85
Time Value of Money Lakehead University Fall 2004
Time Value of Money
Lakehead University
Fall 2004
Outline of the Lecture
• Future Value and Compounding
• Present Value and Discounting
• More on Present and Future Values
2
Future Value and Compounding
Future valuerefers to the amount of money an investment would
grow to over some length of time at a given rate of interest.
To determine this value, it is important to know when interest is
calculated. Is it once a year? Every six months? Each month?
When many payments are involved, it is also important to know
the timing of these payments.
3
Future Value and Compounding
Investing for a Single Period
Suppose $100 is invested in an account that pays 10% per year.
This investment will then be worth
100 + 0.10×100 = (1+0.10)×100
= 1.1×100
= $110
after one year.
4
Future Value and Compounding
Investing for More than One Period
Suppose $100 is invested in an account that pays 10% per year.
After one year, this investment will be worth $110. If the interest
payment is reinvested, this investment will be worth, after two
years,
110 + 0.10×110 = 1.1×110
= 1.1×1.1×100
= (1.1)2×100 = $121.
5
5.1 Future Value and Compounding
Decomposing(1.1)2×100 gives us
1.1×1.1×100 =(1+ .2+ .12)×100
= 100︸︷︷︸Capital
+ 20︸︷︷︸Interest
on capital⇓
Simpleinterest
+ 1︸︷︷︸Interest
on interest⇓
Compoundinterest
6
Future Value and Compounding
More generally, $m invested at a period interest rater will grow
to
(1+ r)t×m = m︸︷︷︸Capital
+ t× r×m︸ ︷︷ ︸Simpleinterest
+ Compound interest
aftert periods.
7
Future Value and Compounding
Compound interestcan be significant over the long run.
Take $100 invested forT years at 10% compounded annually:
Ending Simple CompoundT Amount Capital Interest Interest
1 110.00 = 100 + 10 + 0.00
5 161.05 = 100 + 50 + 11.05
10 259.37 = 100 + 100 + 59.37
15 417.72 = 100 + 150 + 167.72
20 672.75 = 100 + 200 + 372.75
8
Future Value and Compounding
Examples of Future Value Calculations
1. $2,250 invested for 30 years at 18% compounded annually
gives
2,250× (1.18)30 = $322,583.94.
2. $9,310 invested for 15 years at 6% compounded annually
gives
9,310× (1.06)15 = $22,311.96.
9
Future Value and Compounding
One More Example
3. You are scheduled to receive $22,000 in two years. When
you receive it, you will invest it for six more years at 6
percent per year. How much will you have in eight years?
Answer:
22,000× (1.06)8−2 = 22,000× (1.06)6 = $31,207.42.
10
5.2 Present Value and Discounting
Present valuerefers to the amount of money that has to be
invested today to obtain a specific amount of money after a
specific length of time at a given rate of interest.
If, for example, we want to know how much to invest to obtain
$1 after one year at 10% interest, we need to solve
Present value×1.1 = $1 ⇒ Present value=1
1.1= $0.909.
11
Present Value and Discounting
More generally, the amount of money that needs to be invested
today to obtain $1 int years at the annual rate of interestr is
PV =1
(1+ r)t .
This amount is the present value, as of today, of $1 to be received
in t years discounted at the annual rater.
12
Present Value and Discounting
Examples of Present Value Calculations
1. The present value of $15,000 to be received in 5 years,
discounted at the annual rate 12%, is
PV =15,000(1.12)5 = $8,511.40.
2. The present value of $25,000 to be received in 10 years,
discounted at the annual rate 8%, is
PV =25,000(1.08)10 = $11,579.84.
13
5.3 More on Present and Future Values
With a period rate of interestr and a number of periodst, we can
define
Future value factor = (1+ r)t
Present value factor=1
(1+ r)t
14
More on Present and Future Values
Let PV0 denote the present value, as of today (date 0), of an
investment that will grow to the future value FVt in t periods, the
period interest rate beingr. Then
PV0× (1+ r)t = FVt and, equivalently PV0 =FVt
(1+ r)t .
This result is thebasic present value equation.
15
More on Present and Future Values
Determining the Discount Rate
What mustr be for PV0 to grow to FVt in t periods?
PV0× (1+ r)t = FVt
(1+ r)t =FVt
PV0
1+ r =(
FVt
PV0
)1/t
r =(
FVt
PV0
)1/t
−1.
