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FIN 3701 Chapter 2 :The Time Value of Money
1
Assumption University of Thailand
FIN3701CorporateFinance
Chapter 2The Time Value
of Money
Dr. Chainarin Srinutchasart
1
After studying this chapter,• You will understand the concept of future
value, with both annual and intra-yearcompounding.
• Your will be able to distinguish betweenfuture value and present value concepts.
• Your will be able to calculate the futurevalue and present value of a singlepayment and an annuity.
• You will see how to utilize future valueand present value tables.
2
Principles Applied in This Chapter
• Principle 1: Money Has a Time Value
• Principle 3: Cash Flows Are the Sourceof Value.
3
Corporate Finance addresses thefollowing 3 questions:
1. What long-term investments should thefirm engage in?
2. How can the firm raise money for therequired investments? (Alternatives:Bonds, Stocks, Preferred Stocks=whatis the appropriate price?)
3. How much short-term cash flow does acompany need to pay its bills? and howto raise it
4
We know that receiving $1 today is worth morethan $1 in the future. This is due toopportunity costs.The opportunity cost of receiving $1 in thefuture is the interest (based upon inflation,economy and other risks) we could haveearned if we had received the $1 sooner.
Today Future
So, interest rate = Rf + Inflation + Risk Premium
5
Intuition Behind Present ValueThere are three reasons why a dollar tomorrow isworth less than a dollar today.• Individuals prefer present consumption to
future consumption. To induce people to give uppresent consumption you have to offer them morein the future.
• When there is monetary inflation. the value ofcurrency decreases over time. The greater theinflation the greater the difference in valuebetween a dollar today and a dollar tomorrow.
• If there is any uncertainty (risk) associated withthe cash flow in the future, the less that cash flowwill be valued.
Interest rate = Rf + Inflation rate + Risk premium6
FIN 3701 Chapter 2 :The Time Value of Money
2
Using Timelines to Visualize Cashflows
• A timeline identifies the timing andamount of a stream of cash flows alongwith the interest rate.
• A timeline is typically expressed in years,but it could also be expressed as months,days or any other unit of time.
7
Time Line Example
• The 4-year timeline illustrates the following:• The interest rate is 10%.• A cash outflow of $100 occurs at the beginning
of the first year (at time 0), followed by cashinflows of $30 and $20 in years 1 and 2, a cashoutflow of $10 in year 3 and cash inflow of $50in year 4.
The end ofperiod
0 1 2 3 4
YearsCash flow -$100 $30 $20 -$10 $50
i=10%
8
The Time Value of MoneyCompounding
andDiscounting Single Sums
9
If we can measure thisopportunity cost, we can:• Translate $1 today into its equivalent in the future
(compounding).
• Translate $1 in the future into its equivalent today(discounting).
Today Future
?
Today Future
?
10
Simple Interest and CompoundInterest• What is the difference between simple
interest and compound interest?• Simple interest: Interest is earned only
on the principal amount.• Compound interest: Interest is earned
on both the principal and accumulatedinterest of prior periods.
11
Simple Interest and CompoundInterest (cont.)• Example: Suppose that you deposited
$500 in your savings account that earns5% annual interest. How much will youhave in your account after two yearsusing (a) simple interest and (b)compound interest?
12
FIN 3701 Chapter 2 :The Time Value of Money
3
Simple Interest and CompoundInterest (cont.)Simple Interest• Interest earned
• = 5% of $500 = .05×500 = $25 per year• Total interest earned = $25×2 = $50• Balance in your savings account:
• = Principal + accumulated interest• = $500 + $50 = $550
13
Simple Interest and CompoundInterest (cont.)Compound interest• Interest earned in Year 1
• = 5% of $500 = $25• Interest earned in Year 2
• = 5% of ($500 + accumulated interest)• = 5% of ($500 + 25) = .05×525 =
$26.25• Balance in your savings account:
• = Principal + interest earned• = $500 + $25 + $26.25 = $551.25
14
Compound Interest andFuture Value
15
Future Value - single sums• If you deposit $100 in an account earning
6%, how much would you have in theaccount after 1 year?
16
Future Value - single sums• If you deposit $100 in an account earning 6%,
how much would you have in the accountafter 1 year?
