1
Introduction
While you are shopping for a financial product and/or a mortgage for your first home, you
will usually compare various financial products or mortgage on the basis of interest rate.
For example, if you are about to invest $5,000 into a saving account, you probably want
to choose the bank who offers the highest interest rate assuming that everything else is the
same. On the other hand, if you want to apply for a house mortgage, you want to have the
mortgage rate as low as possible assuming that everything else is the same. The lower the
interest rate is, the less interest you pay or receive. The notion of interest rate is widely
understood and considered as a common sense.
However, how do we consider the notion of interest and/or interest rate from the theoretical point of view? In the economics theory, interest is the opportunity cost of lending/borrowing, which means that the interest is the compensation to the lender for the
loss of use of money while it is loaned to the borrower, and the cost born by the borrower
to enjoy the benefit of extra money which he does not own. How do we translate this
economic-theoretic concept into a concrete mathematical language? In order to develop
further mathematical theories, we need to understand the ”time value” of money.
2
Time Value of Money
Money has its time value, and it is the fundamental principle of theory of interest. $1
today does not have the same value in term of purchasing power as it was 10 years ago
or it will be in 10 years. The whole theory of interest is built on this very fundamental
concept by defining equivalence among past, present, and future dollar values. Therefore,
it is imperative to understand the mathematical formulations which take the time value of
money into account through accumulation, amount, and present–value functions. Moreover,
this equivalence relationship will also be utilized in evaluation of many different financial
products.
2.1
Accumulation Function
An accumulation function, denoted by a(t; j), is an non-negative function of time t and
the real–valued parameter of j such that a(0; j) = 1 for j. The accumulation function tells
1
us how much $1 worths at time t if it is invested/deposited at time 0. I would like to call the
parameter j “growth” parameter to avoid later confusion with interest rate. Given a(·; j),
$a(t; j) at time t is said to be equivalent to $1 at time 0, i.e. ($1 @ time = 0) ∼ ($a(t; j) @
time = t).
2.2
Amount Function
Given the accumulation function a(t; j), the amount function, which gives how much the
principal P worths at time t if it is invested/deposited at time 0, is denoted by A(t; j). In
the modern financial environment, it is almost always the case that the growth parameter j
depends on the size of the principal P . For example, the interest rate of an $100,000 loan is
definitely different from a loan of million dollar. In other words, j = j(P ). The interaction
between j and P complicates the mathematical theories of interest and is beyond the scope
of this course. Therefore, we have to assume the independence of P and j in order to simplify
the mathematical formulations for an intuitive understanding.
By assuming the independence of j and P , we can express the relationship between the
accumulation and amount functions by the following equation:
A(t; j, P ) = P × a(t; j)
From the above equation, it is obvious that A(t; j, P ) = a(t; j) in the case of P = 1. Again,
through the use of amount function, $A(t; j, P ) at time t is equivalent to $P at time 0, i.e.
($P @ time = 0) ∼ ($A(t; j, P) @ time = t).
The accumulative/amount function gives the accumulated value of $P invested/deposited
at time 0. Then it is nature to ask what if $P is not deposited at time 0, say time s > 0,
then what will be the accumulated value at time t? For example, $100 is deposited into
an account at time 2, how much does the $100 grow by time 4 provided the accumulation
function a(t; j)?
Consider that a deposit of $X is made at time 0 such that the $X grows to $100 at time
2 (the same as a deposit of $100 made at time 2).
A(2; j, X) = Xa(2; j) = 100
100
X =
a(2; j)
2
Then the accumulated value of $X at time 4 is given by A(4; j, X) =
In other words, since $100 at time 2 is equivalent to $X(=
100
)
a(2;j)
100
×a(4; j)
a(2;j)
= 100 a(4;j)
.
a(2;j)
at time 0, and $X at time
at time 4, $100 at time 2 is equivalent to $100 a(4;j)
at time 4.
