Chapter 4 The Time Value of Money
Jan 01, 2016
Chapter 4The Time Value of Money
Essentials ofChapter 4
Why is it important to understand and apply time value to money concepts?
What is the difference between a present value amount and a future value amount?
What is an annuity?
What is the difference between the Annual Percentage Rate and the Effective Annual Rate?
What is an amortized loan?
How is the return on an investment determined?
Time Value of Money
The most important concept in finance
Used in nearly every financial decisionBusiness decisions
Personal finance decisions
Time 0 is todayTime 1 is the end of Period 1 or the beginning of Period 2.
Graphical representations used to show timing of cash flows
Cash Flow Time Lines
Time:
FV = ?
0 1 2 36%
PV=100Cash Flows:
Future Value
The amount to which a cash flow or series of cash flows will grow over a period of time when compounded at a given interest rate.
PV = Present value, or the beginning amount that is invested
r = Interest rate the bank pays on the account each year
INT = Dollars of interest you earn during the year
FVn = Future value of the account at the end of n periods
n = number of period interest is earned
Terms Used in Calculating Future Value
Future Value
In general, FVn = PV (1 + r)n
Four Ways to Solve Time Value of Money Problems
Use Cash Flow Time Line
Use Equations
Use Financial Calculator
Use Electronic Spreadsheet
Time(t):
133.10
0 1 2 310%
100.00Account balance: x (1.10) x (1.10) x (1.10)
110.00 121.00
The Future Value of $100 invested at 10% per year for 3 years
Time Line Solution
Numerical (Equation) Solution
FVn PV(1 r)n
FV3 =$100(1.10)3
=$100(1.331)=133.10
Financial Calculator Solution
INPUTS
OUTPUT
3 10 -100 0 ? N I/YR PV PMT FV
133.10
Spreadsheet Solution
Click on Function Wizard and choose Financial/FV
Set up Problem
Spreadsheet SolutionReference cells:
Rate = interest rate, r
Nper = number of periods interest is earned
Pmt = periodic payment
PV = present value of the amount
Present Value
Present value is the value today of a future cash flow or series of cash flows.
Discounting is the process of finding the present value of a future cash flow or series of future cash flows; it is the reverse of compounding.
What is the PV of $100 due in 3 years if r = 10%?
Time(t):
100.00
0 1 2 310%
75.13Account balance: x (1.10) x (1.10) x (1.10)
82.64 90.90
What is the PV of $100 due in 3 years if r = 10%?
$75.13 =0.7513$100 =
PV = FVn 1
(1+r)n
PV = $100 1
(1.10)3
Financial Calculator Solution
INPUTS
OUTPUT
3 10 ? 0100
N I/YR PV PMT FV
-75.13
Spreadsheet Solution
What interest rate would cause $100 to grow to $125.97 in 3 years?
100 (1 + r )3 = 125.97
100.00 = FV
0 1 2 3
r = ?
PV = 100 100 = 125.97/ (1 + r )3 ] [
$100 (1 + r )3 = $125.97
INPUTS
OUTPUT
3 ? -100 0 125.97
8%
N I/YR PV PMT FV
What interest rate would cause $100 to grow to $125.97 in 3 years?
Spreadsheet Solution
How many years will it take for $68.30 to grow to $100 at an interest rate of 10%?
68.30 (1.10 )n = 100.00
100.00 = FV
0 1 2 n-1
10%
PV = 68.30 68.30 = $100.00 (1.10 )n ] [
. . .
n = ?
How many years will it take for $68.30 to grow to $100 if interest of 10% is paid each year?
FVn = PV(1+r)n
$100.00 = $68.30 (1.10)n
FV4 = 68.30(1.10)4 = $100.00
INPUTS
OUTPUT
? 10 -68.30 0 100.00
4.0
N I/YR PV PMT FV
How many years will it take for $68.30 to grow to $100 if interest of 10% is paid each year?
Spreadsheet Solution
Future Value of an Annuity
Annuity: A series of payments of equal amounts at fixed intervals for a specified number of periods.