PV = FV =
0 1Calculator Solution:P/Y = 1 I = 6N = 1 PV = -100FV = $106
-100 106
17
Future Value - single sums• If you deposit $100 in an account earning 6%,
how much would you have in the accountafter 1 year?
PV = FV =
0 1Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .06, 1 ) (use FVIF table, or)FV = PV (1 + i)n
FV = 100 (1.06)1 = $106
-100 106
18
FIN 3701 Chapter 2 :The Time Value of Money
4
19
Future Value - single sums• If you deposit $100 in an account earning
6%, how much would you have in theaccount after 5 years?
20
Future Value - single sums• If you deposit $100 in an account earning 6%, how
much would you have in the account after 5years?
PV = FV =0 5
Calculator Solution:P/Y = 1 I = 6N = 5 PV = -100FV = $133.82
-100 133.82
21
Future Value - single sums• If you deposit $100 in an account earning 6%, how
much would you have in the account after 5years?
PV = FV =0 5
-100 133.82
Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .06, 5 ) (use FVIF table, or)FV = PV (1 + i)nFV = 100 (1.06)5 = $133.82
22
Future Value - single sums• If you deposit $100 in an account earning
6% with quarterly compounding, howmuch would you have in the account after5 years?
23
Future Value - single sums• If you deposit $100 in an account earning 6% with
quarterly compounding, how much would youhave in the account after 5 years?
PV = FV =
0 20Calculator Solution:P/Y = 4 I = 6N = 20 PV = -100FV = $134.68
-100 134.68
24
FIN 3701 Chapter 2 :The Time Value of Money
5
Future Value - single sums• If you deposit $100 in an account earning 6% with
quarterly compounding, how much would youhave in the account after 5 years?
PV = FV =
0 20
-100 134.68
Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .015, 20 ) (can’t use FVIF table)
FV = PV (1 + i/m) m x n
FV = 100 (1.015)20 = $134.6825
Future Value - single sums• If you deposit $100 in an account earning
6% with monthly compounding, howmuch would you have in the account after5 years?
26
Future Value - single sums• If you deposit $100 in an account earning 6% with
monthly compounding, how much would youhave in the account after 5 years?
PV = FV =0 20
Calculator Solution:P/Y = 12 I = 6N = 60 PV = -100FV = $134.89
-100 134.89
27
Future Value - single sums• If you deposit $100 in an account earning 6% with
monthly compounding, how much would youhave in the account after 5 years?
PV = FV =0 20
Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .005, 60) (use FVIF table, or)FV = PV (1 + i)nFV = 100 (1.005)60 = $134.89
-100 134.89
28
Future Value - continuouscompounding
• What is the FV of $1,000 earning 8% withcontinuous compounding, after 100years?
29
Future Value - continuouscompounding• What is the FV of $1,000 earning 8% with
continuous compounding, after 100 years?
PV = FV =
0 100Mathematical Solution:FV = PV (e in)FV = 1000 (e .08x100) = 1000 (e 8)FV = $2,980,957.99
-1000 $2.98m
30
FIN 3701 Chapter 2 :The Time Value of Money
6
Present Value and Annuities
31
Present Value - single sums• If you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
32
Present Value - single sums• If you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
PV = FV =0 1
Calculator Solution:P/Y = 1 I = 6N = 1 FV = 100PV = -94.34
100-94.34
33
Present Value - single sums• If you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
PV = FV =0 1
Mathematical Solution:PV = FV (PVIF i, n )PV = 100 (PVIF .06, 1 ) (use PVIF table, or)PV = FV / (1 + i)n
PV = 100 / (1.06)1 = $94.34
100-94.34
34
35
Present Value - single sums• If you receive $100 five years from now, what is
the PV of that $100 if your opportunity cost is 6%?
PV = FV =
0 5Calculator Solution:P/Y = 1 I = 6N = 5 FV = 100PV = -74.73
100-74.73
36
FIN 3701 Chapter 2 :The Time Value of Money
7
Present Value - single sums• If you receive $100 five years from now, what is
the PV of that $100 if your opportunity cost is 6%?
