THE TIME VALUE OF MONEY

The concept of the time value of money is based on the availability of investment
opportunities. Investors loan their money to businesses that use the funds for capital
expenditures, and the businesses in turn pay interest to the investors for the use of their
money. Interest is the cost of debt; it is the price businesses pay for borrowing money.
The payment of interest makes the investment more valuable in the future than the
principal amount of the loan. Similarly, a cash flow expected at some point in the future
has less value in the present, for if that cash were available in the present, it could be
invested to earn interest, and thereby increase in value until it reached its future value in
the appropriate time period. This chapter deals with the mathematics associated with
calculating both future value and present value.

FUTURE VALUE – DISCRETE TIME

The future value (FVn) of an investment is literally the value of that investment at some
future point in time. The subscript n represents the number of periods into the future
included in the calculation (unless otherwise indicated, in this text the periods represented
by n will be years). The investment has more value in the future due to the payment of
interest by the borrower. The rate of interest paid for the invested money, the length of
time of the loan (its maturity), and the frequency of compounding (assuming interest is
compounded) determine the future value of the investment. Since the investment has
more value in the future, any multiplier used to calculate future value must be greater
than 1.

The mathematics for simple interest and compound interest are different, reflecting the
difference in the future value of these two methods. Compound interest can be calculated
two ways: discretely or continuously. The mathematics are different for these two
methods of compounding as well. This section will focus on discrete time; continuous
time will be discussed later in the chapter.

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Simple Interest
Simple interest is calculated purely on the principal (as opposed to compound interest,
which pays interest on both the principal and on accrued interest). Mathematically, this
means that the multiplier used to calculate the future value based on simple interest is
purely a function of the principal. It also means that the interest is calculated using
discrete time periods, because using smaller time periods does not affect the interest paid
as it does when interest is compounded. This will be made clear in the section on the
Effective Annual Rate

If i% simple interest is paid annually, at the end of n years, the principal P will have
earned simple interest n times, as in the following equation:

Since the amount of interest earned each year does not change, the formula can be written
more efficiently by multiplying the interest term by the number of years:

To further simplify the formula, the principal can be factored out of the equation. The
resulting formula calculates the future value based on an interest multiplier:

(7.1)

The term (1 + ni) is the future value simple interest factor (FVSIF i,n); that is, the term (1 +
ni) calculates the future value of an investment when simple interest is paid at i percent
annually for n years. Thus equation 7.1 can be rewritten as:

2
The following table shows the future value simple interest factors for various interest
rates and maturities. These factors can be compared to those calculated in the section on
compound interest to see the effects of compounding on the future value of an
investment.

2 Years 4 Years 6 Years 8 Years 10 Years

Interest Rate (i) FVSIFi,2 FVSIFi,4 FVSIFi,6 FVSIFi,8 FVSIFi,10
1.0% 1.0200 1.0400 1.0600 1.0800 1.1000
3.0% 1.0600 1.1200 1.1800 1.2400 1.3000
5.0% 1.1000 1.2000 1.3000 1.4000 1.5000
7.0% 1.1400 1.2800 1.4200 1.5600 1.7000
9.0% 1.1800 1.3600 1.5400 1.7200 1.9000
Table 7.1. Future Value Simple Interest Factors

Example
You have $100 available to invest in a project that will pay 5% simple interest annually
for 5 years. How much money will you have at the end of the investment’s maturity?

Compound Interest
Compound interest is calculated on both the principal and on the accrued interest.
Mathematically, this means that interest cannot be summed as in the simple interest
example, but must be multiplied, since the amount upon which the interest is calculated
increases each period, as interest from prior periods is added to the principal amount.
Compound interest can be calculated using discrete time periods or continuously. This
section deals with discrete compounding. Continuous compounded is treated later in the
chapter.

If i% interest is compounded annually, the principal will effectively grow by i% each

year, due to the compounding of accrued interest. At the end of n years, the principal P
will have grown at i percent per year n times, as per the following equation:

3
Since the principal is multiplied by the same amount each year, the equation can be
written more efficiently by using an exponent, with the value of the exponent equal to the
number of years included in the future value calculation:

(7.2)

The term (1 + i)n is the future value interest factor (FVIF i,n); that is, the term (1 + i)n
calculates the future value of an investment when interest is compounded annually at i
percent for n years (the term ‘compound’ is not included in the multiplier’s description
because interest is typically paid on a compounded basis). Thus equation 7.2 can be
rewritten as:

The following table gives the future value interest factors for various interest rates and
maturities. Note that each multiplier is higher than the equivalent value in table 7.1. This
is the result of compounding.