0 is equivalent to $100 a(4;j)
a(2;j)
a(2;j)
In general, if $P is deposited at time t1 , then the accumulated value of P at time t2 > t1
a(t2 ;j)
is P × a(t
, and
1 ;j)
factor
a(t2 ;j)
a(t1 ;j)
a(t2 ;j)
a(t1 ;j)
is called the accumulation factor. In other words, the accumulation
gives the dollar value at time t2 of $1 deposited at time t1 . Note that X is said
to be the present value at time 0 of $100 at time 2. The notion of present value is discussed
in the later section.
3
Interest and Effective Rate per Period
The interest It1 ,t2 is the amount generated/earned between time t1 and t2 by the $P invested/deposited at time 0.
It1 ,t2 = A(t2 ; j, P ) − A(t1 ; j, P )
The interest rate between time t1 and t2 , denoted by it1 ,t2 , is the ratio of the amount of
interest earned during time t1 and t2 to the amount of principal at time t1 , which P is
invested/deposited at time 0.
A(t2 ; j, P ) − A(t1 ; j, P )
A(t1 ; j, P )
P a(t2 ; j) − P a(t1 ; j)
=
P a(t1 ; j)
a(t2 ; j) − a(t1 ; j)
=
a(t1 ; j)
it1 ,t2 =
Note that it1 ,t2 is independent of the initial investment P . The effective interest rate
per period, denoted by jt+1 , is defined as the ratio of the amount of interest earned during
the period to the amount of principal invested at the beginning of the period. For example
jt+1 =
A(t+1)−A(t)
.
A(t)
In contrast to the effective interest rate per period, the effective discount
rate per period, denoted by dt+1 , is defined as the ratio of the amount of interest earned
during the period to the amount of principal invested at the end of the period. For example
dt+1 =
A(t+1)−A(t)
.
A(t+1)
3
Example 3.0.1 If $100 is deposited into an account, whose accumulation function is a(t; j) =
√
1 + 0.03 t, at time 0,
Q.1 Find the amount of interest generated at time 4.
Q.2 Find the interest rate between time 0 and 3.
Q.3 Find the amount of interest generated between time 1 and 4.
Q.4 Find the effective interest rate for the first (between time 0 and time 1) and second
period (between time 1 and time 2).
Q.5 Find the effective discount rate for the first and second period.
√
Note that A(t; j, P ) = P × a(t; j) = 100(1 + 0.03 t)
A.1 t1 = 0 and t2 = 4.
√
√
I0,4 = A(4; j, P ) − A(0; j, P ) = 100(1 + 0.03 4) − 100(1 + 0.03 0) = 106 − 100 = 6
A.2 t1 = 0 and t2 = 3.
i0,3
√
√
√
1 + 0.03 3 − (1 + 0.03 0)
a(3; j) − a(0; j)
√
=
= 0.03 3
=
a(0; j)
1 + 0.03 0
√
√
A.3 I1,4 = A(4; j, P ) − A(1; j, P ) = 100(1 + 0.03 4) − 100(1 + 0.03 1) = 106 − 103 = 3
A.4 The first period is between time 0 and time 1: j1 =
A(1)−A(0)
A(0)
=
100(1.03)−100
100
= 3%
A(1)−A(0)
A(1)
=
100(1.03)−100
103
= 2.9126%
The second period is between time 1 and time 2:
j2 =
A(2)−A(1)
A(1)
=
√
100(1+0.03 2)−103
103
= 1.206447269%
A.5 The first period is between time 0 and time 1: d1 =
The second period is between time 1 and time 2:
d2 =
4
A(2)−A(1)
A(2)
=
√
100(1+0.03 2)−103
√
100(1+0.03 2)
= 1.1920656258%
Common Types of Accumulation Functions
There are several common types of accumulation functions a(t; j), whose parameter j has a
specific name. They are introduced as follow:
4
4.1
Simple Interest Accumulation Function
If an accumulation function is of the form a(t) = 1+js t, then it is said to be a simple–interest
accumulation function, where js is called the simple interest rate. Note that time t is usually
measured by year and between 0 and 1, and hence js is reported as the ”annualized” interest
rate.