PV = FV =
0 5Mathematical Solution:PV = FV (PVIF i, n )PV = 100 (PVIF .06, 5 ) (use PVIF table, or)PV = FV / (1 + i)n
PV = 100 / (1.06)5 = $74.73
100-74.73
37
Present Value - single sums• What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
PV = FV =
0 15Calculator Solution:P/Y = 1 I = 7N = 15 FV = 1,000PV = -362.45
1000-362.45
38
Present Value - single sums• What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
PV = FV =0
Mathematical Solution:PV = FV (PVIF i, n )PV = 100 (PVIF .07, 15 ) (use PVIF table, or)PV = FV / (1 + i)n
PV = 100 / (1.07)15 = $362.45
-362.
15
100045
39
Present Value - single sums• If you sold land for $11,933 that you bought 5
years ago for $5,000, what is your annual rate ofreturn?
Calculator Solution:P/Y = 1 N = 5PV = -5,000 FV = 11,933I = 19%
PV = FV =0 5
-5,000 11,933
40
Present Value - single sums• If you sold land for $11,933 that you bought 5
years ago for $5,000, what is your annual rate ofreturn?
Mathematical Solution:PV = FV (PVIF i, n )5,000 = 11,933 (PVIF ?, 5 )PV = FV / (1 + i)n
5,000 = 11,933 / (1+ i)5
.419 = ((1/ (1+i)5)2.3866 = (1+i)5
(2.3866)1/5 = (1+i) i = .19
41
Present Value - single sums• Suppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How longwill it take for your account to grow to $500?
Calculator Solution:• P/Y = 12 FV = 500• I = 9.6 PV = -100• N = 202 months
PV = FV =
0 5
-100 500
42
FIN 3701 Chapter 2 :The Time Value of Money
8
Present Value - single sums• Suppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How longwill it take for your account to grow to $500?
Mathematical Solution:PV = FV / (1 + i)n
100 = 500 / (1+ .008)N
5 = (1.008)N
ln 5 = ln (1.008)N
ln 5 = N ln (1.008)1.60944 = .007968 N N = 202 months
43
The Time Value of Money
Compounding and DiscountingCash Flow Streams
0 1 2 3 4
44
Annuities
45
Annuities• Annuity: a sequence of equal cash
flows, occurring at the end of each period.
0 1 2 3 4
46
Examples of Annuities:• If you buy a bond, you will receive equal
semi-annual coupon interest paymentsover the life of the bond.
• If you borrow money to buy a house or acar, you will pay a stream of equalpayments.
47
Future Value - annuity• If you invest $1,000 each year at 8%, how
much would you have after 3 years?
0 1 2 3
Calculator Solution:P/Y = 1 I = 8 N = 3PMT = -1,000FV = $3,246.40
1000 1000 1000
48
FIN 3701 Chapter 2 :The Time Value of Money
9
Future Value - annuity• If you invest $1,000 each year at 8%, how
much would you have after 3 years?Mathematical Solution:FV = PMT (FVIFA i, n )FV = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or)
FV = PMT (1 + i)n - 1i
FV = 1,000 (1.08)3 - 1 = $3246.40.08
4950
Present Value - annuity• What is the PV of $1,000 at the end of
each of the next 3 years, if the opportunitycost is 8%?
0 1 2 3
Calculator Solution:P/Y = 1 I = 8 N = 3PMT = -1,000PV = $2,577.10
1000 1000 1000
51
Present Value - annuity• What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?Mathematical Solution:PV = PMT (PVIFA i, n )PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or)
1PV = PMT 1 - (1 + i)n
i
1PV = 1000 1 - (1.08 )3 = $2,577.10
.0852
53
Growing Annuities,Perpetuities,Growing Perpetuities
54
FIN 3701 Chapter 2 :The Time Value of Money
10
Growing AnnuityA growing stream of cash flows with a fixedmaturity
0 1
C
T
T
r
gC
r
gC
r
CPV
)1(
)1(
)1(
)1(
)1(
1
2
T
r
g
gr
CPV
)1(
11
2
C×(1+g)
3
C ×(1+g)2
T
C×(1+g)T-1
55
Growing Annuity: ExampleA defined-benefit retirement plan offers to pay$20,000 per year for 40 years and increase theannual payment by 3% each year. What is thepresent value at retirement if the discount rateis 10%?
0 1
$20,000
57.121,265$10.1
03.11
03.10.