2 Years 4 Years 6 Years 8 Years 10 Years

Interest Rate (i) FVIFi,2 FVIFi,4 FVIFi,6 FVIFi,8 FVIFi,10
1.00% 1.0201 1.0406 1.0615 1.0829 1.1046
3.00% 1.0609 1.1255 1.1941 1.2668 1.3439
5.00% 1.1025 1.2155 1.3401 1.4775 1.6289
7.00% 1.1449 1.3108 1.5007 1.7182 1.9672
9.00% 1.1881 1.4116 1.6771 1.9926 2.3674
Table 7.2. Future Value Interest Factors

Example
You have $100 available to invest in a project that will pay 5% interest compounded

4
annually for 5 years. How much money will you have at the end of the investment’s
maturity?

Frequent Compounding

If interest is compounded more frequently than once a year, the investment will earn
more interest over the same period of time. That is because the effects of compounding
start earlier in the year, and are amplified with each additional compounding period
added per year.

Suppose that the principal earns the same i percent interest per year, but that interest is

compounded m times per year. This means that every years the current principal

(including prior accrued interest) will increase by %. Over n years, there will be

compounding periods. In this manner equation 7.2 can be modified to create a new
equation that incorporates frequent compounding:

(7.3)

In fact, equation 7.2 is really a special case of equation 7.3; if m = 1 (if there is only one
compounding period per year), the two equations are identical.

Example
You have $100 available to invest in a project that will pay 5% interest compounded
quarterly for 5 years. How much money will you have at the end of the investment’s

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maturity?

Effective Annual Rate

As shown above, if interest is compounded more frequently than once per year, the
investment actually earns more than the quoted annual interest rate. The Effective Annual
Rate (EAR, also known as the Annual Percentage Yield) determines the annual
compound interest rate that would produce the same future value as the more frequent
compounding formula. This is useful when comparing investments that have a different
number of compounding periods per year, as it allows you to standardize the interest rates
for the two investments.

To determine the EAR, first set the future value using the frequent compound rate equal
to the future value using the effective annual compound rate. This is done by using
equation 7.3 for the frequent compounding rate and rewriting equation 7.2 using the EAR
in place of the interest rate:

Dividing both sides by P eliminates the principal and allows you to work with the interest
multipliers directly:

6
Taking the nth root of each side eliminates the exponent for the number of years, and gets
you closer to isolating the EAR:

= ; = 1 + EAR

Finally, by subtracting one from each side, EAR is isolated, and the equation is solved:

– 1 = EAR (7.4)

In the frequent compounding example, you invested $100 for 5 years at 5% annual
interest, compounded quarterly, which had a future value of $128.20. To determine the
Effective Annual Rate for this example, plug the interest rate and compounding period
values into the EAR equation:

EAR = – 1 = 1.01254 – 1 = 0.050945 or 5.0945%

To illustrate that this annual interest rate of 5.0945% is equal to the quarterly interest rate
of 1.25%, use the EAR as the annual compounding rate for 5 years:

FV5 = $100(1.050945)5 = $128.20

When the quoted annual rate is 5%, but you earn compound interest quarterly, you are
effectively earning 5.0945% per year.

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Example
Suppose you want to open a new savings account, and there are two banks in your
neighborhood. Bank A offers an annual interest rate of 4.75% compounded semi-
annually (every 6 months), while bank B offers an annual interest rate of 4.73%
compounded monthly. Which account will earn you more interest? The EAR will help
you determine this:

Bank A: EAR = – 1 = 1.023752 – 1 = 0.048064 or 4.8064%

Bank B: EAR = – 1 = 1.00394212 – 1 = 0.048339 or 4.8339%

The EAR for bank B is higher, meaning that over time your money will earn more
interest in their savings account.

The EAR shows the impact of compounding more frequently than on an annual basis; it
has no relevance when simple interest is paid. Since simple interest applies only to the
principal, it makes no difference how small the units of time are dissected. For example,
if $100 is invested at 5% simple interest that is paid quarterly, at the end of the year the
investment will still be worth the same amount as if interest were compounded annually:

FV = $100 = $105.

Annuities
An annuity is a series of equal payments made at regular intervals for a finite period of
time. For example, you may set up an annuity to pay for your child’s college tuition,

8
which requires you to make fixed monthly payments for a given number of years. If the
payments are made to an interest-bearing account, the future value of the account is
determined by the interest rate paid on the account, the frequency of compounding, and
the number and size of the payments. When the payments are made at the end of the
period it is called a regular annuity, whereas when the payments are made at the
beginning of the period it is referred to as an annuity due.