16
More on Present and Future Values
Example of Discount Rate Determination
You are offered an investment that requires you to put up
$12,000 today in exchange for $40,000 12 years from now. What
is the annual rate of return on the investment?
Answer: In this example, PV0 = 12,000, FVt = 40,000 and
t = 12. Therefore,
r =(
40,00012,000
)1/12
− 1 = 10.55%.
17
More on Present and Future Values
Finding the Number of Periods
What mustt be for PV0 to grow to FVt at a rater?
Note: We will be using the following rules:
ln(ab) = ln(a) + ln(b)
ln(ab) = bln(a)
ln(a
b
)= ln(a) − ln(b).
18
Finding the Number of Periods
PV0× (1+ r)t = FVt
ln(PV0× (1+ r)t) = ln(FVt)
ln(PV0) + ln((1+ r)t) = ln(FVt)
ln(PV0) + t ln(1+ r) = ln(FVt)
t =ln(FVt)− ln(PV0)
ln(1+ r)
t =ln(FVt/PV0)
ln(1+ r)
19
How Long to Double Your Money?
Knowing r, how many periods is needed for PV0 to double?
t =ln(FVt/PV0)
ln(1+ r)
=ln(2PV0/PV0)
ln(1+ r)
=ln(2)
ln(1+ r)
20
The Rule of 72
Note that
• whenr is small, ln(1+ r)≈ r (slightly belowr);
• ln(2) = 0.6931 (slightly below 0.72).
A good approximation of the time it takes to double an
investment is0.72
r=
72100r
.
If r = 8%, PV0 will double in approximately 72/8 = 9 years.
21
The Rule of 72
r ln(1+ r) ln(2)ln(1+r)
72100r
2% 0.01980 35.00 36
4% 0.03922 17.67 18
6% 0.05827 11.90 12
8% 0.07696 9.01 9
10% 0.09531 7.27 7.2
12% 0.11333 6.12 6
14% 0.13103 5.29 5.14
22
The Rule of 72
The rule of 72 holds exactly at around 7.85%.
The rule of 72 will
• overestimate the time it takes to double an investment when
r < 7.85%;
• underestimate the time it takes to double an investment when
r > 7.85%;
23
The Rule of 72
Whenr is small, the error will be insignificant.
The error is significant when using large numbers.
Taker = 72%, for instance. According to the rule of 72, an
investment doubles in approximately one year at this rate.
This makes no sense: it takesr = 100% to double an investment
in one year.
24
Finding the Number of Periods: An Example
You are trying to save to buy a new $120,000 Ferrari. You have
$40,000 today that can be invested at 8% compounded annually.
How long will it take before you have enough money to buy the
car?
Answer:
t =ln(120,000/40,000)
ln(1.08)=
ln(3)ln(1.08)
= 14.27 years.
25
Future and Present Values of Multiple Cash Flows
Future Value with Multiple Cash Flows
Suppose $100 is invested today and another $100 is invested in
one year at an annual rate of 8%. How much will this investment
be worth in two years?
-
$100 -×1.08 $108
$100
$208 -×1.08 $224.64
Time
0 1 2
26
Future Value with Multiple Cash Flows
The same example, put differently:
-
$100 -×1.08 $108 -×1.08 $116.64
$100 -×1.08 $108.00
$224.64
Time
0 1 2
That is,
FV = $100× (1.08)2 + $100×1.08 = $224.64.
27
Future Value with Multiple Cash Flows
Suppose now that the two payments are made at theendof each
period. This gives us
-
$100 -×1.08 $108
$100
$208
Time
0 1 2
and thus, in this case
FV = $100×1.08 + $100 = $208.
28
Future Value with Multiple Cash Flows
More generally, let
dt ≡ payment made in periodt;
r ≡ period interets rate;
T ≡ the total number of periods.
Then
FV = (1+ r)Td0 + (1+ r)T−1d1 + . . . + (1+ r)dT−1 + dT
=T
∑t=0
(1+ r)T−tdt .
29
An example withT = 4.
d0 -×(1+ r) ×(1+ r) ×(1+ r) ×(1+ r) (1+ r)4d0
d1 -×(1+ r) ×(1+ r) ×(1+ r) (1+ r)3d1
d2 -×(1+ r) ×(1+ r) (1+ r)2d2
d3 -×(1+ r) (1+ r)d3
d4
4
∑t=1
(1+ r)4−tdt
0 1 2 3 4
30
Present Value with Multiple Cash Flows
What is the present value of $100 to be received one year from
now and another $100 to be received in two years, the annual rate
of interest being 8%?
