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. |