Future Value of an Ordinary Annuity

The geometric expansion technique discussed in section 3.4 is needed to determine the
multiplier used to calculate the future value of an ordinary annuity. The structure of the
cash flows in an ordinary annuity is as follows: if the payments are made annually for n
years, then the first payment will earn interest for n-1 years (since it is made at the end of
the first year), the second payment will earn interest for n-2 years, and so on until the last
payment, which will earn no interest at all (since it will be paid at the end of the last year,
or at the maturity of the annuity). If PMT is used to represent the fixed payment and i is
used to represent the interest rate (compounded annually), the formula for the future
value of the ordinary annuity would be:

FVA = PMT

If n is large, determining the future value of this account would require many steps and
would be cumbersome even with a computer. The geometric expansion technique,
however, allows you to simplify the formula. The first step is to multiply both sides of the
equation by 1 + i:

FVA(1 + i) = PMT

The second step is to subtract the first equation from this newly created equation:

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 FVA(1 + i) – FVA = PMT

The subtraction cancelled out all of the duplicate terms in the two equations, leaving only
the first term in the second equation minus the last term in the first equation (the only two
terms for which there were no duplicates). Now if you multiply the first two terms on the
left side of the equation, you can eliminate the third term on the left side:

FVA(1) + FVA(i) – FVA = PMT ; FVA(i) = PMT

The third step is to divide both sides of the equation by i in order to isolate the FVA:

FVA =

Finally, the PMT term can be stripped out of the right-hand term of the equation, in order
to separate the payment from the multiplier:

FVA = PMT (7.5)

The term is the future value interest factor of an ordinary annuity (FVIFA i,n);

that is, the term calculates the future value of an ordinary annuity investment

when interest is compounded annually at i percent for n years. Therefore equation 7.5 can
be rewritten as:

FVAn = PMT(FVIFAi,n)

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A simple example will prove that the formula is correct. Suppose you are investing $100
at the end of each year for 5 years, and that the payments will earn 5% interest
compounded annually until the maturity date (when the last payment is made). This
means that the first payment (made at the end of year 1) will earn 4 years of interest, the
second payment will earn 3 years of interest, the third payment will earn 2 years of
interest, the fourth payment will earn 1 year of interest, and the final payment will earn
no interest at all. A straightforward mathematical structuring of this annuity would be:

FVA5 = $100(1.05)4 + $100(1.05)3 + $100(1.05)2 + $100(1.05) + $100

= $121.55 + $115.76 + $110.25 + $105 + $100 = $552.56

If you plug the values of PMT, n and i from this problem into equation 7.5, you get the
same answer as above, demonstrating that the formula is accurate:

FVA5 = $100 = $100(5.5256) = $552.56

Example

You want to buy the new model convertible that comes out one year from now. The car
will cost $25,000, including fees and taxes. You currently have $5,704 in an account that
pays 9% interest compounded monthly. How much do you need to deposit in that account
at the end of each month to be able to pay cash for the car in one year?

First you have to determine how much you will already have in the account based on
your current balance, then back that out of the $25,000 needed to get at the future value
of the annuity. Since the current balance will earn 9% compounded monthly for one year,
the calculation is straightforward:

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So the future value of the annuity is $18,761. Now you need to solve the FVA formula
for the payment;

Therefore you will need to deposit $1,500 at the end of each month for the next year.

Future Value of an Annuity Due

As mentioned previously, an annuity due is an annuity that is paid at the beginning of the
period. As with the ordinary annuity, the geometric expansion technique is needed to
determine the multiplier. The structure of the cash flows in an annuity due is: if the
payments are made annually for n years, then the first payment will earn interest for n
years (since it is made at the beginning of the first year), the second payment will earn
interest for n-1 years, and so on until the last payment, which will earn interest for one
year. If PMT is used to represent the fixed payment and i is used to represent the interest
rate (compounded annually), the formula for the future value of the annuity due would
be:

FVAD = PMT

Utilizing the geometric expansion method, you must multiply both sides of the equation
by 1 + i:

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 FVAD(1 + i) = PMT

Subtracting the first equation from the new equation gets rid of most of the terms in the
brackets:

FVAD(1 + i) – FVAD = PMT

The next step is to simplify the term on the left side of the equation:

FVAD(1) + FVAD(i) – FVAD = PMT ;

FVAD(i) = PMT

Divide both sides by i to isolate the FVAD term:

FVAD =

And last, strip out the PMT term in order to separate the payment from the multiplier:

FVAD = PMT

The multiplier for the future value of an annuity due, , looks very

much like the multiplier for the future value of an annuity, . The difference is

that in the FVAD multiplier, both terms in the numerator have been increased by the term

13
(1 + i). You can strip this term out of the brackets as well, showing that the FVAD
multiplier is merely the FVA multiplier increased by 1 + i:

FVAD = PMT = PMT(FVIFAi,n)

