November 21st, 2008
1. You think you will spend $40,000 a year for 20 years once you
retire in 40 years. If the interest rate is 6% per year, how much must
you save each year until retirement to meet your retirement goal? How
do you calculate this scenario?
2. If I put aside $3,000 each year in a savings plan that earns 8%
interest, and in 5 years receive a gift of $10,000 that can also be
invested, (a) how much money will I accumulate 30 years from now? and
(b) If my goal is to retire with $800,000 of savings, how much extra
do I need to save every year? What computations are necessary to
arrive to the answer?
3. If you insulate your office for $10,000, you will save $1,000 a
year in heating expenses. (these savings will last forever). How do
you calculate the NPV, the IRR, and the payback period for this
scenario?
(a) What is the NPV of the investment when the cost of capital is 8%? 10%?
(b) What is the IRR of the investment?
(c) What is the payback period on this investment?
4. The internal rate of return for the following project is 13.1%.
Should I accept or reject the project if the discount rate is 12%?
Please provide computations that helped arrive to your answer.
Year Cash Flow
Year 0 has +$100
Year 1 has -60
Year 2 has -60
5. Here are the cash flow forecasts for two mutually exclusive projects:
Year Project A Project B
0 -$100 -$100
1 30 49
2 50 49
3 70 49
(a) Which project would you choose if the opportunity cost of capital is 2%?
(b) Which would you chooise if the opportunity cost of capital is 12%?
(c) Why does your answer change?
* Please show computations that helped you arrive to your answer.
6. Copy Company is thinking about buying a new high-volume copier. The
machine costs $100,000 and will be depreciated straight-line over 5
years to salvage value of $20,000. The company anticipates that the
machine can be actually sold in 5 years for $30,000. Also, the machine
will save $20,000 a year in labor costs but will require an increase
in working capital, mainly paper supplies of $10,000. The company's
marginal tax rate is 35% and the discount rate is 8%. Should Copy
Company buy the machine? Please provide calculations that relate to
whether or not Copy Company should purchase the machine.
7. I am evaluating an expansion for a business. The cash-flow
forecasts (in millions) for the project are:
Years Cash Flow
0 -100
1-10 + 15
Based on the behavior of the company's stock, I believe that the beta
of the firm is 1.4. Assuming that the rate of return on the market
portfolio is 12%, what is the net present value of the project? How do
you calculate the NPV for this scenario?
#8. In which of the following situations would you get the largest
reduction in risk by spreading your portfolio across two stocks? Why?
(a) The stock returns vary with each other.
(b) The stock returns are independent.
(c) The stock returns vary against each other.
9.Capital Structure:
* Company A's Capital Structure consists of:
Debt at 12% = $600,000; Common Stock, $10 per share at $400,000;
Total: $1,000,000; Common Shares: $40,000.
* Company A's Operating Plan consists of: Sales ($50,000 units @ $20
each) $1,000,000; Less Variable costs of $800,000 and fixed costs of
0. EBIT is $200,000.
* Company B's Capital Structure consists of:
Debt at 12% = 0; Common stock, $10 per share = $1,000,000; Common
shares of $100,000.
* Company B's operating plan consists of Sales 50,000 units at $20
each) = $1,000,000; Less variable costs of $500,000 and fixed costs of
$300,000. EBIT is $200,000
(a) If you combine Company A's capital structure with Company B's
operating plan, what is the degree of combined leverage?
(b) If you combine Company B's capital structure with Company A's
operating plan, what is the degree of combined leverage?
(c) Please explain why you got the results you did in part (b)
(d) In part (b), if sales double, by what percent will EPS increase?Dear greeneyes1,
My answers follow. I hope these calculations will be instructive for you.
Regards,
leapinglizard
1.
We must save $13,220.52 a year to meet the retirement goal. The first
step in calculating this answer is to determine the total amount we will
need to save if it is to last for 20 years at 6% interest with annual
withdrawals of $40,000. The target amount T is given by the following
formula, which relies on specified values for the annual debit d,
interest rate p, and number of years n.
T = d * (1 - 1/(1+p/100)^n) / (1 - 1/(1+p/100))
For d = 40000, p = 6, and n = 20, we have
T = $40,000 * (1 - 1/1.06^20) / (1 - 1/1.06)
= $48,6324.66 .
