Citibank: basic of corporate finance
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PRACTICE EXERCISE 3.3
Directions: Calculate the answer to each question and enter the correct answer.
Check your solution with the Answer Key on the next page.
13. If the annual discount rate is 8.5%, how much will an investor pay to receive
$100,000 at the end of 1 year?
$_____________________
14. What is the present value of $250, to be received in two years time, using an annual
discount rate of 9%?
$_____________________
15. What is the present value of $10,000, due five years from now, using a discount
rate of 10%?
$_____________________
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ANSWER KEY
13. If the annual discount rate is 8.5%, how much will an investor pay to receive
$100,000 at the end of 1 year?
$92,166
PV = FV
T
[ 1 / (1 + R) ]
T
PV = $100,000 [ 1 / (1 + 0.085)]
1
PV = $100,000 [ 1 / 1.085]
PV = $100,000 [0.92166]
PV = $92,166
14. What is the present value of $250, to be received in two years time, using an annual
discount rate of 9%?
$210.42
PV = FV
T
[ 1 / (1 + R) ]
T
PV = $250 [ 1 / (1 + 0.09)]
2
PV = $250 [ 1 / (1.09)]
2
PV = $250 [ 0.84170]
PV = $210.42
15. What is the present value of $10,000, due five years from now, using a discount
rate of 10%?
$6,209.20
PV = FV
T
[ 1 / (1 + R) ]
T
PV = $10,000 [ 1 / (1 + 0.10)]
5
PV = $10,000 [ 1 / (1.10)]
5
PV = $10,000 [ 0.62092]
PV = $6,209.20
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PRACTICE EXERCISE 3.3
(Continued)
16. What is the present value of $2,000 to be received after 10 years, given an annual
discount rate of 12% discounted continuously?
$_____________________
17. Discounting continuously, what is the present value of $2,000, received after 10
years, given an annual discount rate of 10%?
$_____________________
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ANSWER KEY
16. What is the present value of $2,000 to be received after 10 years, given an annual
discount rate of 12% discounted continuously?
$602.38
PV = FV
T
[ e
-RT
]
PV = $2,000 [ e
-0.12 x 10
]
PV = $2,000 [ e
-1.2
]
PV = $2,000 [0.3011918]
PV = $602.38
17. Discounting continuously, what is the present value of $2,000, received after 10
years, given an annual discount rate of 10%?
$735.75
PV = FV
T
[ e
-RT
]
PV = $2,000 [ e
-0.10 x 10
]
PV = $2,000 [ e
-1.0
]
PV = $2,000 [0.367877]
PV = $735.75
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ANNUITIES
Series of
evenly
spaced, equal
payments
Many financial market securities involve more than one cash flow.
An annuity is a security with a series of equal payments (or
receipts) spaced evenly over a determined period. There are two
types of annuities:
• Ordinary annuity (also called deferred annuity)
• Annuity due
An ordinary annuity payment is received at the end of the payment
period, while an annuity due requires payment at the beginning of
the period. In this section, we focus on the valuation of ordinary
annuities, which are the most common. The calculations for an annuity
due are almost the same, but they include an adjustment for the timing
of cash payments.
Future Value of an Annuity
Compoundin
g annuity
payments
Calculating the future value of an annuity is a fairly straight-forward
exercise. However, it can be tedious as the following example
illustrates. Suppose that an investor invests $100 each December
31st for the next ten years. The account pays an annual interest rate of
5%. What will be the value of the account at the end of the ten years?
Let's build a table that outlines the compounding of the cash flows.
Year Deposit x FVIF = Payments
1 $100 x (1.05)
9
= 155.14
2 $100 x (1.05)
8
= 147.75
3 $100 x (1.05)
7
= 140.71
4 $100 x (1.05)
6
= 134.01
5 $100 x (1.05)
5
= 127.63
6 $100 x (1.05)
4
= 121.55
7 $100 x (1.05)
3
= 115.76
8 $100 x (1.05)
2
= 110.25
9 $100 x (1.05)
1
= 105.00
10 $100 x (1.05)
0
= 100.00
FV = $1,257.80
Figure 3.4: Future Value of an Annuity
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As you can see from the table, the interest on the deposit at the end of
Year One is compounded for nine years, interest on the deposit at the
end of Year Two for eight years, and so forth. The deposit at the end
of Year Ten earns no interest. Thus, at the end of ten years, the total
future value of this annuity is $1,257.80.
Using a
financial
calculator
Most financial calculators can perform this type of calculation.
The basic idea is to input the interest rate, the number of payments
and the size of the payments, and push the future value key to
calculate. Many calculators allow you to specify how many
payments per year and whether the payment is made at the beginning
of the period (annuity due) or at the end of the period (ordinary
annuity).