-
$100�×1/1.08$92.59
$100�×1/1.08 ×1/1.08$85.73
$178.32
Time
0 1 2
That is, PV = $1001.08 + $100
(1.08)2 = $178.32.
31
Future Value with Multiple Cash Flows
More generally, let
dt ≡ payment made in periodt;
r ≡ period interets rate;
T ≡ the total number of periods.
Then
PV = d0 +d1
1+ r+
d2
(1+ r)2 + . . . +dT
(1+ r)T
=T
∑t=0
dt
(1+ r)t .
32
A Note on Cash Flow Timing
Unless specified otherwise, cash flows are assumed to take place
at the end of each period.
A cash flow in year 2, for instance, means a cash flow to be
received two years from now, and thus at the end of the second
year.
33
A Note on Cash Flow Timing
If you are told that a three-year investment has a first-year cash
flow fo $100, a second-year cash flow of $200 and a third-year
cash flow of $300, then the timing of cash flows is as follows:
-
$100 $200 $300
0 1 2 3
34
6.2 Valuing Level Cash Flows
A ordinaryannuityis a series of constant, or level, cash flows that
occur at theendof each period for some fixed number of periods.
An annuity dueis a series of constant, or level, cash flows that
occur at thebeginningof each period for some fixed number of
periods.
A perpetuityis an annuity in which the cash flows continue
forever.
35
A Note on How to Value Level Cash Flows
Let
S =T
∑t=1
qt = q + q2 + q3 + . . . + qT−1 + qT .
Then
qS = qT
∑t=1
qt = q2 + q3 + q4 + . . . + qT + qT+1,
and thus, ifq≥ 0 andq 6= 1,
S− qS = q − qT+1 ⇒ S =q−qT+1
1−q=
q1−q
(1−qT) .
36
A Note on How to Value Level Cash Flows
If q = 1, thenS= ∑Tt=1qt = T.
What happens whenT is arbitrarily large?
limT→∞
q1−q
(1−qT) =
q
1−q if 0 ≤ q < 1,
∞ if q > 1.
37
A Note on How to Value Level Cash Flows
Suppose that we have
S =1
1+ r+(
11+ r
)2
+(
11+ r
)3
.
Let q = 11+r , wherer > 0 is a discount rate. Then
S = q + q2 + q3 =q
1−q
(1−q3) .
38
A Note on How to Value Level Cash Flows
Replaceq with 11+r in the last equation. This gives
S =1/(1+ r)
1−1/(1+ r)
(1−(
11+ r
)3)
=1
1+ r−1
(1−(
11+ r
)3)
=1r
(1−(
11+ r
)3)
39
A Note on How to Value Level Cash Flows
So if we have
S =T
∑t=1
(1
1+ r
)t
=1
1+ r+(
11+ r
)2
+ . . . +(
11+ r
)T
,
then
S =1r
(1−(
11+ r
)T)
.
40
Present Value of Annuity Cash Flows
Consider an ordinary annuity that pays $C each period forT
periods, the first payment being made one period from now.
What is the present value of this annuity if the period rate of
interest isr?
41
PV =C
1+ r+
C(1+ r)2 +
C(1+ r)3 + . . . +
C(1+ r)T
= C
(1
1+ r+(
11+ r
)2
+(
11+ r
)3
+ . . . +(
11+ r
)T)
= C× 1r
(1−(
11+ r
)T)
=Cr
(1−(
11+ r
)T)
42
Present Value of Annuity Cash Flows
The term1−( 1
1+r
)Tr
is often referred to as thepresent value interest factor for
annuitiesand abbreviated PVIFA(r,T).
Note that
PVIFA(r,T) =1−( 1
1+r
)Tr
=1−Present Value factor
r
43
Present Value of Annuity Cash Flows
Consider now an annuitydue involving T payments of $C, theperiod interest rate beingr. Then
PV = C +C
1+ r+
C
(1+ r)2 + . . . +C
(1+ r)T−1
= (1+ r)C
(1
1+ r+(
11+ r
)2
+(
11+ r
)3
+ . . . +(
11+ r
)T)
= (1+ r)C× 1r
(1−(
11+ r
)T)
= (1+ r)×C×PVIFA(r,T).
44
Present Value of Annuity Cash Flows
The present value of an annuity due is equal to 1+ r times its
ordinary counterpart.