Now, to compute the amount we must save annually to reach this goal,
we use the formula
c = T * p/100 / ((1+p/100)^n - 1).
We plug in the previously obtained value of T, along with p and n.
c = $486324.66 * 0.06 / (1.06^20 - 1)
= $13,220.52 .
2.
(a)
Let us first determine how much we will save in the first five years of
the savings plan.
T = c * ((1+p/100)^n - 1) / (p/100)
= $3,000 * (1.08^5 - 1) / 0.08
= $17,599.80 .
The gift of $10,000 augments the five-year total to $27,599.80 . In the
subsequent 25 years of the plan, cumulative interest will increase this
amount to
1.08^25 * $27,599.80
= $189,016.57
But in those 25 years, the annual $3,000 credits will further contribute
$3,000 * (1.08^25 - 1) / 0.08
= $219,317.82
which leads to total accumulated savings of
$189,016.57 + $219,317.82
= $408,334.39
(b)
The difference between the desired savings and the actual savings is
$800,000 - $408,334.39
= $391,665.61
To accumulate this target amount over 30 years, we must contribute an
extra annual deposit of
c = $391,665.61 * 0.08 / (1.08^30 - 1)
= $3,457.40
to the savings plan.
3.
(a)
With a timeline of n years and a capital cost of p percent, the present
value of $10,000 is
$10,000 * (1+p/100)^n
while the annual savings of $1,000 yield a return of
$1,000 * ((1+p/100)^n - 1) / (p/100) .
The Net Present Value (NPV) is the latter value less the former
value. Thus, with a capital cost of 8%, the NPV of the investment would be
$1,000 * (1.08^n - 1) / 0.08 - $10,000 * 1.08^n .
With a capital cost of 10%, it would be
$1,000 * (1.10^n - 1) / 0.10 - $10,000 * 1.10^n .
(b)
The IRR is the interest rate at which the NPV of the investment is equal
to zero. Thus, it is the value of p for which
$1,000 * ((1+p/100)^n - 1) / (p/100) = $10,000 * (1+p/100)^n .
To compute such a value for a given number of years n, we can use a
sequence of successively closer approximations to arrive at the IRR.
(c)
The payback period is the amount of time it takes to recover the cost
of the investment. In this case, with an initial outlay of $10,000 and
annual inflows of $1,000, the payback period is
$10,000 / $1,000 = 10
years.
4.
The basic IRR rule tells us to accept a project if its IRR is higher
than the prevailing discount rate.
http://www.marketvolume.com/glossary/b0076.asp
The difference between the IRR and the discount rate indicates the degree
to which the project is profitable. A positive difference indicates a
money-making project, while a negative difference indicates a losing
proposition. In this case, we have a positive difference of
13.1% - 12% = 1.1%
which tells us that we should accept the project.
5.
The initial investment is $100 for each of Project A and Project B. In
subsequent years, we divide each project's annual cash flow by
(1+p/100)^n ,
where p is the cost of capital and n is the year number, in order to
arrive at its present value. The sum of the annual present values less
the initial investment is the Net Present Value (NPV) of the project. We
shall choose the project with the higher NPV.
(a)
We calculate the PV of the revenues from Project A as follows.
year divisor present value of cash flow
1 1.02^1 = 1.0200 $30 / 1.0200 = $29.41
2 1.02^2 = 1.0404 $50 / 1.0404 = $48.06
3 1.02^3 = 1.0612 $70 / 1.0612 = $65.96
-------
total = $143.43
The total PV of the cash flows is $156.90 .
For Project B, we make the following calculations.
year multiplier present value of cash flow
1 1.02^1 = 1.0200 $49 / 1.0200 = $48.04
2 1.02^2 = 1.0404 $49 / 1.0404 = $47.10
3 1.02^3 = 1.0612 $49 / 1.0612 = $46.17
-------
total = $141.31
After subtracting the $100 initial investment, we see that the NPV of
Project A is $43.34, while that of Project B is lower at $41.31 . We
should therefore choose Project A.
(b)
Project A:
year multiplier present value of cash flow
1 1.12^1 = 1.1200 $30 / 1.1200 = $26.79
2 1.12^2 = 1.2544 $50 / 1.2544 = $39.86
3 1.12^3 = 1.4049 $70 / 1.4049 = $49.83
-------
total = $116.48
Project B:
year multiplier present value of cash flow
1 1.12^1 = 1.1200 $49 / 1.1200 = $43.75
2 1.12^2 = 1.2544 $49 / 1.2544 = $39.06
3 1.12^3 = 1.4049 $49 / 1.4049 = $34.88
-------
total = $117.69
Project A now has a higher NPV at $16.48 than Project B at $17.69 .