Example 4.1.1 (Textbook Example 1.4) On January 31, Smith borrows $500 from Brown
and gives Brown a promissory note. The note states that the loan will e repaid on April 30
of the same year, with interest at 12% per annum. On March 1 Brown sells the promissory
note to Jones, who pays Brown a sum of money in return for the right to collect the payment
from Smith on April 30. Jones pays Brown an amount such that Jones’ yield (interest rate
earned) from March 1 to the maturity date can be stated as an annual rate of interest of
15%.
1. Determine the amount that Jones paid to Brown and the yield rate (interest rate)
Brown earned quoted on an annual basis. Assume all calculations are based on simple
interest and a 365-day year.
2. Suppose instead that Jones pays Brown an amount such that Jones’ yield is 12%.
Determine the amount that Jones paid.
4.2
Compound Interest Accumulation Function
If an accumulation function is of the form a(t) = (1 + jc )t , then it is said to be a compound–
interest accumulation function, where jc is called the compound interest rate. It can be
shown that jc is equivalent to the effective interest rate per period.
Example 4.2.1 (Textbook Example 1.1) The current rate of interest quoted by a bank
on its savings account is 9% per annum, with interest credited annually. Smith opens an
account with a deposited of $1,000. Assuming that there are no transactions on the account
other than the annual crediting of interest, determine the account balance just after interest
is credited at the end of 3 years.
Example 4.2.2 (Textbook Example 1.3) Smith deposits $1,000 into an account on January 1, 2000. The account credits interest at an effective annual rate of interest of 5% every
5
December 31. Smith withdraws $200 on January 1, 2002, deposits $100 on January 1, 2003,
and withdraws 250 on January 1, 2005. What is the balance in the account just after interest
is credited on December 31, 2006?
Example 4.2.3 (Textbook Exercise 1.1.4) Joe deposits $10 today and another $30 in
five years into a fund/account paying simple interest of 11% per year. Tina will make the
same two deposits, but the $10 will be deposited n years from today, and the $30 will be
deposited 2n years from today. Tina’s deposits earn an effective annual rate of 9.15%. At the
end of 10 years, the accumulated amount of Tina’s deposits equals the accumulated amount
of Joe’s deposits. Calculate n.
4.3
Simple Discount Accumulation Function
If an accumulation function is of the form a(t) =
1
,
1−ds t
then it is said to be a simple–discount
accumulation function, where ds is called the simple discount rate. Just like js , Time t is
usually measured by year and between 0 and 1, and ds is reported as the ”annualized”
discount rate.
4.4
Compound Discount Accumulation Function
If an accumulation function is of the form a(t) = (1−dc )−t , then it is said to be a compound–
discount accumulation function, where dc is called the compound discount rate. It can be
proved that dc is equivalent to the effective discount rate per period.
4.5
Force of Interest
The force of interest, denoted by i(∞) or δt , is interpreted as the instantaneous rate of growth
of the investment per dollar at time point t. It can be derived from A(t).
δt =
d
A0 (t)
=
ln(A(t))
A(t)
dt
6
Integrating this equation from time 0 to time t, we get
Z t
Z n
d
δs ds =
ln(A(s)) ds
0 ds
0
= ln(A(t)) − ln(A(0))
A(t)
= ln
A(0)
Z t
A(t) = A(0) × exp
δs ds
0
Z n
δs ds
= P × exp
0
In general, A(t2 ) = A(t1 ) × exp
nR
t2
t1
o
δs ds or
A(t2 )
A(t1 )
= exp
nR
t2
t1
δs ds
o
Example 4.5.1 (Textbook Example 1.13) Derive an expression for δt if accumulation
function is of the form of
1. simple interest
2. compound interest
3. simple discount
4. compound discount
Example 4.5.2 (Textbook Example 1.14) Given δt = 0.08 + 0.005t, calculate the accumulated value over five years of an investment of $1000 made at each of the following
times:
1. at time 0
2. at time 2
4.6
Rate Equivalence
Two rates are equivalent if they result in the same pattern of growth over the same time
period. In other words, two accounts with different accumulation functions, a1 (t; j1 ) and
a2 (t; j2 ) respectively, have the identical balances at time 0 and at time n. The j1 and j2 are
equivalent over the time period of n. Note that a1 (t; j1 ) is not necessarily of the same form
as a2 (t; j2 ). The following example illustrates the notion of equivalence; further examples
are provided in the later section.