Using the above equations, we can answer questions similar to
those in chapter 5, such as finding the fixed payment that will
repay a loan, or finding the number of periods necessary to repay
a loan, etc..
45
Present Value of Annuity Cash Flows
Example 1
An investment offers $2,250 per year for 15 years, with the first
payment occurring one year from now. If the required return is
10 percent, what is the value of the investment?
Answer:
PV =2,2500.1
(1−(
11.1
)15)
= $17,113.68.
46
Present Value of Annuity Cash Flows
Example 2
Betty’s Bank offers a $25,000, seven-year loan at 11 percent
annual interest payable in equal annual amounts. What will the
annual payment be?
Answer:
C =PV
1r
(1−( 1
1+r
)T) =25,000
10.11
(1−( 1
1.11
)7) = $5,305.38.
47
Present Value of Annuity Cash Flows
Example 3
How long does it take to repay a $25,000 loan with fixed annual
payments of $4,000 at an 11% annual interest rate?
We will be using the “ln” trick to solve this problem.
48
Example 3
Answer:
PV =Cr
(1−(
11+ r
)T)
rPVC
= 1−(
11+ r
)T
(1
1+ r
)T
= 1− rPVC
T ln
(1
1+ r
)= ln
(1− rPV
C
)
T =ln(
1− rPVC
)ln( 1
1+r
)
49
Example 3
Answer:
T =ln(
1− rPVC
)ln( 1
1+r
)=
ln(
1− 0.11×25,0004,000
)ln( 1
1.11
)= 11.15 years,
and thus it takes 12 years to repay such a loan, the last payment
being less than $4,000.
50
Present Value of Annuity Cash Flows
Example 4
What must the annual rate of interest be in order to fully repay a$25,000 loan in 10 years with fixed annual payments of $4,000?Answer:
PV =Cr
(1−(
11+ r
)T)
25,000 =4,000
r
(1−(
11+ r
)10)
.
Can’t solve this equation analytically.
51
Example 4
The solution can be found by trial-and-error or by using a
computer.
In Excel:
RATE(NPER,PMT,PV)= RATE(10,-4000,25000)= 9.61%.
52
Future Value of Annuity Cash Flows
Consider an ordinary annuity that pays $C each period forT
periods, the first payment being made one period from now.
What is the future value of this annuity if the period rate of
interest isr?
53
Future Value of Annuity Cash Flows
Answer:
FV = (1+ r)T−1C + (1+ r)T−2C + . . . + (1+ r)C + C
= C((1+ r)T−1 + (1+ r)T−2 + . . . + (1+ r) + 1
)= C
(1 + (1+ r) + . . . + (1+ r)T−2 + (1+ r)T−1
)= C
(1− (1+ r)T
1− (1+ r)
)
= C
(1− (1+ r)T
−r
)
= C
((1+ r)T −1
r
)
54
Future Value of Annuity Cash Flows
The term(1+ r)T −1
ris often referred to as thefuture value interest factor for annuities
and abbreviated FVIFA(r,T).
Note that
FVIFA(r,T) =(1+ r)T −1
r= (1+ r)T 1−
(1
1+r
)T
r= (1+ r)TPVIFA(r,T).
55
Perpetuities
A perpetuity is an annuity with perpetual cash flows.
The future value of a perpetuity is always infinite.
The present value of a perpetuity paying $C forever at theendof
each period, the period interest rate beingr > 0, is
limT→∞
C
(1−( 1
1+r
)Tr
)=
Cr,
since limT→∞( 1
1+r
)T = 0.
56
Perpetuities
The present value of a perpetuity paying $C forever at the
beginningof each period, the period interest rate beingr > 0, is
limT→∞
C(1+ r)
(1−( 1
1+r
)Tr
)=
(1+ r)Cr
.
57
Relationship between Annuities and Perpetuities
Consider the following perpetuities:
Perpetuity P1: Pays $C forever, the first payment being made
one period from now.
Perpetuity P2: Pays $C forever, the first payment being made
at timeT +1 (i.e. at the end of periodT, which is the
beginning of periodT +1).
. . . -0 1 2 3 4 5 T−1 T T+1 T+2 T+3
P1 :
P2 :
C C C C C C C C C C
C C C
58
Relationship between Annuities and Perpetuities
Note that
P1 − P2 = A(C,T),
whereA(C,T) denotes the (ordinary) annuity that pays $C for T
periods.