000,20$40
PV
2
$20,000×(1.03)
40
$20,000×(1.03)39
56
Growing Annuity: Example
• You are evaluating an income generatingproperty. Net rent is received at the end ofeach year.
• The first year's rent is expected to be$8,500, and rent is expected to increase7% each year.
• What is the present value of theestimated income stream over the first 5years if the discount rate is 12%?
57
Growing Annuity: Example (Cont.)
0 1 2 3 4 5
500,8$ )07.1(500,8$
2)07.1(500,8$
095,9$ 65.731,9$ 3)07.1(500,8$
87.412,10$
4)07.1(500,8$
77.141,11$
$34,706.26
58
Perpetuities
59
Perpetuities• Suppose you will receive a fixed payment
every period (month, year, etc.) forever.This is an example of a perpetuity.
• You can think of a perpetuity as anannuity that goes on forever.
60
FIN 3701 Chapter 2 :The Time Value of Money
11
Present Value of a Perpetuity• When we find the PV of an annuity, we
think of the following relationship:
PV = PMT (PVIFA i, n )
61
Mathematically,
(PVIFA i, n ) =
We said that a perpetuity is an annuitywhere n = infinity. What happens tothis formula when n gets very, verylarge?
1 – (1 + i)
i
1n
62
Mathematically,
1 – (1 + i)
i
1n
this becomes zero.
So we’re left with PVIFA =1i
63
Present Value of a Perpetuity• So, the PV of a perpetuity is very simple
to find:
PMTiPV =
64
What should you be willing to pay in orderto receive $10,000 annually forever, if yourequire 8% per year on the investment?
PMT $10,000i .08
PV = =
= $125,000
65
Growing Perpetuities
66
FIN 3701 Chapter 2 :The Time Value of Money
12
Growing Perpetuity• A growing stream of cash flows that lasts
forever
0…
1
C
2
C×(1+g)
3
C ×(1+g)2
3
2
2 )1(
)1(
)1(
)1(
)1( r
gC
r
gC
r
CPV
gr
CPV
67
Growing Perpetuity: Example• The expected dividend next year is $1.30,
and dividends are expected to grow at 5%forever.
• If the discount rate is 10%, what is the valueof this promised dividend stream?
0…
1
$1.30
2
$1.30×(1.05)
3
$1.30 ×(1.05)2
00.26$05.10.
30.1$
PV
68
Annuities Due and UnevenCash Flows
69
Ordinary Annuity vs.Annuity Due
$1000 $1000 $1000
4 5 6 7 8
70
Annuities Due• Annuity due is an annuity in which all the
cash flows occur at the beginning of theperiod. For example, rent payments onapartments are typically annuity due asrent is paid at the beginning of the month.
71
Earlier, we examined this“ordinary” annuity:
• Using an interest rate of 8%, we find that:• The Future Value (at 3) is $3,246.40.• The Present Value (at 0) is $2,577.10.
0 1 2 3
1000 1000 1000
72
FIN 3701 Chapter 2 :The Time Value of Money
13
Earlier, we examined this“ordinary” annuity:
• Same 3-year time line,• Same 3 $1000 cash flows, but• The cash flows occur at the beginning of each
year, rather than at the end of each year.• This is an “annuity due.”
0 1 2 3
1000 1000 1000
73
Future Value - annuity due
0 1 2 3
• If you invest $1,000 at the beginning of eachof the next 3 years at 8%, how much wouldyou have at the end of year 3?
-1000 -1000 -1000
Calculator Solution:Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = -1,000FV = $3,506.11
74
Future Value - annuity due• If you invest $1,000 at the beginning of each
of the next 3 years at 8%, how much wouldyou have at the end of year 3?Mathematical Solution: Simply compound theFV of the ordinary annuity one more period:FV = PMT (FVIFA i, n ) (1 + i)FV = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or)
FV = PMT (1 + i)n - 1i
FV = 1,000 (1.08)3 - 1 = $3,506.11.08
(1 + i)
(1.08)
75
Calculator Solution:Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = 1,000PV = $2,783.26
Present Value - annuity due
0 1 2 3
• What is the PV of $1,000 at the beginning ofeach of the next 3 years, if your opportunitycost is 8%?