This should make sense intuitively; the difference between an annuity due and an
ordinary annuity is that, in an annuity due, the payments are compounded for one more
period (since they are made at the beginning rather than at the end of the period).
Therefore to calculate the future value of an annuity due, it is merely necessary to
calculate the future value of an ordinary annuity, and then compound that value for one
more period. So for the 5-period ordinary annuity in the previous section the value would
be increased by 5%:

FVA5 = $100 = $100(5.5256)(1.05) = $552.56(1.05) = $580.19

PRESENT VALUE – DISCRETE TIME

The present value (PV) of an expected future cash flow is the value an investor would
assign to that cash flow if it were available today. The cash has less value now than in the
future because, if it were available today, it could be invested to earn interest. The term
discounting is used to describe the process of calculating the present value of a future
cash flow. The discount rate used and the number of periods discounted determine the
present value of the cash flow. Since the cash flow has less value in the present, any
multiplier used to calculate present value must be less than 1.

Mathematically, discounting is the inverse of compounding. In other words, the process

of adding value over time through compound interest is the exact opposite of the process
of removing value when going backward in time. More precisely, the multiplier used to
calculate the future value of a cash flow is the inverse of the multiplier used to calculate
the present value of a cash flow for the same number of periods and at the same interest

14
rate (and vice versa). Simple interest is not considered when calculating present value
multipliers as it is assumed that an investor will have the ability to invest at interest rates
that are compounded over time.

Discounting
The present value of a cash flow is calculated by dividing the future value by one plus the
discount rate raised to the power of how many years in the future the cash flow is
scheduled to arrive.

If FVn is used to represent the future value in n years and k is used to represent the
discount rate, the formula can be written in generalized form:

If you remove FVn from the numerator of the fraction, the present value multiplier can be
isolated:

(7.6)

The term is the present value interest factor (PVIF k,n); that is, the term

calculates the present value of a future cash flow when discounted at k percent annually
for n years. Thus equation 7.6 can be rewritten as:

Note the similarities between equations 7.2 and 7.6; if you substitute i for k and P for PV
(or vice versa), you can use one equation to derive the other:

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 1. ;

2. ;

3. ;

4. Substituting k for i and PV for P, .

This confirms the fact that compounding and discounting are inverse operations. This fact
can be further proven by choosing one of the compound interest factors and inverting it
(dividing it into 1). For example, if you use a discount rate of 5% for 6 years, the present
value interest factor is:

Inverting this number gives you the future value interest factor for 5% for 6 years (which
can be confirmed in Table 7.2):

Example
Your uncle offers you a choice: you can have $300 now or $500 six years from now. If
your discount rate is 7%, which cash flow should you choose?

The $500 in 6 years is worth more than the $300 now.

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Present Value of a Series of Cash Flows
The present value of a series of cash flows is simply the sum of the present values of the
individual cash flows. Therefore each cash flow must be discounted at the appropriate
rate and for the required number of periods, and then the present values of those cash
flows can be summed.

Assume that you are scheduled to receive $100 at the end of each year for 4 years. If your
opportunity cost (the discount rate) is 5%, the PV of the cash flows is calculated as
below:

If CFt is used to represent the cash flow in a given time period, the above equation can be
written in a generalized form:

(7.7)

Example
You have the opportunity to invest in a project that is expected to pay you $500 at the end
of the first year, $550 at the end of the second year, $600 at the end of the third year,
$650 at the end of the fourth year, and $750 at the end of the fifth year. Assuming your
annual discount rate is 6%, what should you pay for this investment?

You should pay no more for this investment than the present value of its expected future
cash flows. Therefore you need to use equation 7.7 to determine the present value of the
cash flows given above.

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 = $471.70 + $489.50 + $503.77 + $514.86 + $560.44 = $2,540.27

Present Value of an Ordinary Annuity

As with the future value multiplier, deriving the present value multiplier of an ordinary
annuity requires the use of the geometric expansion technique. The first payment will be
discounted for one period, the second payment will be discounted for two periods, and so
on until the last payment, which will be discounted for n periods. Using k to represent the
discount rate, the series of cash flows is represented as below:

Multiply both sides of the above equation by (1 + k) to derive the second equation:

Subtract the first equation from the second to reduce the term in brackets:

The left side of the equation can be simplified:

Divide both sides of the equation by k to isolate the PVA:

18
Finally, separate PMT from the fraction to isolate the multiplier:

(7.8)

The term is the present value interest factor of an ordinary annuity

(PVIFAk,n); that is, the term calculates the present value of an ordinary

annuity investment when discounted at k percent for n years. Therefore equation 7.8 can
be rewritten as:

Example
You want to buy a bond that has a par value of $1,000 and pays 6% in coupons annually.
The bond has 8 years until maturity, and the current market rate for similar bonds is
5.5%. What should you pay for this bond?

