FINANCE 351 Corporate Finance – John Graham
Things to do before the first class meeting
C
C
C
Read the Gifford and Brealey and Myers material (see class schedule)
Read over the syllabus and class schedule.
REVIEW the following:
C
where C is the cash flow and r is the discount rate.
r
Q: What is the present value of an investment that pays $100 each year forever, starting next year,
assuming the interest rate is 10%?
A: $100/.1=$1000
1) Perpetuity formula:
Note: the first payment is one year from today, not today.
Q: Displaced Perpetuity. What is the present value of an investment that pays $100 each year
forever, starting in three years, assuming that the interest rate is 10%?
A: The present value as of year 2 is $1000, like in the previous problem. This is the value in year
2 (NOT year 3) because the perpetuity formula is designed to value cash flows that start one year
hence. That is, the formula calculates the value as of t=2 of cash flows that start in t=3.
To answer the question, we need to value the lump sum of $1000 as of today. That is, we need to
1000
discount the $1000 back two periods:
= $826.44.
11
.2
2) Growing perpetuity formula:
C
where g is the rate of growth of the cash flow.
r−g
Q: What is the present value of an investment that pays $100 next year, $110 in year 2, $121 in year
3, growing at 10% each year forever, assuming that the interest rate is 15%?
A:
100
= $2000
.15−.10
3) Present value of an annuity formula:
1 
1 
x 1 −
 where n is the number of years the annuity
r  (1 + r) n 
payments continue.
Q: What is the present value of an investment that pays $200 each year for the next 12 years, starting
next year, assuming that the interest rate is 15%?
A: $200 x (annuity factor, n=12 years, r=15%) = $200(5.4206) = $1084.12
NOTE: An annuity is an investment that pays the same cash flow (like $200) for a specific number
of periods (like 12 years). During the second class session you will learn about deferred annuities
which are investment vehicles that allow an investor to “defer” paying tax on
income/interest/dividends. Although there are some similarities, it would be best if you thought of
an annuity as being a completely different concept/formula than a deferred annuity; they just happen
to have similar names. The next item, a displaced annuity is simply an annuity with a first payment
that occurs more than one year in the future. Don’t confuse “displaced” with “deferred”.
Q: Displaced Annuity. What is the present value of an investment that pays $200 each year for 12
years, with the first payment starting in year 5, assuming that the interest rate is 15%?
A: From the previous question we know that, as of year 4, the annuity is worth $1084.12. This is the
value in year 4 (NOT year 5) because the annuity formula is designed to value cash flows that start
one year hence.
To answer this question, we need to discount the lump-sum back to year 0. The present value of this
deferred annuity is:
108412
.
=$619.85
115
. 4
[
]
1
x (1 + r ) n − 1 where n is the number of years the annuity
r
payments continue. In this case, this is the value of the annuity as of year n. Notice that this is just
the present value annuity formula times (1+r)n. You might need this formula to do problem 2.4 in
the homework set. This formula is sometimes called the ‘sum of an annuity.’
4) Future value of an annuity formula:
Q: What is the future value (as of year 12) of an annuity that pays $200 each year for the next 12
years, starting next year, assuming that the interest rate is 15%?
A: $200 x (future value of an annuity factor, 12 years, 15%) = $200(29.002) = $5800.40
Note: this is the answer to the first question in 3) times (1+r)n: 1084.12(1.15)12 = $5800.4
That is, the FV of an annuity formula is (1+r)n times the PV of an annuity formula.
5) Cash flows: The introductory accounting course teaches that cash flows are not the same thing
as accounting income. Here, we review some basics related to calculating cash flows. These cash
flows in turn can be used in project valuation. There is nothing magic going on here. You can always
figure out the additional cash flows using accounting-like statements (as in the tables) or using the
short-cut formula.
a) Depreciation short-cut: ($depreciation)x(tax rate)
Q: How much cash flow is added if a firm gets an extra dollar of depreciation and the tax rate is 40%
Base Case
Extra $1 depreciation
Oper. Income
$20
$20
-Depreciation
-10
-11
Taxable Income
$10
$9
- taxes paid
-4
-3.6
Net Income
$6
$5.4
+ Depreciation
+10
+11
Cash Flow
$16
$16.4
A: An extra dollar of depreciation adds $1(tax rate) = $1(.40) = $0.40 extra cash flow.
b) Reducing expenses short-cut: ($expense reduction)x(1 - tax rate)
Q: How much cash flow is added if a firm reduces expenses by $1 and the tax rate is 40%?
Base Case
Expenses reduced by $1
Sales Revenue
$40
$40
-Expenses
$20
-$19
Oper. Income
$20
$21
-Depreciation
-10
-10
Taxable Income
$10
$11
- taxes paid
-4
-4.4
Net Income
$6
$6.6
+ Depreciation
+10
+10
Cash Flow
$16
$16.6
A: Reducing expenses by $1 adds $1(1-tax rate) = $1(1-0.40) = $0.60 extra cash flow.
Note: The depreciation is not needed in this example. It is included only for completeness.
c) Revenue short-cut: ($revenue) x (1-tax rate)
Q: How much cash flow is added by increasing revenue by $1 if the tax rate is 40%?
Base Case
Extra $1 sales revenue
Sales Revenue
$40
$41
-Expenses
$20
-$20
Oper. Income
$20
$21
-Depreciation
-10
-10
Taxable Income
$10
$11
- taxes paid
-4
-4.4
Net Income
$6
$6.6
+ Depreciation
+10
+10
Cash Flow
$16
$16.6
A: Increasing revenue by $1 adds $1(1-tax rate) = $1(1-0.40) = $0.60 extra cash flow.
Note: this is the exact same concept as reducing expenses by $1.
d) Interest deduction short-cut: ($interest)x(tax rate)
Q: How much extra cash flow do investors get if a firm takes an extra dollar of interest deduction
(relative to paying out the dollar to equity investors out of net income)? Tax rate is 40%
Base Case
Extra $1 interest
Oper. Income
$30
$30
- Depreciation
-10
-10
-Interest expense
-0
-1
Taxable Income
$20
$19
- taxes paid
-8
-7.6
Net Income
$12
$11.4
CF to stockholders
CF to bondholders
$22 =$10 dep+ $12
NI
$0
$21.4= $10 dep + $11.4 NI
$1
Total Cash Flow
$22
$22.4
A: An extra dollar of interest adds $1(tax rate) = $1(.40) = $0.40 extra cash flow to investors.
Note: depreciation included in this example for completeness only.
Note: we usually use the short-cut to account for interest benefits in the cash flows, rather than using
this long income-statement approach.
This example might look fairly simple. However, if I told you that “you can either account for the
tax benefits of debt in the cash flows or in the discount rate, not both” you might become
bewildered. “Accounting for the tax benefits of debt in the cash flows” amounts to knowing that
the cash flow is $22 base case + $0.40 financing benefits= $22.40 in this example. If you use a cash
flow of $22.40, DO NOT discount with an after tax cost of debt (like you would when you use
WACC.) Alternatively, you could consider the cash flow as $22 and instead discount using the aftertax cost of debt (like you would when you use WACC).