7
Example 4.6.1 A $100 is deposited into an account with accumulation function a1 (t; 3.5%) =
1 + 0.035t, and another $100 is deposited into a different account with accumulation function
a2 (t; j) = (1 + j)t . Suppose that two account have the same balance at the end of second
period. Find j
5
Present Value
5.1
Present Value Function
Present value function is a function which gives you the amount to invest/deposit
at time 0 if you want to receive $1 at time t, provided the accumulation function a(t; j).
Comparing with the definition of the accumulation function, it is not difficult realize that
the present value function is the inverse function of the accumulation function. For example,
if the accumulation function a(t; j) is given, then the amount you have to invest/deposit
1
a(t;j)
P =
1
a(t;j)
at time 0 because, if you do so, you will receive A(t; j, P ) = P × a(t; j) =
× a(t; j) = 1 at time t. The amount of
1
a(t;j)
is said to be present value at time 0 of
1
at time 0 is
$1 at time t, denoted as PV0 {$1 @ t}. Moreover, it is understood that $ a(t;j)
1
equivalent to $1 at time t, i.e. ($ a(t;j)
@ time = 0) ∼ ($1 @ time = t).
Likewise, given that the present value function is the inverse function of the accumulation
function, the present value at time t1 for $1 at time t2 > t1 is simply the reciprocal of the
accumulation factor, namely
a(t1 ;j)
a(t2 ;j)
. Since a deposit of
a(t1 ;j)
a(t2 ;j)
has to be made at time t1 in
order to receive $1 at time t2 ,
5.2
5.2.1
Common Types of Present–Value Functions
Simple Interest
If an accumulation function is of the form a(t; js ) = 1 + js t, then the present–value at time
t1 of $1 at time t2 is PVt1 {$1 @ t2 } =
of $1 @ time t is
1
.
1+js t
a(t1 ;js )
a(t2 ;js )
=
1+js t1
.
1+js t2
For example, present value at time 0
Read Textbook Example 1.6 (Canadian Treasure Bills).
8
5.2.2
Compound Interest
If an accumulation function is of the form a(t) = (1 + jc )t , then the present–value at time
t1 of $1 at time t2 is PVt1 {$1 @ t2 } =
value at time 0 of $1 @ time t is
5.2.3
a(t1 ;jc )
a(t2 ;jc )
1
,
(1+js )t
=
(1+jc )t1
(1+jc )t2
= (1 + jc )t1 −t2 . For example, present
denoted by νjs or simply ν.
Simple Discount
If an accumulation function is of the form a(t) =
$1 at time t2 is PVt1 {$1 @ t2 } =
a(t1 ;jc )
a(t2 ;jc )
=
1
,
1−ds t
1−ds t2
.
1−ds t1
then the present–value at time t1 of
For example, present value at time 0 of
$1 @ time t is 1 − ds t. Read Textbook Example 1.11 (U.S. Treasure Bills).
5.2.4
Compound Discount
If an accumulation function is of the form a(t) = (1 − dc )−t , then the the present–value at
time t1 of $1 at time t2 is PVt1 {$1 @ t2 } =
a(t1 ;jc )
a(t2 ;jc )
=
(1−dc )t2
(1−dc )t1
= (1 − dc )t2 −t1 . For example,
present value at time 0 of $1 @ time t is (1 − dc )t .
5.2.5
Force of Interest
Since the present–value at time t1 of $1 at time t2 is the reciprocal of the accumulation
factor, the the present–value at time t1 of $1 at time t2 is
A(0) × exp
nR
t1
δs ds
o
Z t2
0
a(t1 ; δs )
P × a(t1 ; δs )
A(t1 ; δs )
o = exp −
nR
PVt1 {$1 @ t2 } =
δs ds
=
=
=
t
a(t2 ; δs )
P × a(t2 ; δs )
A(t2 ; δs )
t1
A(0) × exp 0 2 δs ds
5.3
Equation of Value
The equation of value is the mathematical representation of the dated cash flow diagram. It
is imperative to understand the following
• It is necessary to choose a reference/focal time point (t), also called valuation date;
• At the reference time point t,
PVt {Initial Amount}+PVt {Subsequent Deposits} = PVt {Final Amount}+PVt {Subsequent Withdrawals}
Remarks:
9
1. In general, the equation of value depends the sizes of deposits/withdrawals, the times
of deposits/withdrawals, the choice of reference time point, and the form of accumulation/present value functions.