1000 1000 1000
76
Present Value - annuity dueMathematical Solution: Simply compound the FVof the ordinary annuity one more period:PV = PMT (PVIFA i, n ) (1 + i)PV = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or)
1PV = PMT 1 - (1 + i)n
i
1PV = 1000 1 - (1.08 )3 = $2,783.26
.08(1.08)
(1 + i)
77
Uneven Cash Flows
78
FIN 3701 Chapter 2 :The Time Value of Money
14
• Is this an annuity?• How do we find the PV of a cash flow stream
when all of the cash flows are different? (Usea 10% discount rate.)
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
79
• Sorry! There’s no quickie for this one. Wehave to discount each cash flow backseparately.
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
80
Period CF PV (CF)0 -10,000 -10,000.001 2,000 1,818.182 4,000 3,305.793 6,000 4,507.894 7,000 4,781.09
PV of Cash Flow Stream: $ 4,412.95
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
NPV
81
Examples
82
Example• Cash flows from an investment are
expected to be $40,000 per year at theend of years 4, 5, 6, 7, and 8. If yourequire a 20% rate of return, what is thePV of these cash flows?
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
83
• This type of cash flow sequence is oftencalled a “deferred annuity.”
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
84
FIN 3701 Chapter 2 :The Time Value of Money
15
How to solve:• 1) Discount each cash flow back to
time 0 separately.
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
Or,85
• 2) Find the PV of the annuity:• PV : End mode; P/YR = 1; I = 20; PMT =
40,000; N = 5PV = $119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
3
86
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
$119,624
Then discount this single sum back to time 0.PV: End mode; P/YR = 1; I = 20;
N = 3; FV = 119,624;Solve: PV = $69,226
87
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
$119,624
• The PV of the cash flow stream is $69,226.
69,226
88
Retirement Example• After graduation, you plan to invest $400
per month in the stock market. If youearn 12% per year on your stocks, howmuch will you have accumulated whenyou retire in 30 years?
0 1 2 3 . . . 360
400 400 400 400
89
0 1 2 3 . . . 360
400 400 400 400
Using your calculator,
P/YR = 12N = 360
PMT = -400I%YR = 12
FV = $1,397,985.65
90
FIN 3701 Chapter 2 :The Time Value of Money
16
Retirement ExampleIf you invest $400 at the end of each month for thenext 30 years at 12%, how much would you have atthe end of year 30?
Mathematical Solution:
FV = PMT (FVIFA i, n )FV = 400 (FVIFA .01, 360 ) (can’t use FVIFA table)
FV = PMT (1 + i)n - 1i
FV = 400 (1.01)360 - 1 = $1,397,985.65.01
91
House Payment Example• If you borrow $100,000 at 7% fixed
interest for 30 years in order to buy ahouse, what will be your monthly housepayment?
92
0 1 2 3 . . . 360
? ? ? ?
Using your calculator,
P/YR = 12N = 360
I%YR = 7PV = $100,000
PMT = -$665.30
93
House Payment ExampleMathematical Solution:PV = PMT (PVIFA i, n )100,000 = PMT (PVIFA .07, 360 ) (can’t use PVIFA table)
1PV = PMT 1 - (1 + i)n
i
1100,000 = PMT 1 - (1.005833 )360 PMT=$665.30
.00583394
Example• Upon retirement, your goal is to spend 5
years traveling around the world. To travel instyle will require $250,000 per year at thebeginning of each year.
• If you plan to retire in 30 years, what are theequal monthly payments necessary toachieve this goal? The funds in yourretirement account will compound at 10%annually.
95
• How much do we need to have by the endof year 30 to finance the trip?
• PV30 = PMT (PVIFA .10, 5) (1.10) == 250,000 (3.7908) (1.10) == $1,042,470
27 28 29 30 31 32 33 34 35
250 250 250 250 250
96
FIN 3701 Chapter 2 :The Time Value of Money
17
27 28 29 30 31 32 33 34 35
250 250 250 250 250
Using your calculator,
Mode = BEGINPMT = -$250,000
N = 5I%YR = 10P/YR = 1
PV = $1,042,466
97
27 28 29 30 31 32 33 34 35
250 250 250 250 250
• Now, assuming 10% annualcompounding, what monthly paymentswill be required for you to have$1,042,466 at the end of year 30?