Hence, the present value ofP1−P2 must be equal to the present
value ofA(C,T).
59
Relationship between Annuities and Perpetuities
Let r denote the period interest rate. Then
PV(P1) =Cr
and PV(P2) =C/r
(1+ r)T ,
and thus
PV(P1−P2) =Cr− C/r
(1+ r)T
=Cr
(1−(
11+ r
)T)
= PV(A(C,T)).
60
Growing Annuities
Consider an annuity in which the payment grows at the rateg
from one period to the other. That is, the cash flows from this
annuity are as follows:
. . . -0 1 2 3 T−1 T
C (1+g)C (1+g)2C (1+g)T−2C (1+g)T−1C
61
The present value of this annuity is
PV =C
1+ r+
(1+g)C(1+ r)2 +
(1+g)2C(1+ r)3 + . . . +
(1+g)T−1C(1+ r)T
=C
1+g
(1+g1+ r
+(
1+g1+ r
)2
+(
1+g1+ r
)3
+ . . . +(
1+g1+ r
)T)
=C
1+g× (1+g)/(1+ r)
1− (1+g)/(1+ r)
(1−(
1+g1+ r
)T)
=C
1+g× 1
(1+ r)/(1+g)−1
(1−(
1+g1+ r
)T)
= C× 11+ r− (1+g)
(1−(
1+g1+ r
)T)
=C
r−g
(1−(
1+g1+ r
)T)
62
Growing Annuities
The present value of an annuity in which the payments grow at
the constant rateg, the first payment beingC, is
PV =C
r−g
(1−(
1+g1+ r
)T)
,
whereT is the number of payments andr is the period discount
rate.
What is the present value of a growing perpetuity?
63
Growing Annuities
If g < r, then1+g1+ r
< 1 and limT→∞
(1+g1+ r
)T
= 0.
If g > r, then1+g1+ r
> 1 and limT→∞
(1+g1+ r
)T
= ∞.
Therefore,
limT→∞
Cr−g
(1−(
1+g1+ r
)T)
=
Cr−g if g < r,
∞ if g≥ r.
64
Growing Annuities
Example
Problem 77. Consider a firm that is expected to generate a net
cash flow of $10,000 at the end of the first year. The cash flows
will increase by 3 percent a year for seven years and then the firm
will be sold for $120,000. The relevant discount rate for the firm
is 11 percent. What is the present value of the firm?
65
Answer: The total cash flows generated by this firm are the 8cash flows from its operations and the terminal value of$120,000, which will materialize eight years from now. Thepresent value of the firm is then (numbers in 000’s)
PV =10
1.11+
1.03×10(1.11)2 +
(1.03)2×10(1.11)3 + . . . +
(1.03)7×10(1.11)8 +
120(1.11)8
=10
0.11−0.03
(1−(
1.031.11
)8)
+120
(1.11)8
= $108,360.
66
The Effect of Compounding
Interest rates can be quoted in many different ways.
How rates are quoted may come from tradition or regulation.
Very often, rates are quoted in a misleading manner.
What’s under a quoted rate?
67
Effective Annual Rates and Compounding
Suppose a rate is quoted at 10% compounded semiannually.
This means that 5% is charged every six months.
10%, thequoted rate, is the interest charged on theprincipal
during the year, it does not include the interest on interest
(compound interest).
The rate that takes into account compound interest is called the
effective annual rate (EAR).
What is the EAR in the above example?
68
Effective Annual Rates and Compounding
With a 10% interest rate compounded semiannually, the EAR is
EAR = (1.05)2 − 1 = 1.1025− 1 = 10.25%.
Note that
0.25% = 5%×5%
is the interest on interest charged during the year.
69
Effective Annual Rates and Compounding
More generally, the EAR of a quoted annual rate compoundedm
times during the year is
EAR =(
1+Quoted Rate
m
)m
− 1.
Compare the following rates:
Bank A: 15% compounded daily
Bank B: 15.5% compounded quarterly
Bank C: 16% compounded annually
70
Effective Annual Rates and Compounding
Bank A:
EARA =(
1+0.15365
)365
− 1 = 16.18%.
Bank B:
EARB =(
1+0.155
4
)4
− 1 = 16.42%.
Bank C:
EARC =(
1+0.16
1
)1
− 1 = 16.00%.
71
Quoting a Rate
What is the quoted rate, compounded monthly, that provides an
effective return of 15%?