2. The equation of value does not depend on the choice of reference time point.
Example 5.3.1 (Textbook Example 1.7) Every Friday in February (the seventh, fourteenth, twenty-first, and twenty-eighth) Walt places $1,000 bet on credit with his off-track
bookmaking service, which charges an effective weekly interest rate of 8% on all credit extended. Unfortunately for Walt, he loses each bet and agrees to repay his debt to the bookmaking service in for installments, to be made on March 7, 14, 21, and 28. Walt pays $1,100
on each of March 7, 14, and 21. How much must Walt pay on March 28 to completely repay
his debt?
Example 5.3.2 (Textbook Exercise 1.2.3) A magazine firm offers a one-year subscription $15 with renewal the following year at $16.50. The firm also offers a two-year subscription at $28. What is the effective annual interest rate that makes the two-year subscription
equivalent to two successive one-year subscriptions?
6
Effective and Nominal Annual Rate
It is quite often that the time period is not measure by year. Nevertheless, it is standard practice to compare interest rate in some ”annualized” forms. There are two ways to annualize
the interest rates, which will be introduced as follow.
6.1
Effective and Nominal Annual Interest Rate
Suppose that the time unit of period is month, and we are given the effective rate per period
(month) be 1%, i.e. the interest is credited at the end of month. Then after n months, the
accumulated value is a(n) = (1.01)n . In other words, if $P is deposited at the beginning of
year, then there will P (1.01)12 at the end of year. Therefore,
the effective annual rate =
A(12) − A(0)
= 1.0112 = 12.682503013%
A(0)
Remarks:
10
1. the effective annual interest rate is often denoted by i;
2. receiving 1% interest at the end of each month is equivalent to receiving 12.682503013%
at the end of year.
In practice, the effective annual interest rate is seldom quoted. Instead, the nominal
interest rate is more commonly reported in the financial statements and/or investment advertisements. The rationale behind the use of nominal rate is that since the monthly interest
is calculated based on 1% and there are 12 months in one year, the nominal annual interest
rate is (roughly) 12% (= 1% × 12), denoted by i(12) in this case. In a more precise language,
it should be stated as ”annual interest rate of 12%, compounded/convertible monthly.”
Remarks:
1. keep in mind that the nominal rate is a notational convenience; it should be converted
into effective rate per period before any meaningful calculations.
2. the nominal annual interest rate is denoted by i(m) , where m is the number of (compounding) periods within one year;
3. the length of one period is
1
-year;
m
4. the effective rate per period is simply
i(m)
;
m
5. the following relationship establishes the rate equivalence between effective and nominal
annual interest rates:
m
i(m)
1+i= 1+
m
Example 6.1.1 Given the nominal interest rate of 12%, compounded monthly. Find the
equivalent
1. effective annual interest rate.
2. nominal annual interest rate, convertible semi-annually.
11
6.2
Effective and Nominal Annual Discount Rate
6.2.1
Effective Discount Rate per Period
In the previous section, we have defined the compound discount rate in the context of
accumulation functions. Here, I would like to elaborate the discount rate in the financial
context.
The interest rate, defined above, assumes that the amount of interest is credited at the
end of each period. It is sometimes referred as interest payable in arrears. Contrary to the
interest payable in arrears, there are also types of financial transactions (can you think of
any?), interest of which is payable in advance.
Suppose that you borrow $100, and you are required to pay 10% interest at the time of
borrowing, i.e. the net amount you receive is $90 (= 100 - 10). Then the rate of 10% payable
in advance is called the rate of discount or discount rate.