1,042,466
98
27 28 29 30 31 32 33 34 35
250 250 250 250 250
Using your calculator,
Mode = ENDN = 360
I%YR = 10P/YR = 12
FV = $1,042,466PMT = -$461.17
1,042,466
99
• So, you would have to place $461.17 inyour retirement account, which earns10% annually, at the end of each of thenext 360 months to finance the 5-yearworld tour.
100
Amortized Loans,Making Interest RatesComparable,Some Complications
101
Amortized Loans• An amortized loan is a loan paid off in
equal payments – consequently, the loanpayments are an annuity.
• Examples: Home mortgage loans, Autoloans
102
FIN 3701 Chapter 2 :The Time Value of Money
18
Amortized Loans (cont.)• Example Suppose you plan to get a
$9,000 loan from a furniture dealer at18% annual interest with annualpayments that you will pay off in over fiveyears.
• What will your annual payments be onthis loan?
103
Amortized Loans (cont.)• Using a Financial Calculator• Enter
• N = 5• i/y = 18.0• PV = 9000• FV = 0• PMT = -$2,878.00
104
The Loan Amortization Schedule
Year AmountOwed onPrincipal atthe Beginningof the Year (1)
AnnuityPayment(2)
InterestPortionof theAnnuity(3) = (1) ×18%
Repayment of thePrincipalPortion oftheAnnuity(4) =(2) –(3)
Outstanding LoanBalance atYear end,After theAnnuityPayment (5)=(1) – (4)
1 $9,000 $2,878 $1,620.00 $1,258.00 $7,742.00
2 $7,742 $2,878 $1,393.56 $1,484.44 $6,257.56
3 $6257.56 $2,878 $1,126.36 $1,751.64 $4,505.92
4 $4,505.92 $2,878 $811.07 $2,066.93 $2,438.98
5 $2,438.98 $2,878 $439.02 $2,438.98 $0.00
105
The Loan Amortization Schedule(cont.)
• We can observe the following from thetable:• Size of each payment remains the
same.• However, Interest payment declines
each year as the amount owed declinesand more of the principal is repaid.
106
Making Interest RatesComparable
107
Which is the better loan:• 8% compounded annually, or• 7.8% compounded annually?
Which is the better loan:• 8% compounded annually, or• 7.8% compounded monthly?
108
FIN 3701 Chapter 2 :The Time Value of Money
19
Annual Percentage Rate (APR)• The annual percentage rate (APR)
indicates the amount of interest paid orearned in one year without compounding.APR is also known as the nominal orstated interest rate. This is the raterequired by law.
109
Comparing Loans using EAR(or APY> Annual Percentage Yield)
• We cannot compare two loans based on APR ifthey do not have the same compounding period.
• To make them comparable, we calculate theirequivalent rate using an annual compoundingperiod. We do this by calculating the effectiveannual rate (EAR)
110
Comparing Loans using EAR(cont.)• Example Calculate the EAR for a loan that
has a 5.45% quoted annual interest ratecompounded monthly.
• EAR = [1+.0545/12]12 - 1= 1.0558 – 1= .05588 or 5.59%
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Which is the better loan:• 8% compounded annually, or• 7.85% compounded quarterly?• We can’t compare these nominal (quoted)
interest rates, because they don’t include thesame number of compounding periods peryear!
We need to calculate the APY or EAR.
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• Find the APY for the quarterly loan:
• The quarterly loan is more expensive than the8% loan with annual compounding!
APY = ( 1 + ) m - 1quoted ratem
APY = ( 1 + ) 4 - 1.07854
Annual Percentage Yield (APY)
APY = .0808, or 8.08%
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Complications:Inflation,Compounding Frequency,Currencies
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FIN 3701 Chapter 2 :The Time Value of Money
20
Complication 1Inflation• How does inflation affect DCF analysis?
NPV = CF +0
CF1
(1+r)
CF2
(1+r)2
CF3
(1+r)3
CF4
(1+r)4
CF5
(1+r)5+ + + + + …
Discounting Rule• Treat inflation consistently: Discount real
cashflows at the real interest rate andnominal cashflows at the nominal interestrate.