72
Quoting a Rate
Answer:
0.15 =(
1+Quoted rate
12
)12
− 1
1.15 =(
1+Quoted rate
12
)12
(1.15)1/12 = 1+Quoted rate
12
(1.15)1/12 − 1 =Quoted rate
12
12×((1.15)1/12−1
)= Quoted rate = 14.06%.
73
Mortgages
Regulations for Canadian institutions require that mortgage rates
be quoted with semiannual compounding. Payments, however,
are made each month.
How to calculate monthly payments from a quoted mortgage
rate?
(i) When quoting a rate, a financial institution is thinking EAR,
so the first step is to find the EAR implied by the quoted rate.
(ii) Calculate the monthly rate prodiving the EAR in (i).
(iii) Using the annuity formula, find the monthly payment.
74
Mortgages
Example 1
Find the monthly payment on a $300,000 mortgage quoted at 14percent and amortized over 25 years.
EAR =(
1+Quoted rate
m
)m
−1 =(
1+0.14
2
)2
−1 = 14.49%.
Find the monthly rate that gives an EAR of 14.49%:
(1+Monthly Rate)12−1 = 14.49%
⇒ Monthly Rate= (1.1449)1/12−1 = 1.13%.
75
Mortgages
Example 1 (continued)
Find the monthly payment (T = 25×12= 300):
PV =Cr
(1−(
11+ r
)T)
300,000 =C
0.0113
(1−(
11.0113
)300)
C =0.0113×300,000
1−(
11.0113
)300
C = $3,510.61.
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Mortgages
Example 2
An entrepreneur is considering the purchase of an office in a new
high-rise complex. The office is worth $1,000,000 and a bank is
offering a mortgage for the whole amount at 8 percent APR. If
the entrepreneur’s budget allows payments of $7,000 a month,
how long will it take to pay off the purchase?
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EARs and APRs
Cost of borrowing disclosure regulations in Canada require that
lenders disclose anannual percentage rate (APR)in a prominent
and unambiguous manner.
By law, the APR is the interest rate per period multiplied by the
number of periods in a year. This is indeed the quoted rate
mentioned earlier.
For example, the APR on a loan at 1.5% monthly interest rate is
12×1.5% = 18%.
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Continuous Compounding
What is the EAR when the quoted rate is compounded every
nanosecond?
Take, for example, a 12% APR:
Compounded EAR
Annually 12.00%
Quarterly 12.55%
Monthly 12.68%
Weekly 12.73%
Daily 12.75%
Continuously ?
79
Continuous Compounding
The more often a quoted rate is compounded, the greater the
EAR.
Continuous compounding thus yields the maximum EAR from a
given APR.
Given a quoted rateq,
limm→∞
(1+
qm
)m− 1 = eq − 1.
Note thateq−1 is thehighestEAR that can be obtained with an
APR ofq.
80
Loan Types and Loan Amortization
We will see three types of loan in this section:
Pure Discount Loans: Usually short-term loans, such as T-bills.
Interest-Only Loans: Usually long-term loans, such as
government and corporate bonds.
Amortized Loans: Majority of individual loans.
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Pure-Discount Loans
In a pure discount loan, the borrower receives money today and
makes one lump-sum payment at some time in the future.
Consider, for example, a T-bill that promises to pay $1,000 in one
year. When the interest rate is 3.48%, the value of this T-bill is
PV =1,0001.0348
= $966.37.
If the repayment (L) takes place aftert periods, the present value
of the loan is
PV =L
(1+ r)t .
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Interest-Only Loans
With this type of loan, the borrower pays interest each period and
repays the principal at some point in the future.
Take, for example a 5-year loan of $1,000 at an 8% annual
interest rate.
Each year the borrower pays $80 in interest and the principal
($1,000) is repaid after 5 years. Cash flows to the lender are then
-0 1 2 3 4 5
Interest
Principal
$80 $80 $80 $80 $80
$1,000
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Interest-Only Loans
The present value of the above loan, at a discount rater, is
PV =80r
(1−(
11+ r
)5)
+1,000
(1+ r)5 .
Note that
PV
> $1,000 if r < 8%,
= $1,000 if r = 8%,
< $1,000 if r > 8%.
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Amortized Loans
An amortized loan is such that interest and principal are repaid
each period.
This type of loan can be such that a constant amount of the
principal is repaid each period, or can be such that a constant
payment is made each period.
How long would it take to repay a $5,000 loan with an APR of
10% compounded monthly if $500 in principal has to be repaid
each month?
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