Present Value of an Annuity
Discounting
annuity
payments
The calculation of the present value of an annuity is the reversal of
the compounding process for the future value. The idea is to discount
each individual payment from the time of its expected receipt to the
present. We can illustrate by building another table
to calculate the present value of the annuity in Figure 3.4: ten
payments of $100, made on December 31st of each year, discounted
at a rate of 5%.
Year Payment x PVIF = Present Value
10 $100 x 1/(1.05)
10
= 61.39
9 $100 x 1/(1.05)
9
= 64.46
8 $100 x 1/(1.05)
8
= 67.68
7 $100 x 1/(1.05)
7
= 71.07
6 $100 x 1/(1.05)
6
= 74.62
5 $100 x 1/(1.05)
5
= 78.35
4 $100 x 1/(1.05)
4
= 82.27
3 $100 x 1/(1.05)
3
= 86.39
2 $100 x 1/(1.05)
2
= 90.70
1 $100 x 1/(1.05)
1
= 95.24
PV = $772.17
Figure 3.5: Present Value of an Annuity
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The present value of this annuity is $772.17. This means that an
investor with opportunities to invest in other investments of equal
risk at 5% would be willing to pay $772.17 for this annuity. Notice
that because the payments are received at the end of each period, we
discount them for the entire period.
Formula for
present value
of annuity
The formula for calculating the present value of an annuity is:
PV
a
= A[1/(1+R)]
1
+ A[1/(1+R)]
2
+ ... + A[1/(1+R)]
T
Where:
PV
a
= Present value of the annuity
A = Amount of each annuity payment
R = Discount rate
T = Number of periods that the annuity is to be received
The idea is to discount each individual payment from the year of its
expected receipt to the present. After discounting each cash flow, add
up the discounted cash flows to arrive at the present value. The table in
Figure 3.5 is a good way to organize the calculations.
Using a
financial
calculator
The quickest method for calculating the present value of an annuity
is to use a financial calculator. Input the number of payments, the
size of the payment, and the interest rate. Push the present value
key to derive the present value.
Perpetuities
Infinite
number of
payments
A perpetuity is special type of annuity where the time period for the
payments is infinity. Some examples of perpetuities are payments
from an education endowment fund or the dividends paid on
preferred stock.
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Forumula for
present value
of a perpetuity
The future value of a perpetuity is always infinity and, therefore,
pointless to calculate. The present value of a perpetuity is easily
calculated. The formula is:
PV
p
= A x (1 / R)
Where:
PV
p
= Present value of the perpetuity
A = Amount of each payment
R = Discount rate (required rate of return)
Example
Let's use the formula with an example. An investor requires 16%
return on investment. What will the investor be willing to pay for a
share of preferred stock with a $10 annual dividend? To calculate,
use these values: A = $10 and R = 16%.
PV
p
= A x (1 / R)
PV
p
= $10 x (1 / 0.16)
PV
p
= $62.50
If the investor pays $62.50 per share, she will receive a $10 dividend
per year, which equals a 16% return on her investment.
UNEVEN PAYMENT STREAMS
Unequal
size or
spacing
Sometimes an investment has payments of unequal size or spacing.
The present and future value formulas can be used to accommodate
these irregularities. For instance, suppose that you deposit $100 in an
investment at the end of Year One, $350 at the end of Year Three,
$250 at the end of Year Four, and $200 at the end of Year Five. If the
investment earns 6% compounded annually, what is the value of this
investment at the end of Year Six? Let's build a table to illustrate the
cash flows and their future values.
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Year Deposit x FVIF = Payments
1 $100 x (1.06)
5
= 133.82
2 $000 x (1.06)
4
= 0.00
3 $350 x (1.06)
3
= 416.86
4 $250 x (1.06)
2
= 280.90
5 $200 x (1.06)
1
= 212.00
6 $000 x (1.06)
0
= 0.00
FV = $1,043.58
Figure 3.6: Future Value of Annuity with Unequal Payments
You can see that the first deposit is in the account for five years,
earning compounded interest of $33.82 during that time; no deposit is
made in the second year; the third deposit earns compound interest for
three years in the amount of $66.86; and so on. The total interest the
account will earn for the six years is $1,043.58 – $900.00 (the total
amount of all the deposits) = $143.58.
Formula for
present value
of unequal
payments
The formula for calculating the present value of unequal payments is a
modification of the formula for the present value for an annuity.
PV = CF
1
[1/(1+R)]
1
+ CF
2
[1/(1+R)]
2
+ ... + CF
T
[1/(1+R)]
T
Where:
PV = Present value of the stream of cash flows
CF
i
= Cash flow (payment) in period i
R = Discount rate
T = Number of periods in the stream of cash flows
The idea is to discount each payment or cash flow from the time of
receipt back to the present time. The present value is the sum of all of
the discounted cash flows.
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