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Complication 1
CashflowsNorminal = actual cashflowsReal = cashflows expressed in today's purchasing power
real CF = normal CF / (1+ inflation rate)
Discount ratesNorminal = actual interest ratesReal = interest rates adjusted for inflation1 + real int. rate = (1+ nominal int. rate)/(1+ inflation rate)Approximation: real int. rate normal int.rate - inflationrate
Terminology
t tt
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ExampleThis year you earned $100,000. You expect your earningsto grow 2% annually, in real terms, for the remaining 20years of your career. Interest rates are currently 5% andinflation is 2%. What is the present value of your income?Real interest rate = 1.05/ 1.02 - 1 = 2.94%Real cashflows
Cashflow 102,000 104,040 ….. 148,595
1.0294 1.02942 ….. 1.029420
PV 99,086 98,180 ….. 83,219
Year 1 2 ….. 20
_●●
Present value= $1,818,674117
Complication 2Compounding frequencyOn many investments or loans, interest is credited orcharged more often than once a year.ExamplesBank accounts - dailyMortgages and leases - monthlyBonds – semiannuallyImplicationEffective annual rate (EAR) can be much differentthan the stated annual percentage rate (APR)
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ExampleCar Loan'Finance charge on the unpaid balance, computeddaily, at the rate of 6.75% per year.‘If you borrow $10,000 to be repaid in one year, howmuch would you owe in a year?Daily interest rate = 6.75 / 365 = 0.0185%Day 1: balance=10,000.00 x 1.000185=10,001.85Day 2: balance=10,000.85 x 1.000185=10,003.70Day 365: balance=10,000.00 x (1.000185)365=10.698.50
EAR=6.985%119
Effective annual rateEAR = (1+APR/k)k – 1APR = quoted annual percentage ratek = number of compounding intervals each yearWhat happens as k gets big?In the limit as k → ∞, interest is 'continuouslycompounded‘EAR = eAPR – 1'e' is the base of the natural logarithm2.7182818
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FIN 3701 Chapter 2 :The Time Value of Money
21
Complication 2, cont.Discounting ruleIn applications, interest is normallycompounded at the same frequency aspayments.If so, just divide the APR by number ofcompounding intervals.BondsMake semiannual payments, interestcompounded semiannually Discountsemiannual cashflows by APR/2MortgagesMake monthly payments, interest compoundedmonthly Discount monthly cashflows by APR/12
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Complication 3CurrenciesHow do we discount cashflows in foreign currencies?
Discounting ruleDiscount each currency at its own interest rate: discount $'sat the U.S. interest rate, €'s at the U.K. interest rate, ....This gives PV of each cashflow stream in its own currency.Convert to domestic currency at the current exchange rate.
PV = CF +0
CF1
(1+r)
CF2
(1+r)2
CF3
(1+r)3
CF4
(1+r)4
CF5
(1+r)5+ + + + + …
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Currencies, cont.LogicYou have $1 now. How many pounds can youconvert this to in one year? The current exchangerate is 1.6$/€ and the U.K.interest rate is 5%.Today: $1 = €0.625One year: €0.625 x 1.05=€0.6563Implication: $1 today is worth 0.6263 pounds in oneyear.The discounting rule simply reverses this procedure.It starts with pounds in one year, then converts to $today.
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ExampleYour firm just signed a contract to deliver 2,000batteries in each of the next 2 years to a customer inJapan, at a per unit price of ¥800. It also signed acontract to deliver 1,500 in each of the next 2 yearsto a customer in Britain, at a per unit price of £6.2.Payment is certain and occurs at the end of the year.
The British interest rate is r£ = 5% and the Japaneseinterest rate is r¥=3.5%. The exchange rates are s¥/$= 118 and s$/£ = 1.6.
What is the value of each contract?
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ExampleJapanCFt = 2,000 x 800 = ¥ 1,600,000
PV contract = 3,039,511 x (1/118 ¥/$ ) = $ 25,759
BritainCFt = 1,500 x 6.2 = £ 9,300
PV contact = 17,293 x 1.6$/£ = $ 27,668
PV contract = = ¥ 3,039,5111,600,000 1,600,000
1.035 1.0352
+
PV contract = = ¥ 17,2939,3001.05 1.052
+ 9,300
Source: NYU125
Reference• Sheridan Titman, Arthur J. Keown, John D. Martin,
Financial Management: Principles andApplications(12thed). New Jersey: Pearson &Prentice Hall Inc, 2014.
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