From the definition of effective discount rate per period in the previous section, we can
derive the relationship between the interest and discount rates:
A(t + 1) − A(t)
A(t + 1)
A(t)
= 1−
A(t + 1)
A(t) = A(t + 1) × (1 − dc )
dt+1 =
On the other hand,
A(t + 1) − A(t)
A(t)
A(t + 1)
=
−1
A(t)
1
A(t) = A(t + 1) ×
1 + jt+1
= A(t + 1) × ν
jt+1 =
This implies that
1 − dt+1 = νjt+1 or dt+1 =
jt+1
1 + jt+1
Therefore, given the effective interest rate per period, we can use the above relationship to
find the effective discount rate per period.
12
6.2.2
Effective and Nominal Annual Discount Rates
Just like the way the interest rate being ”annualized”, we would also like to annualize the
discount rate.
• From the definition of discount rate, the effective annual discount rate, denoted by d,
is the ratio of the difference between balances at the beginning and end of year to the
balance at the end of year, i.e.
d=
A(end of year) − A(beginning of year)
A(end of year)
• the nominal annual discount rate, denoted by d(m) , is the product of the number of
(compounding) period and the effective discount rate per period, i.e. d(m) = m × d. In
other words, the effective discount rate per period is d =
d(m)
;
m
• The equivalence between the effective and nominal discount rate:
m
d(m)
1−d= 1−
m
Example 6.2.1 Given the nominal interest rate of 12%, compounded monthly. Find the
equivalent
1. effective annual discount rate.
2. nominal annual discount rate, convertible quarterly.
Example 6.2.2 Suppose that $100 is deposited into a saving account, earning at a discount
rate of 0.15% biweekly, at the beginning of year 2006.
Q.1 Find the nominal annual discount rate.
Q.2 Find the effective annual discount rate.
Q.3 Find the amount of interest generated during the year.
Q.4 Find the equivalent effective annual interest rate.
Q.5 Find the equivalent nominal annual interest rate, convertible monthly.
A.1 Since there are 26 bi-weeks in one year, d(26) = 26 × 0.0015 = 0.039
13
A.2 d = 1 − 1 −
d(26)
26
26
= 1 − (1 − 0.0015)26 = 0.03827744981
A.3
I0,26 = A(26; 0.0015, 100) − A(0; 0.0015, 100)
= 100(1 − 0.0015)−26 − 100
= 3.980092783
A.4 Two ways to find the effective annual interest rate:
(a) i =
A(end
of year)−A(beginning of year)
=
A(beginning of year)
A(26)−A(0)
A(0)
=
3.980092783
100
= 3.980092783%
(b)
i
1+i
i
0.03827744981 =
1+i
i = 0.03980092783
d =
A.5 Since there are 12 months in one year, the nominal annual interest rate, convertible
monthly is i(12) .
i(m)
1+
m
m
i(12)
1+
12
12
1+i =
1.03980092783 =
i(12) = 0.03909281832
7
7.1
Applications
Varying Interest Rates
Example 7.1.1 (1 + 1 GIC offered by TD Canada Trust) TD Canada Trust is offering the following GIC (locked-in saving account)
14
Product Name
Terms
Rates
1+1 GIC
2 years
Year 1: 3.4%; Year 2: 3.7%
Effective Annual Yield: 3.55%
Triple Value
3 years
Year 1: 3.4%; Year 2: 3.7%; Year 3: 4.1%
Effective Annual Yield: 3.583%
Interest is compounded annual and paid at maturity or redemption, and the minimum investment amount is $1,000.
Q.1 If $1,000 is deposited into the 1+1 GIC, how much interest is generated during the tw0
years?
Q.2 Instead of this 1+1 GIC, $1,000 is deposited into another 2–year locked-in saving account, which generate the same amount of interest as the 1+1 GIC at maturity. This
saving account offers the constant effective annual interest rate of i. Find i.
Q.3 $X is invested into the 1+1 GIC. The balance at maturity is $5,000. Find X.
A.1 To find the amount of interest generated during the 2 years, we need to find A(2). Since
the interest is compounded annually, the effective annual interest rates are 3.4% for the
first year and 3.7% for the second year. Hence,
A(2) = A(1) × 1.037
= (A(0) × 1.034) × 1.037
= 1000(1.034)(1.037) = 1072.258
I0,2 = A(2) − A(0) = 72.258
A.2 Since the interest amount is the same, we know A(2) = 1072.258 and A(2) = A(0) ×
q
2
(1 + i) . Then i = 1072.258
− 1 = 0.03549891357 = 3.549891357%.
