Here are the answers to your questions.
In order to answer this questions and the other ones related to
present value, we'll use the present value formula explained in the
following link (it's the same I stated in my previous answer to you)
Calculating the Present and Future Value of Annuities
In this question, we must first calculate the present value of this
project if the discount rate is 10%. The present value would be:
PV = -3000 + 800/1.1 + 800/1.1^2 + 800/1.1^3 + 800/1.1^4 + 800/1.1^5 + 800/1.1^6
Using the formula mentioned in the link under the title of
"Calculating the Present Value of an Ordinary Annuity", this equation
is the same as:
PV = -3000 + 800*[1 - (1+0.1)^(-6)]/0.1
PV = -3000 + 3484.2
PV = 484.2
Since the persent value is greater than zero, then the project should be pursued.
In order to find the "cut-off point" for the interest rate, we must
find the interest rate that makes the present value of the project
equal to zero. This is precisely the IRR, the rate at which the
present value of the project costs ($3000) equal the present value of
the project's cash flow. You can use any financial calculator or
Microsoft Excel in order to calculate the IRR of this project. We get
that the value is 0.1534 [please let me know through a clarification
request if you need help on how to get this value in Excel]. To verify
that this is the right value, let's calculate the PV using 0.1534 as
the discount rate:
PV = -3000 + 800*[1 - (1+0.1534)^(-6)]/0.1534
PV = -3000 + 3000.06
PV = 0.06 [the fact that it's not exactly equal to zero is due to rounding]
Therefore, the project should be pursued only if the discount rate
were smaller than 15.34%
You can find the definition and formula for the payback period in the
Using the formula provided there, the payback period for this project
is 2500/600=4.16 years. Since the firm accepts projects with payback
periods of less than 5 years, the project wil be accepted.
In ordert ofind whether it shoud be pursued at a 2% disocunt rate, we
find its present value using the same formula as in the previous
PV = -2500 + 600*[1 - (1+0.02)^(-6)]/0.02
PV = -2500 + 3360.85
PV = 860.85
Since PV>0, the project should be pursued if the discount rate is 2%.
If the discount rate is 12%:
PV = -2500 + 600*[1 - (1+0.12)^(-6)]/0.12
PV = -2500 + 2466.84
PV = -33.15
Since PV<0, the project should not be pursued if the discount rate is 12%.
The firm's decision will change just as before as the discount rate
changes. In order to find the cut-off value, we calculate the IRR of
this project, obtaining that it is 11.53%. The project should only be
accepted if the discount rate is smaller than that value.
Let's call PVa and PVb to the present value of project's A and B
respectively. We then have that:
PVa = -20000 + 8000*[1 - (1+i)^(-3)]/i
PVb = -20000 + 25000/(1+i)^3
Now, if we equate these formulas, we'll find the value of i that makes
the present value of both projects equal.
-20000 + 25000/(1+i)^3 = -20000 + 8000*[1 - (1+i)^(-3)]/i
25000/(1+i)^3 = 8000*[1 - (1+i)^(-3)]/i
3.125 = [(1+i)^3 - 1]/i
3.125*i = (1+i)^3 - 1
This is a cubic equation that yields the relevant result of i=0.0411.
In case you don't know how to solve a cubic equation, I've written an
Excel spreadsheet that shows the NPV profile of each project. As you
can see in this sheet, the value of both project is approximately the
same at i=0.04. Below this value, project B is better; while above it,
project A is better. You can find the spreadsheet here:
I've also included the IRR calculation for both projects in the spreadsheet.
Here's the definition of the Profitability Index
The PV of the cash flows of project A is 2000/1.22 + 1200/1.22^2 = $2445.57
The PV of the cash flows of project B is 1440/1.22 + 1728/1.22^2 = $2341.30
The profitability index (PI) of project A is then 2445.57/2100 = 1.16;
and tehy PI of project B is 2341.30/2100 = 1.11.
If you could undertake both projects, then the PI rule, which is to
accept projects with PI greater than one, would say to accept both,
since both PI's are greater than one.
If you could undertake only one project, then, given an equal
investment, we should pursue the project with the greatest PI. In this
case, given the same investment ($2100 for each), project A has the
largest PI, so that's the project that should be pursued if we had to
choose only one.
The proper cash flow to use here is the Incremental Cash Flow, which
represent the CHANGE in the firm's total cash flow that happens as a
result of undertaking a project.
