FIN 470 Exam 2
1.
The profitability index is the PV of the future cash flows divided by the initial
investment. The cash flows for both projects are an annuity, so:
PII = $25,800(PVIFA11%,3) / $57,000 = 1.106
PIII = $10,050(PVIFA11%,3) / $18,700 = 1.313
The NPV of each project is:
NPVI = –$57,000 + $25,800(PVIFA11%,3) = $6,047.84
NPVII = –$18,700 + $10,050(PVIFA11%,3) = $5,859.33
The point of indifference occurs where the two projects have the same NPV. At this point the NPV of the
difference is zero. Therefore, the IRR of the difference between the cash flows of the two projects is the rate
that makes the investor indifferent.
Year
0
1
2
3
Cash Flow (I)
–$ 57,000
25,800
25,800
25,800
Cash Flow (II)
–$ 18,700
10,050
10,050
10,050
IRR =
Cash flow (I - II)
-38,300
15,750
15,750
15,750
11.28%
At a required rate of 11% project I has the higher NPV and the indifference rate is 11.28%. Therefore, for
any rate below 11.28% project I is preferred.
2. To find the bid price, we need to calculate all other cash flows for the project, and then solve
for the bid price. The aftertax salvage value of the equipment is:
Aftertax salvage value = $60,000(1 − 0.34) = $39,600
Now we can solve for the necessary OCF that will give the project a zero NPV. The equation
for the NPV of the project is:
NPV = 0 = −$770,000 − 65,000 + OCF(PVIFA10%,5) + [($65,000 + 39,600) / 1.105]
Solving for the OCF, we find the OCF that makes the project NPV equal to zero is:
OCF = $770,051.63 / PVIFA12%,5 = $203,137.68
The easiest way to calculate the bid price is the tax shield approach, so:
OCF = $203,137.68 = [(P − v)Q − FC](1 − T) + TD
$203,137.68 = [(P − $9.30)(110,000) − $315,000](1 − 0.34) + 0.34($770,000/5)
P = $14.24
3. At the cash break-even, the OCF is zero. Setting the tax shield equation equal to zero and
solving for the quantity, we get:
OCF = 0 = [($300 – 200)QC – $700,000](0.70) + 0.30($6,000,000/6)
QC= 2,714
The accounting breakeven is:
QA= [$700,000 + ($6,000,000/6)]/($300 – 200)
QA= 17,000
Using the tax shield approach, the OCF is:
OCF = [($300 – 200)(40,000) – $700,000](0.70) + 0.30($6,000,000/6)
OCF = $2,610,000
And the NPV is:
NPV = –$6,000,000 – 580,000 + $2,610,000(PVIFA18%,6) + [$580,000 + $680,000(1 –
.30)]/1.186
NPV = $2,939,918.39
To calculate the sensitivity to changes in quantity sold, we will choose a quantity of
41,000. The OCF at this level of sale is:
OCF = [($300 – 200)(41,000) – $700,000](0.70) + 0.30($6,000,000/6)
OCF = $2,680,000
The sensitivity of changes in the OCF to quantity sold is:
ΔOCF/ΔQ = ($2,610,000 – 2,680,000)/(40,000 – 41,000)
ΔOCF/ΔQ = +$70.00
The NPV at this level of sales is:
NPV = –$6,000,000 – $580,000 + $2,680,000(PVIFA18%,6) + [$580,000 + $680,000(1 –
0.30)]/1.186
NPV = $3,184,750.57
And the sensitivity of NPV to changes in the quantity sold is:
ΔNPV/ΔQ = ($2,939,918.39 – 3,184,750.57))/(40,000 – 41,000)
ΔNPV/ΔQ = +$244.83
You wouldn’t want the quantity to fall below the point where the NPV is zero. We know
the NPV changes $244.83 for every unit sold, so we can divide the NPV for 40,000 units
by the sensitivity to get a change in quantity. Doing so, we get:
$2,939,918.39 = $244.83(ΔQ)
ΔQ = 12,008
For a zero NPV, we need to decrease sales by 12,008 units, so the minimum quantity is:
QMin = 40,000 – 12,008
QMin = QF = 27,992
4. R1 = ($48.31 – 43.33 + 0.54) / $43.33 = 0.1274, or 12.74%
R2 = ($57.23 – 48.31 + 0.57) / $48.31 = 0.1964, or 19.64%
R3 = ($45.31 – 57.23 + 0.80) / $57.23 = –0.1943, or –19.43%
R4 = ($52.23 – 45.31 + 0.85) / $45.31 = 0.1715, or 17.15%
R5 = ($61.31 – 52.23 + 0.93) / $52.23 = 0.1917, or 19.17%
The arithmetic average return was:
RA = (0.1274 + 0.1964 – 0.1943 + 0.1715 + 0.1917)/5 = 0.0985, or 9.85%
And the geometric average return was:
RG = [(1 + 0.1274)(1 + 0.1964)(1 – 0.1943)(1 + 0.1715)(1 + 0.1917)]1/5 – 1 = 0.0869, or
8.69%
The Standard Deviation was:
σ = {(1/4)[(.1274-.0985)2 + (.1964-.0985)2 + (-.1943-.0985)2 + (.1715-.0985)2 + (.1917-.0985)2]}.5 =
0.165955 ≈ 16.6%
5. We need to find the return of the portfolio in each state of the economy. To do this, we
will multiply the return of each asset by its portfolio weight and then sum the products
to get the portfolio return in each state of the economy. Doing so, we get:
Boom: E(RP) = 0.30(0.38) + 0.30(0.50) + 0.4(0.50) = 0.4640, or 46.40%
Normal: E(RP) = 0.30(0.16) + 0.30(0.14) + 0.4(0.12) = 0.1380, or 13.80%
Bust: E(RP) = 0.30(0.00) + 0.30(−0.30) + 0.4(−0.50) = −0.2900, or −29.00%
And the expected return of the portfolio is:
E(RP) = 0.20(0.4640) + 0.55(0.1380) + 0.25(−0.2900) = 0.0962, or 9.62%
To calculate the standard deviation, we first need to calculate the variance. To find the
variance, we find the squared deviations from the expected return. We then multiply each
possible squared deviation by its probability, than add all of these up. The result is the
variance. So, the variance and standard deviation of the portfolio is:
σP2 = 0.20(0.4640 − 0.0962)2 + 0.55(0.1380 − 0.0962)2 + 0.25(−0.2900 − 0.0962)2
σP2 = 0.06530
σP = (0.06530)1/2 = 0.2555, or 25.55%
To find the covariance we first need to find the expected return for both A & B:
E[RA] = .2(.38) + .55(.16) + .25(0) = 0.164
E[RB] = .2(.5) + .55(.14) + .25(-.3) = 0.102
σAB = .2(.38 - .164)(.5 - .102) + .55(.16 - .164)(.14 - .102) + .25(0 - .164)(-.3 - .102) = 0.0336
To find the portfolio beta we use the CAPM:
.0962 = .036 + βP(0.081) → βP = (.0962 - .036)/0.081 = 0.74
6.
Here we have the expected return and beta for two assets. We can express the returns
of the two assets using CAPM. If the CAPM is true, then the security market line holds
as well, which means all assets have the same risk premium. Setting the risk premiums
of the assets equal to each other and solving for the risk-free rate, we find:
(0.195 − Rf)/1.85 = (0.168 − Rf)/1.54
1.54(0.195 − Rf) = 1.85(0.168 − Rf)
0.30030 − 1.54Rf = 0.3108 − 1.85Rf
0.31Rf = 0.01050
Rf = 0.0339, or 3.39%
Now using CAPM to find the expected return on the market with both stocks, we find:
0.195 = 0.0339 + 1.85(RM − 0.0339)
RM = 0.1210, or 12.10%
7.
0.168 = 0.0339 + 1.54(RM − 0.0339)
RM = 0.1210, or 12.10%
Here we have a stock that pays no dividends for 10 years. Once the stock begins
paying dividends, it will have a constant growth rate of dividends. We can use the
constant growth model at that point. It is important to remember that general constant
dividend growth formula is:
Pt = [Dt × (1 + g)] / (R − g)
This means that since we will use the dividend in Year 10, we will be finding the stock price
in Year 9. The dividend growth model is similar to the PVA and the PV of a perpetuity: The
equation gives you the PV one period before the first payment. So, the price of the stock in
Year 9 will be:
P9 = D10 / (R − g) = $10 / (0.115 − 0.05) = $153.85
The price of the stock today is simply the PV of the stock price in the future. We simply
discount the future stock price at the required return. The price of the stock today will be:
P0 = $153.85 / 1.1159 = $57.76
8.
You get to come with an answer on your own..
9.
For tax purposes, a firm would choose MACRS because it provides for larger depreciation deductions
earlier. These larger deductions reduce taxes, but have no other cash consequences. Notice that the choice
between MACRS and straight-line is purely a time value issue; the total depreciation is the same, only the
timing differs.
10. While that fact that the worst-case NPV is positive is interesting, it also indicates that there is likely a problem with
the inputs and/or analysis. While we would like all of our projects to be guaranteed to make money, as a practical
matter, it doesn’t seem likely that these types of projects are very prevalent. In a perfectly competitive market that is
functioning efficiently, even the expected NPV should be zero.
11. Ignoring trading costs, on average, such investors merely earn what the market offers; stock investments
all have a zero NPV. If trading costs exist, then these investors lose by the amount of the costs.
12. If the market expected the growth rate in the coming year to be 2 percent, then there would be no change in
security prices if this expectation had been fully anticipated and priced. However, if the market had been
expecting a growth rate other than 2 percent and the expectation was incorporated into security prices,
then the government’s announcement would most likely cause security prices in general to change; prices
would drop if the anticipated growth rate had been more than 2 percent, and prices would rise if the
anticipated growth rate had been less than 2 percent.
13. False. The variance of the individual assets is a measure of the total risk. The variance on a welldiversified portfolio is a function of systematic risk only.
14. Bond ratings tell the relative probability of default. In other words, a AAA bond is less likely to default
than a AA bond.
15. The value of anything (except a wife) is the present value of the expected future cash flows.