1000
Note that it is exactly the advertised effective annual yield.
A.3 Here, we are looking for the present value at time 0 of $5,000 at time 2. Since A(0) = X,
A(2) = 5000 and
A(2) = A(0)(1.034)(1.037)
X = 4663.05683893242
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In general, A(n) = A(n−1)×(1+in ) = A(n−2)×(1+in−1 )×(1+in ) = · · · = A(0)×
Qn
k (1+ik ),
where ik is the effective interest rate for period k. It is not hard to realize that A(k) is also
the present value at time k of $A(n) at time n.
7.2
Inflation and Real Interest Rate
Inflation is used to describe the rising cost of goods. A widely used measure of inflation is
the change in the Consumer Price Index, generally quoted on an annual basis. As the price
of goods is increasing, the value of money is decreasing. In other words, the inflation erodes
the purchasing power of money as time goes on.
The purchasing power of money is another way to describe the value of money. For
example, $100 can be used to purchase 50 Big Mac’s, at $2.00 each, now. However, in 10
years, $100 could only buy 25 Big Mac’s at $4.00 each. Then it is said that the purchasing
power of $100 is 50 Big Mac’s now and 25 Big Mac’s in 10 years.
Consider the following scenario. $100 is deposited into a saving account, earning interest
at 15.5% per year, for two years. At the time of deposit, one Big Mac costs $2.50. Two
years later, the Big Mac costs $2.75 (10% increase). What is the effective annual growth
rate in purchasing power with respect to the Big Mac? At time of deposit, $100 can buy 40
Big Mac’s. At the end of two years, $100 grows to $133.4025 (= 100(1 + 0.155)2 ) inside the
saving account. However,$133.4025 can only by 48.51 Big Mac’s in two years. Therefore the
effective annual growth rate can be found by solving
48.51 = 40 × (1 + igrowth )2 and igrowth = 10.124929058%
In summary, 15.5% of interest rate can only increase the purchasing power of money by
10.125%. It is said that the real rate of interest is approximately 10.125%.
Let r denote the annual inflation rate, i denote the effective annual interest rate, and ireal
denote the ”real” effective annual interest rate or inflation-adjusted effective annual interest
rate. Then
ireal =
i−r
1+r
Keep in mind that the real interest rate is used to describe the growth rate of money-value
in term of purchasing power. Therefore, we have to take the increase in price into account
when we calculate the real interest rate.
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7.3
After-Tax Interest Rate
In life, nothing is certain, except tax and death. Everyone has to pay tax. However, you do
not pay taxes on the money you already have; you only pay taxes on the money you earn.
The key point to remember is that taxes are paid on the interest, not on the principal. In
order to calculate the after-tax interest rate, we would have to consider how much we are
really getting back after paying taxes.
Example 7.3.1 A person’s saving earn an annual interest rate of 12% on which 45% income
tax is paid. What is the annual after-tax interest rate?
Suppose this person deposits $100 into the saving account. Then after one year, $100
grows to $112 (A(1) = A(0) × 1.12), and the amount of interest is $12. In other words, this
person earns $12 by saving $100 (money he has). Therefore, he would have to pay taxes on
the interest of $12. Since the tax rate is 45%, the amount of tax is $5.4. The amount he
really receives after paying the tax is $106.6 (= 112 - 5.4). The annual after-tax interest rate
can be found by
106.6 = 100 × (1 + iaf ter−tax )
8
Basic Textbook Questions to Practice
• 1.2.2 ˜ 1.2.4, 1.2.7, 1.2.10, 1.2.11
• 1.3.1 ˜ 1.3.4, 1.3.6
• 1.4.3, 1.4.6 ˜ 1.4.9
• 1.5.1 ˜ 1.5.3, 1.5.5, 1.5.7, 1.5.9
• 1.6.1, 1.6.2, 1.6.5
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