In this case, if the company introduces the new chip, the firm's total
cash flow will rise by 12*(25-8) = $204 million (12 million chips sold
at $25, cost $8 each). However, since sales of the old chip will fall
by 7 million units. Therefore, the firm's cash flow, as a result of
undertaking this project, will also fall 7*(20-6) = $98 million (7
million chips sold at $20, cost $6 each). Therefore, the incremental
annual cash flow of this project is 204-98 = $106 million; and this is
the value that should used in order to evaluate the project.
By the very definition of Incremental Cash Flow, D is the only valid
option that is an incremental cash flow associated to the donation of
the painting. The other options are not even cash flows.
a. The initial investment in this project are the $45,000 of plant and
equipment plus the working capital of Year 1's sales, which amount to
0.2*40000=$8,000. Therefore, the initial investment is $53,000
b. The working capital required each year becomes smaller each year
until it reaches 0 in Year 4. Since this question doesn't specify that
the working capital depreciates over time, I'll assume it does not
depreciate. Furthermore, I'll assume that this firm uses exactly the
required working capital each year. This means that, each time a year
passes, the firm will be left with excess working capital. I'll assume
that the firm can sell this excess. For example, the required working
capital in Year 0 is $8,000. The required working capital in Year 1 is
0.2*30000=$6,000. I'll assume then that the firm has an extra $2,000
in its cash flow this year, as a result of the sale of excess working
capital. I'll also assume that no taxes are levied on this sale.
Regarding depreciation, since we're using straight-line depreciation
over 4 years to a salvage value of $0, and the initial value is
$45,000, then depreciation each year is 45000/4 = $11,250.
Since the tax rate is 40%, then the firm only gets to keep 60% of its
income. The taxable income is Revenues - Expenses - Depreciation.
Finally, depreciation should be added back to the cash flow.
So these are the project cash flows:
Year 0 -45,000 - 8,000 = -53,000
Year 1 0.6*[40,000 - 0.4*40,000 - 11,250] + 11,250 + 2,000 = $20,900
Year 2 0.6*[30,000 - 0.4*30,000 - 11,250] + 11,250 + 2,000 = $17,300
Year 3 0.6*[20,000 - 0.4*20,000 - 11,250] + 11,250 + 2,000 = $13,700
Year 4 10,000 - 0.4*10,000 + $2,000 = $8,000
Since in year 4 income net of depreciation is negative (10,000 -
0.4*10,000 - 11,250 = -$5,250), I assumed that no tax is levied; hence
the different calculations.
As you can see, I've made several assumptions throughout this
question, which may be not what you had in mind. Please check that
these assumptions are correct or if you're curently working with
different ones. In any case, I think the reasoning is clear enough in
order to re-calculate the results under different assumptions. Please
request clarification otherwise.
c. The project NPV is:
NPV = -53,000 + 20,900/1.12 + 17,300/1.12^2 + 13,700/1.12^3 + 8,000/1.12^4
At this discount rate, the project should not be undertaken.
d. Plugging the cash flow values in Excel, we get that the IRR is 5.96%
If you buy the car, the cash flow would be something like this:
Year 0 -25,000
Year 1 0
Year 2 0
Year 3 0
Year 4 0
Year 5 5,000
The present value of the purchase is, then,
PV = -25000 + 5000/1.12^5
PV = -22162
If you choose to lease the car, the cash flow will be something like:
Year 0 -5,000
Year 1 -5,000
Year 2 -5,000
Year 3 -5,000
Year 4 -5,000
I'm assuming here that lease payments are made at the beggining of
each year rather than at the end. Therefore, in order to find the
present value of this cashflow, we should use the formula provided in
the first link I mentioned, under the title of "Present Value of an
Annuity Due", which is different to the one we've been using:
PV = -5000*(1+0.12)*[1 - (1+0.12)^(-5)]/0.12
PV = -5000*4.037
PV = -20186
Therefore, we should lease the car rather than buy it, because the
present value of the lease is higher (the cost is smaller)
b. In order to find the cut-off value for the lease payment, we equate
the present value of the lease to the present value of the purchase.
Let's call X to the lease payment:
22162 = X*(1+0.12)*[1 - (1+0.12)^(-5)]/0.12
22162 = X*4.037
X = 5489.72
Therefore, if the lease payment is less than $5,489.72, then you
should choose to lease the car; otherwise you should purchase it.