The bond’s coupon payments are an annuity, so the present value of the coupon payments
can be calculated using equation 7.8, and that must be added to the present value of the
par value received at maturity (in 8 years). Therefore the formula will contain two
components: the present value of the coupon payments, and the present value of the par
value.

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Present Value of an Annuity Due
As mentioned earlier, an annuity due is an annuity that is paid at the beginning of the
period. As with the ordinary annuity, the geometric expansion technique is needed to
determine the multiplier. The structure of the cash flows in an annuity due is: if the
payments are made annually for n years, the first payment will not be discounted (since it
is paid at the beginning of the period), the second payment will be discounted for 1 year,
the third payment will be discounted for 2 years, and so on until the last payment, which
will be discounted for n – 1 years. If PMT is used to represent the fixed payment and k is
used to represent the discount rate (compounded annually), the formula for the present
value of the annuity due would be:

Now you must perform the geometric expansion technique to simplify the equation:

1.

2.

3.

4.

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 5.

The multiplier for the present value of an annuity due, , looks very

much like the multiplier for the present value of an annuity, . The

difference is that in the PVAD multiplier, both terms in the brackets have been increased
by the term (1 + k). You can strip this term out of the brackets as well, showing that the
PVAD multiplier is merely the PVA multiplier increased by 1 + k:

This occurs because, in an annuity due, the payments are discounted for one less period
(since they are made at the beginning rather than at the end of the period). Therefore to
calculate the present value of an annuity due, it is merely necessary to calculate the
present value of an ordinary annuity, and then compound that value for one period at the
discount rate.

Amortization
The schedule of payments associated with a loan is called the loan’s amortization
schedule. It shows how much of each payment goes to the principal of the loan and how
much is used to pay interest. Since each payment is of equal amount, the loan is
structured like an annuity. The lending institution will require that the present value of
the loan payments be at least as much as the present value of the loan amount. Assuming
the payments are made at the end of the period, the loan payment formula is a form of the
equation for the present value of an ordinary annuity.

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If PVL is substituted in equation 7.8 for PVA, the equation can be solved for PMT, which
gives the formula for calculating the payment amount:

1.

Multiply both sides by k to remove the first fraction;

2.

Divide both sides by the term in brackets to isolate PMT;

3.

Example
You want to borrow $100,000 from your bank. The bank requires that the money be
repaid in annual payments over 6 years, and they charge 5% interest on the loan. What
would your payments be?

The amortization schedule shows how much of each payment goes to the principal and
how much is for interest. Interest is charged on the outstanding principal at the end of
each year, so as more payments are made and less principal remains to be paid off, the
amount of each payment that goes to pay interest decreases (and the amount that goes to
pay off the principal increases), as can be seen in Table 7.3.

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 Principal Principal
Year Outstanding Payment Interest Principal Remaining
1 100,000.00 19,701.75 5,000.00 14,701.75 85,298.25
2 85,298.25 19,701.75 4,264.91 15,436.83 69,861.42
3 69,861.42 19,701.75 3,493.07 16,208.68 53,652.74
4 53,652.74 19,701.75 2,682.64 17,019.11 36,633.63
5 36,633.63 19,701.75 1,831.68 17,870.07 18,763.57
6 18,763.57 19,701.75 938.18 18,763.57 0.00

Table 7.3. Amortization Schedule

The amount of interest in each payment is 5% of the amount in the Principal Outstanding
column for the year. The remainder of the payment goes to pay off the principal.

Present Value of a Perpetuity

A perpetuity is an annuity that has no end (i.e., the payments continue forever). This
means that, in the PVA model, n goes to infinity:

As n goes to infinity, the term goes to zero (the denominator gets infinitely

large, which makes the value of the fraction infinitely small). Therefore the formula can
be rewritten as follows:

(7.9)

Example

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If you were offered a perpetuity of $5,000 per year, what would you be willing to pay for
that perpetuity if your discount rate is 8%?

The present value of the cash flows sets the maximum you should pay for the perpetuity.
Therefore you should pay no more than $62,500.

