1M
(300,000)
t = 2 to 4
13M
Revenue
t=1
t=5
1M
t=6
1M
Operating profit
(300,000)
39M
39M
39M
Taxes
120,000
(40%*39M)
+0.1M
(40%*39M)
+0.1M
(40%*39M)
+0.1M
Net Income
(180,000)
23.5M
23.5M
23.5M
Depreciation
-1M
-1M
-1M
Accounts
Payable
-70%*25M
-70%*25M
70%*25M
-70%*25M
70%*25M
70%*25M
Accounts
Receiv.
-90%*13M
-90%*13M
90%*13M
-90%*13M
90%*13M
90%*13M
(excluding profit from sale of equipment) Adjustments:
Purchase/Sale
at t = 0
-105M
+34M
Purchase/Sale at t = 5
Net cash flow
MGCR 341
97.5M-23M
-71.18M
-6.7M
22.5M
97M
29.2M
Page 18 of 17
Use this page to show your work for Q10.
Extra 300,000 at t = 0 to consultants:
If you implement the project, you have to pay an extra 300,000 at t = 0. This, however, is a taxdeductible expense, which saves you from paying 40%(300,000) = 120,000 in taxes. (We assume the company has enough revenue to make the tax savings matter). Also, the $3M you already paid is a sunk cost and should be ignored.
Incremental depreciation for t = 1 to 5:
Previous depreciation expenses were (50 – 0)/20 = 2.5
New depreciation expense = (105 – 90)/10 = 1.5
So incremental depreciation expense is 1.5 – 2.5 = -1
(This is 1 less depreciation, which implies a positive contribution to operating profit. But since this is a non-cash contribution, you have to subtract it afterwards in the adjustments.)
Purchase of new machine and sale of old machine at t = 0:
If you implement project, it will cost you 105 at t = 0 to buy the new machine. You will sell the old furnace for 40, on which you have to pay taxes. The taxes equals the tax rate times the profit, where the profit equals the sale price minus the book value at time of sale, and where the book value at the time of sale = purchase price minus accumulated depreciation.
Net cash flow from sale at t = 0
= 40 – 40%(40 – (50 – 10(2.5))) = 34
Sale of new machine at t = 5
If you implement the project, then at t = 5, you sell the new machine for 97.5, on which you don’t have to pay taxes since the BV at t = 5 is also 97.5.
Opportunity Cost at t = 5
Note, if you implement the project, there is an opportunity cost of not being able to sell the old machine at t = 5, since you will have sold it at t = 0.
Opportunity cost net cash flow
= 30 – 40%(30 - (50 – 15(2.5))) = 23
Account Payable (AP) adjustments:
For t = 1 to 5 there is a savings of 25M in expenses. But not all those savings are cash savings since some of your expenses are paid on credit. So, for t = 1 to 5 you have to subtract 70% of that 25M. (25M less expenses implies 70% of 25M fewer AP. A decrease in AP needs to be subtracted from net income.)
But, the decreased 25M in expenses from t = 1 to 5 also implies that there will be 70% of 25M less cash being paid out from t = 2 to t = 6. This needs to be added back for t = 2 to 6.
Account Receivable (AR) adjustments:
For t = 1 to 5 there is 13M in extra revenue, but not all that increase in revenue is a cash increase since some of your sales are made on credit. So, for t = 1 to 5 you have to subtract off 90% of the 13M.
The increase of 13M in revenue from t = 1 to 5 also implies that there will be 90% of 13M more cash being received from clients for t = 2 to t = 6. This needs to be added for t = 2 to 6.
MGCR 341
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11) (18 points, 20 minutes) True or False: write True or False directly to the left of the statements below.
No explanation is required.
a) It is possible for markets to be efficient even if a stock has a statistically significant positive alpha relative to the CAPM. True. The CAPM could be the wrong model of risk control.
b) The value effect, also known as the book-to-market effect, is a reference to the fact that historically stocks with high book-to-market ratios have, on average, outperformed stocks with low book-to-market ratios, even after controlling for beta risk. True
c) If markets are weak form efficient, then strategies based on momentum should be able to generate superior risk-adjusted returns. False
d) According to the single index model, firm-specific shocks are correlated to the market return.
False
e) If two stocks have a positive correlation, then there are no benefits of diversification.
False. There are benefits as long as the correlation is less than 1.
f)
Stock A has an abnormally low PE ratio at the moment because earnings are high, but are expected to drop significantly in the future. This suggests that stock A’s stock price is expected to drop going forward. False. The price at t = 0 already incorporates the fact that earnings will drop. MGCR 341
Page 20 of 17
MGCR 341: Formula Sheet
HPR =
annualized HPR = (1 + HPR )1 / t − 1
ending price
−1
beginning price
ending price
=
beginning price
S
E[r ] = p1r1 + .... + pS rS = ∑ pi ri
T
cov[rA , rB ]
y realized
σ Aσ B
FW
=
beginning price
E ( X + Y ) = E ( X ) + E (Y )
E (aX + bY ) = aE ( X ) + bE (Y )
−1
Var (aX ) = a 2Var ( X )
Cov (aX , bY ) = abCov ( X , Y )
Var (aX + bY ) = a 2Var ( X ) + b 2Var (Y ) + 2abCov ( X , Y )
Var ( X ) = 0 if X is a constant
E (rp ) − r f
Cov ( X , X ) = Var ( X )
Cov ( X , Y ) = 0 if either X or Y is a constant
σp
σ 2 = ∑ w i2 ⋅ σ i2 + 2 ∑ w i w j cov(ri , r j ) p n
E(rp ) = ∑i =1 w i ⋅ E(ri ) n i =1
E[ri ] = rf + β i (E[rm ] − rf ) βi =
1/ t
Var (a + X ) = Var ( X )
E (a + X ) = a + E ( X )
E (aX ) = aE ( X )
S=
−1
PBond (1 + y realized ) = FW
i =1
ρ A, B =
1/ t
cov(rm , ri ) σ i,m σ = 2 = i ρ i, m var(rm ) σm σm
i≠ j
σ 2 = w 2 ⋅ σ 2 + w 2 ⋅ σ 2 + 2w A w B cov(rA , rB ) p A
A
B
B
β P = w 1 × β1 + ... + w N × β N
( F + C n ) / (1 + y )
C1 / (1 + y )
C / (1 + y )
+ 2× 2
+ ... + n ×
P
P
P
2
D = 1×
MD = D / (1 + YTM)
∆P
≈ − MD × ∆y
P
PV (perpetuity) = C/r
PV (annuity ) =
PV (growing perpetuity) = C/(r - g)
PV ( growing _ annuity ) =
n
C
1
1 −
(1 + r ) n r
C (1 + g )
1 − r - g (1 + r ) n
n
APR
1 + EAR = 1 +
k
MGCR 341
k
1 + rr =
1+ r
1+ i
CFreal,t =
CFnominal,t
(1 + i) t
Page 21 of 17