This question can be solved using exactly the same reasoning as in
question 7, depreciating the plant over 3 years.
a. Yes, it is reasonable. Interest rates are procyclic; that is, they
rise when the economy is booming and fall when the economy goes into a
recession. During recessions, the government usually tries to keep
interest rates low in order to stimulate investment. Since bond prices
and interest rates go in opposite directions (when interest rates
rise, bond prices fall, and viceversa), then we would expect bond
prices to go up when the economy hits a recession.
b. The expected rate of return of stocks can be calculated as:
ER = (Prob. of Recession)*(Return on Recession)
+(Prob of Normal)*(Return on Normal)
+(Prob of Boom)*(Return on Boom)
ER = 0.2*(-5) + 0.6*15 + 0.2*25
ER = 13%
Likewise, the expected return on bonds is 8.4%.
The variance is calculated in the following way:
Variance = (Prob. of Recession)*(Return on Recession - ER)^2
+(Prob of Normal)*(Return on Normal - ER)^2
+(Prob of Boom)*(Return on Boom - ER)^2
And the standard deviation is the square root of variance. Therefore, for stocks:
Variance = 0.2*(-5-13)^2 + 0.6*(15-13)^2 + 0.2*(25-13)^2
Variance = 96
So the standard deviation of the return on stocks is 9.79% (square root of 96).
Likewise, the standard deviation of the return on bonds is 3.2%.
c. Since none of the options "dominates" the other two, the answer to
this question is a matter of personal preference. We would say that
one option dominates another one if it has the same expected return
with less variance, or if it has a higher expected return with the
same variance. In this case however, although the portfolio of stocks
has a higher expected return than bonds, it also carries a higher
variance, so there is no clear-cut answer to this question.
Personally, since I am very risk averse, I would choose to invest in
bonds only. But another person with less disliking for risk might very
well choose to invest in a portfolio of stocks only. In any case, the
decision should be based on your risk tolerance, that is, if you feel
that the expected reward (the return) is enough to offset the risk
you'll have to face.
[Answer c. pasted from an answer I've previously given to another
Google Answers customer]
a. If you invest 60% in stocks and 40% in bonds, then the returns in
each scenario are:
Recession = 0.4*(Ret of Bonds in Recession) + 0.6*(Ret of Stocks in Recession)
Normal = 0.4*(Ret of Bonds in Normal) + 0.6*(Ret of Stocks in Normal)
Boom = 0.4*(Ret of Bonds in Boom) + 0.6*(Ret of Stocks in Boom)
Porfolio returns in recession = 2.6%
Porfolio returns in normal = 12.2%
Porfolio returns in boom = 16.6%
b. The expected returns and standard deviation are calculated in
exactly the same fashion as in the previous question. (multiply the
probability of each scenario by the returns in each scenario, etc).
Expected Return = 11.16%
Std. Dev = 4.6%
c. Again, since no investment dominates the other two in the sense
explained above, this is a matter of personal choice. More risk-averse
people will give more weight to bonds (even 100%), while less risk
averse people will give more weight to stocks.
a. FALSE. Calling rk the asset's return, rf the risk-free asset return
and rm the market return, the CAPM equation is:
E(rk)-rf = Beta*[E(rm)-rf]
E(rk)-rf = 2*[E(rm)-rf]
E(rk) = 2*E(rm) - rf
Therefore, clearly E(rk) is not equal to 2*E(rm), so the statement is false.
b. True. In a well-diversified portfolio, consisting of many stocks,
the specific risk of each stock (risk factors affecting only that
company) becomes insignificant. However, a stock contributes to the
risk of the whole portfolio through the covariance of its returns with
the returns of other stocks. The fact that stocks from all sectors
usually fall together in response to particular economic conditions is
the "market risk", the risk that can't be diversified away in a
c. False. In a perfectly efficient market all stocks should lie on the
security market line. If a stock lies below it, it means that it
offers too little an expected return for its risk (beta), thus it's
overpriced rather than underpriced.
d. True. Beta is a measure of the correlation of the portfolio (or
security) returns with the market portfolio. In other words, it
captures the market risk of the portolio (or security) but it does NOT
take in account its specific risk. However, since the statement
mentions a diversified portfolio, it can be assumed that the specific
risk has been completely diversified away; therefore beta "correctly"
measures the volatility of the portfolio, as it is only related to
e. False, for the same reason d is true. An undiversified portfolio
has not diversified away all the specific risk. Therefore, it's very
possible to have a higly volatile portfolio (because of firm- or
sector-specific risk factors) with a low beta, which just measures its
correlation to the market portfolio. In other words, with a beta of 2,
an undiversified portfolio can very well be more than twice as
volatile as the market portfolio.
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I hope this helps! If you have any questions regarding my
answer,please don't hesitate to request a clarification. Otherwise I
await your rating and final comments.