Constant Growth Model

The fundamental principle of valuation states that the value of any asset is equal to the
present value of its expected future cash flows. The Dividend Discount Model, also
known as the Gordon Model, calculates the intrinsic value of a firm’s common stock as
the present value of its expected future dividends. The model assumes that the dividends
will grow at a constant rate over time, and that the discount rate used to calculate the
present value will remain constant over time as well. It also assumes that dividends will
be paid forever, making the dividends in effect a growing perpetuity. If D 1 is used to
represent the next dividend to be paid (one year in the future), g is used to represent the
constant growth rate, and k is used to represent the discount rate, the present value of the
perpetuity of dividends is modeled as follows:

Since the first dividend is received one period in the future, it will grow for one less
period than the number of discounting periods. This creates the difference in the exponent
terms for the fraction. If the exponent terms for the fraction were the same, the right-hand
term on the right side of the equation would be an infinite geometric series, and could be
simplified using the geometric expansion technique. Therefore the right side of the

equation will be multiplied by in order to make use of the technique:

24
 Where g < k

It is important that the growth rate be less than the discount rate for this model, because if

g > k, the term would go to infinity over time. Now the geometric expansion can

be used to simplify the equation. First expand the terms in the equation by removing the
summation symbol:

1.

Multiply this equation by ;

2.

Subtract this from the first equation;

3.

Factor out PV on the left side of the equation;

4.

Isolate PV by dividing both sides of the equation by the right-hand term on the left side
of the equation;

5.

Multiply the right-hand term on the right side of the equation by to simplify the

right-hand term;

25
 6.

Simplify the terms in the denominator of the right-hand term on the right side of the
equation;

7.

Thus the Gordon Model can be seen as a perpetuity in which the growth rate is subtracted
from the discount rate in the denominator. The growth in dividends helps to compensate
investors for the time value of money.

Example
If a firm’s next dividend is expected to be $3.00, and the dividend is expected to grow at
a constant rate of 4% per year, what would be the intrinsic value of the stock if the
discount rate is 10%?

Non-Constant Growth Rate Model

The Gordon Model only works for firms whose dividend is expected to grow at a
constant rate. Despite this, the fundamental principle of valuation can still be used in
conjunction with the Gordon Model to value firms whose dividend growth rate is
expected to change over time. The approach for such firms is a four step process:
1. Calculate the present value of the dividends during the non-constant growth
period;
2. Use the Gordon Model to value the stock at the end of the non-constant growth
period;
3. Calculate the present value of the Gordon Model calculation from step 2;

26
 4. Sum the present values calculated in steps 1 and 3.

Example
Suppose a biotech company that developed a prescription drug has only three years left
on its patent. The firm’s most recent dividend was $5.00, and the dividend growth rate
during the life of the patent has been 12%. After the patent runs out, competitors will be
able to bring generic equivalents of the product to the market. The firm’s marketing
analysts believe that a several firms will enter the market during the first three years after
the patent expires. Based on their analysis, the CFO believes that the dividend growth
rate will be 10% in year 4, 8% in year 5, and 6% in year 6 and beyond. The firm’s
discount rate is 15%. Using these assumptions, the value of the firm can be calculated
using the four step approach given above:

1.

2.

3.

4.

Based on the assumptions made, the intrinsic value of the stock is $71.67.

27
FUTURE VALUE – CONTINUOUS TIME

As discussed in the section on frequent compounding, increasing the number of

compounding periods in a year increases the amount of compound interest earned in that
year. While more frequent compounding leads to more interest, the increase in interest
gets smaller as the number of compounding periods per year gets larger. For example,
suppose you invest $1,000 for one year at an annual interest rate of 5%. The future value
of the investment in one year based on different numbers of compounding periods per
year is given in Table 7.4.

Investment i 1 period 2 periods 4 periods 12 periods 52 periods 365 periods

$1,000 0.05 $1,050.00 $1,050.63 $1,050.95 $1,051.16 $1,051.25 $1,051.27
Table 7.4. The Impact of Increasing Frequency of Compounding

While interest does increase with each rise in the number of compounding periods, the
incremental increases become smaller. This function of increasing frequency has a limit;
that is, as the number of compounding periods increases, the value of the investment
approaches a maximum. The limit can be seen more clearly if we isolate the
compounding function itself, which is equation 7.3 where n = 1. The results can be seen
in Table 7.5.

m y = (1 + 1/m)m y
2 y = (1 + 1/2)2 2.25
5 y = (1 + 1/5)5 2.48832
10 y = (1 + 1/10)10 2.59374246
100 y = (1 + 1/100)100 2.704813829
1,000 y = (1 + 1/1,000)1,000 2.716923932
10,000 y = (1 + 1/10,000)10,000 2.718145927
100,000 y = (1 + 1/100,000)100,000 2.718268237
1,000,000 y = (1 + 1/1,000,000)1,000,000 2.718280469
Table 7.5. The Limit of the Compounding Function

28
As m approaches infinity, y approaches 2.7182818 (rounded to 7 decimal places), which
is known as e. The formula for this limit is written as follows:

If the interest rate (i) and the number of years (n) are included, the formula changes
slightly:

Thus the future value formula is written as:

(7.10)

This is the formula used to calculate the future value of a cash flow with continuous
compounding; it assumes an infinite number of compounding periods per year. The proof
for this formula is contained in Appendix A at the end of this chapter. The term ein in
equation 7.10 is the future value interest factor when interest is compounded
continuously (FVIFCi,n); that is, the term ein calculates the future value of an investment
when interest is compounded continuously at an annual rate of i percent for n years.
Based on this understanding, the formula can be rewritten as below:

29
Example

Suppose you invest $100 for 5 years, and it earns 5% interest compounded continuously.
What is the future value of the loan at maturity?