Thus, we should now choose Project B.
(c)
Due to the non-linear property of exponentiation, a higher cost of
capital has a disproportionately greater effect on cash flow in the
later years of a project. Thus, the higher and later cash flows of
Project A are discounted much more heavily than its lower and earlier
ones. In contrast, the cash flow for Project B is even throughout. So the
higher cost of capital is less damaging to Project B, thereby giving it
a more advantageous NPV in the second case. In the first case, however,
the cost of capital is too low to offset the slightly greater future
cash flows of Project A.
6.
The difference between the purchase cost and salvage value of the
machine is
$100,000 - $20,000 = $80,000 .
With straight-line depreciation over five years, the annual depreciation
is
$80,000 / 5 = $16,000 .
With a marginal tax rate of 35%, the annual tax savings are therefore
0.35 * $16,000 = $5,600 .
After taking into account the $20,000 savings in labor costs and the
$10,000 additional expense in working capital, we arrive at total annual
savings of
$5,600 + $20,000 - $10,000 = $15,600
We can also add the sale value of $30,000 in the fifth year. However,
we must now account for the cost of capital. The present value of the
savings is computed as follows.
year multiplier present value of savings
1 1.08^1 = 1.0800 $15,600 / 1.0800 = $14,444.44
2 1.08^2 = 1.1664 $15,600 / 1.1664 = $13,374.49
3 1.08^3 = 1.2597 $15,600 / 1.2597 = $12,383.90
4 1.08^4 = 1.3605 $15,600 / 1.3605 = $11,466.37
5 1.08^5 = 1.4693 $45,600 / 1.4693 = $31,035.19
-----------
total = $82,704.39
The NPV of the investment is therefore
$82,704.39 - $100,000 = -$17,295.61 .
Since this is a negative value, Copy Company should not buy the machine.
7.
Under the Capital Asset Pricing Model (CAPM), we must take into account
the risk of investing in a particular firm, as expressed by its beta. To
do so in an NPV calculation, we consider that the cost of capital is
the market portfolio's rate of return multiplied by the firm's beta value.
In this case, then, we take the cost of capital to be
1.4 * 12% = 16.8% .
With an annual cash flow of $15 million, the discounted return after
one year will be
$15 million / 1.168
= $12.8425 million .
Over a span of 20 years, the project accumulates
$12.8425 million * (1 - 1/1.168^20) / (1 - 1/1.168)
= $85.28 million
in revenue. We subtract this from the initial outlay to obtain the
project's NPV.
$100 million - $85.28 million
= $14.72 million .
8.
The answer is case (c), when the stock returns vary against each
other. This is because a loss in one stock will be offset by a gain in
the other, thereby providing an ideal hedge against risk.
9.
Combined leverage is the product of operating leverage and financial
leverage. The operating leverage is
(sales - variable costs) / (sales - variable costs - fixed costs)
and the financial leverage is
EBIT / (EBIT - debt * interest rate).
(a)
From Company A's capital structure, we obtain a financial leverage of
$200,000 / ($200,000 - $600,000 * .12)
= 1.5625 .
From Company B's operating plan, we obtain an operating leverage of
($1,000,000 - $500,000) / ($1,000,000 - $500,000 - $300,000)
= 2.5 .
The combined leverage is therefore
1.5625 * 2.5
= 3.90625
(b)
From Company B's capital structure, we obtain a financial leverage of
$200,000 / ($200,000 - $0 * .12)
= 1 .
Company A's operating plan yields an operating leverage of
($1,000,000 - $800,000) / ($1,000,000 - $800,000 - $0)
= 1 .
The combined leverage is therefore
1 * 1
= 1 .
(c)
We obtained the unusual result in part (b) because Company B has zero
debt and Company A has zero fixed costs. The financial leverage and the
operating leverage must therefore each have a value of 1. In consequence,
the combined leverage is also 1.
(d)
With a combined leverage of 1, a doubling in sales results in a doubling
of earnings. This amounts to a 100% increase in EPS.
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