PRESENT VALUE – CONTINUOUS TIME

As with discrete time, discounting in continuous time is the inverse of compounding in

continuous time, given the same annual rate and time to maturity. Based on the proofs
done for equations 7.6 and 7.10, the formula for discounting in continuous time is:

Therefore the formula for calculating the present value of an expected future cash flow is:

(7.11)

The term e-in in equation 7.10 is the present value interest factor when discounting is done
on a continuous time basis (PVIFCi,n); that is, the term e-in calculates the present value of
an expected future cash flow when discounted continuously at an annual rate of i percent
for n years. Therefore the formula can be rewritten as below:

30
Example

Suppose you were expecting a cash flow of $128.40 to arrive in exactly five years. What
would be the present value of that cash flow if it were discounted at a continuous rate of
5% per year?

To show once again how the present value multiplier is the inverse of the future value
multiplier even in continuous time, assume an interest rate of 5% for five years. The
future value interest factor with continuous compounding is:

Dividing that multiplier into 1 gives the present value interest factor with continuous
discounting:

31
 End of Chapter Problems

1. If you invest $2,500 at a simple interest rate of 6% annually, what will your
investment be worth at the end of 5 years?

2. You invest $1,800 for 6 years at an annual compound interest rate of 5.5%. How
much will your investment be worth when you cash out?

3. If you put $16,000 into a 3 year Certificate of Deposit that pays interest at 4.75%,
compounded quarterly, how much will you have when the CD matures?

4. Suppose you put $4,500 into an account that pays 5.15% interest, compounded
continuously. How much will you have in the account after 8 years?

5. Assume that you expect to receive $10,000 in 7 years. What is the present value
of that cash flow if your discount rate is 6.25%?

6. As a settlement from a recent lawsuit, you expect to receive $4,000 in 1 year,

$3,500 in 2 years, $3,000 in 3 years, $2,500 in 4 years, and $2,000 in 5 years.
What is the present value of those expected future cash flows, assuming your
discount rate is 5%?

7. Suppose that you have $5,000 to invest for 8 years at an annual interest rate of
4.25%. What will be the future value of that investment if simple interest is paid?

8. What would be the future value of the investment in problem #7 if interest is

compounded:
a. Annually;
b. Semiannually;
c. Quarterly;
d. Monthly;
e. Daily;
f. Continuously.

32
9. You are offered a $1,000 par value bond that pays 7% interest annually and
matures in 6 years. What should you pay for the bond if the current market rate
for similar bonds is 8%?

10. Your bank offers a Certificate of Deposit that pays an annual interest rate of 3.6%
compounded monthly. Another bank in your neighborhood is offering a CD that
pays an annual interest rate of 3.595% compounded weekly. Which CD will earn
you more money over time?

11. Suppose you invest in an ordinary annuity into which you pay $2,000 each year
for 9 years. If the annuity earns 6.5% interest, how much will you have available
when it matures?

12. If the annuity in problem #11 is an annuity due, how much will be available at
maturity?

13. A firm’s next dividend is expected to be $1.65, and their discount rate is 12%. If
the intrinsic value of their common stock is $82.50, what is the firm’s expected
annual growth rate?

14. Silicon Biotech just paid a dividend of $2.00. Their growth rate over the next 4
years is expected to be 15%, after which they are expecting a constant growth rate
of 8% moving forward. If their discount rate is 11%, what is the intrinsic value of
their stock?

15. You just turned 30 years old, and landed the job of your dreams. It is your
intention to retire on your 55th birthday, at which time you expect to require
$80,000 per year to live comfortably until you reach age 75 (your estimated life
expectancy). On the day you retire, you intend to purchase a mobile home and
drive around the country following your favorite rock band on tour. The mobile
home you like currently costs $65,000, and is expected to increase in price an
average of 5% per year. Your bank has an annuity plan that pays 8.5%
compounded annually. How much will you have to put in the annuity each year to
fund your dream retirement?

33
16. The house you want to buy costs $240,000. Your bank charges 6% interest,
compounded monthly, on a 30-year loan. If you put 10% down on the house up
front, how much will your mortgage payment be?

17. You just received a raise that will increase your income by $100 each month (paid
at the end of the month). You decide to put the extra money in an account to save
for a cruise that will cost $6,000, including airfare and all expenses. If the account
pays 6% compounded monthly, how long will it take you to save enough to pay
for the cruise?

34
 Appendix A: Proof for the Continuous Compounding Formula

This proof will show that as the number of compounding periods increases to infinity, the
formula for compounding becomes a function of e. Start by assuming that the number of
compounding years is one (n = 1):

(A.1)

First note that the equation is clearly true if i = 0 (since e0 = 1). The proof looks at the
case when i ≠ 0. Take the natural log of the left-hand side of the equation:

Recall from the chapter on derivatives that the definition of a derivative is given as:

This equation is known as the difference quotient, and it is necessary to put the natural
log of the continuous compounding formula into this format in order to solve the proof.
There are two steps that must be performed in order to manipulate the equation to reach
that format:

1. The equation must be multiplied by ;

35
 2. The term must be subtracted inside the bracketed term. Since = 0,
this does not change the value of the function.

These steps must be performed separately, as there is one more maneuver that needs to be

done between step 1 and step 2; the term must appear in both the numerator and

denominator of the equation to replicate the term h in the difference quotient. This can be

achieved by noting that multiplying by m is the same as dividing by .

1.

By subtracting the term x is replicated and the equation is in the difference quotient
format:

2.

Now x can be substituted for 1 and h can be substituted for ;

Note that in equation A.1, as ;

36
Since the limit is now in derivative form, it can be rewritten as a derivative. Remember

that x was used to represent 1, and that the derivative of the natural log of x is .

Therefore it has been shown that as m gets infinitely large, goes to i. Since

= x, it follows that:

By extension, if , the same process will show that:

37
 Solutions to End of Chapter Problems

1. FV5 = 2,500(1.30) = $3,250.

2. FV6 = 1,800(1.055)6 = $2,481.92.

3. FV12 = 16,000(1.011875)12 = $18,434.97.

4. FV8 = 4,500e.0515(8) = $6,794.25.

5. PV = = $6,541.80.

6. PV =

= $13,199.44.

7. FV8 = 5,000(1.34) = $6,700.

8. a. FV8 = 5,000(1.0425)8 = $6,975.55.

b. FV16 = 5,000(1.02125)16 = $6,999.76.

c. FV32 = 5,000(1.010625)32 = $7,012.15.

d. FV96 = 5,000(1.003542)96 = $7,020.52.

e. FV2,920 = 5,000(1.00011644)2,920 = $7,024.60.

f. FV8 = 5,000e.0425(8) = $7,024.74.

9. VBond = = $323.60 + $630.17 = $953.77.

10. EARCD1 = = 3.66%; EARCD2 = = 3.66%.

11. FVA9 =

12. FVAD9 = FVA9(1 + i) = 23,463.70(1.065) = $24,988.85.

38
13. V = 82.50 = or 10%.

14. PVD1 =

PVD2 =

PVD3 =

PVD4 =

P4 =

VStock = 2.0721 + 2.1467 + 2.2241 + 2.3042 + 82.9530 = $91.70.

15. A. Cost of mobile home @ 55: .

B. PV of annuity due @ 55:

C. Total required @ age 55: 220,113 + 821,418 = $1,041,513.

D.

16.

17. This situation describes an ordinary annuity in which you will be required to make
monthly payments of $100 for an undetermined but finite period of time. The twist on
this problem is that you are given the future value, and you need to solve for the
number of payments. Using n to represent the number of monthly payments, you can
plug the data into equation 7.5:

39
 $6,000 = $100

Now you must manipulate the equation to isolate the term containing n, at which
point you can use the natural log to solve for n. The steps involved in isolating the
term with the exponent are:

1. Divide both sides by 100 to isolate the term in brackets;

2. Multiply both sides by .005 to eliminate the fraction;

3. Add 1 to both sides to isolate the term with the exponent.

Here are the steps mathematically:

1. = ; 60 =

2. 60(.005) = (1.005)n – 1 ; .3 = (1.005)n – 1

3. 1.3 = (1.005)n

Now you can use the natural log to isolate and solve for n. The steps required to
accomplish that are:

1. Take the natural log of both sides of the equation;

2. Divide both sides of the equation by the natural log on the right of the equation to
isolate n;

3. Solve the fraction created in step 2.

Here are those steps mathematically:

1. ln(1.3) = n ∙ ln(1.005)

2. =n

40
3. = n = 52.48 ≈ 53

It will require 53 monthly payments (or almost 4 years) to save up enough money

for the cruise. To check your answer, plug the data back in to equation 7.5, using 53
for n:

FVA53 = $100 = $100 = $100(61) = $6,100

Thus at the end of the 53rd month you will have all of the expenses covered, plus $100
left over for souvenirs.

41