Ordinary (deferred) Annuity: An annuity whose payments occur at the end of each period.
Annuity Due: An annuity whose payments occur at the beginning of each period.
PMT PMTPMT
0 1 2 3r%
PMT PMT
0 1 2 3r%
PMT
Ordinary Annuity
Annuity Due
Ordinary Annuity Versus Annuity Due
100 100100
0 1 2 35%
105
110.25
FV = 315.25
What’s the FV of a 3-year Ordinary Annuity of $100 at 5%?
FVAn PMT (1 r)n
t0
n1
PMT
(1 r)n 1r
FVA3 $100(1.05)3 1
0.05
$100(3.15250)$315.25
Numerical Solution:
Financial Calculator Solution
INPUTS
OUTPUT
3 5 0 -100 ?
N I/YR PV PMT FV
315.25
Spreadsheet Solution
Find the FV of an Annuity Due
100 100
0 1 2 35%
100
105.00
110.25
115.76
331.01
x[(1.05)0](1.05)
x[(1.05)1](1.05)
x[(1.05)2](1.05)
FVA(DUE)3 =
Numerical Solution
FVA(DUE)n PMT (1 r)t
t1
n
PMT
(1+r)n 1
r
x(1 r)
FVA(DUE)3 $100(1.05)3 1
0.05
x(1.05)
= $100[3.15250x1.05]
= $100[3.310125] = 331.01
Financial Calculator Solution
Switch from “End” to “Begin”.
INPUTS
OUTPUT
3 5 0 -100 ? N I/YR PV PMT FV
331.01
Spreadsheet Solution
Present Value of an Annuity
PVAn = the present value of an annuity with n payments.
Each payment is discounted, and the sum of the discounted payments is the present value of the annuity.
248.69 = PV
100 100100
0 1 2 310%
90.91
82.64
75.13
What is the PV of this Ordinary Annuity?
PVAn PMT1
(1 r)tt1
n
PMT
1- 1(1r)n
r
$248.6985)$100(2.486
0.10
-1$100PVA
3(1.10)1
3
Numerical Solution
Financial Calculator Solution
INPUTS
OUTPUT
3 10 ? -100 0 N I/YR PV PMT FV
-248.69
Spreadsheet Solution
100 100
0 1 2 35%
100
100.00
95.24
90.70
285.94
(1.05)x[1/(1.05)1]x
PVA(DUE)3=
(1.05)x[1/(1.05)2]x
(1.05)x[1/(1.05)3]x
Find the PV of an Annuity Due
Financial Calculator Solution
INPUTS
OUTPUT
3 5 ? -100 0 N I/YR PV PMT FV
285.94
250 250
0 1 2 3r = ?
- 864.80
4
250 250
You pay $864.80 for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of each year. What interest rate will you earn on this investment?
Solving for Interest Rates with Annuities
Use trial-and-error by substituting different values of r into the following equation until the right side equals $864.80.
Numerical Solution
$864.80$250
1- 1(1r)4
r
Financial Calculator Solution
INPUTS
OUTPUT
4 ? -846.80 250 0 N I/YR PV PMT FV
7.0
Spreadsheet Solution
Annuities that go on indefinitely
PVP Payment
Interest rate
PMTr
Perpetuities
Uneven Cash Flow Streams
A series of cash flows in which the amount varies from one period to the next:Payment (PMT) designates constant cash
flows—that is, an annuity stream.
Cash flow (CF) designates cash flows in general, both constant cash flows and uneven cash flows.
0
100
1
300
2
300
310%
-50
4
90.91
247.93
225.39
-34.15530.08 = PV
What is the PV of this Uneven Cash Flow Stream?
Numerical Solution
PVCF1
1(1 r)1
CF2
1(1 r)2
...CFn
1(1 r)n
4321 (1.10)
150)(
(1.10)
1300
(1.10)
1300
(1.10)
1100PV
$100(0.90909)$300(0.82645)$300(0.75131) ( $50)(0.68301)
$530.09
Financial Calculator Solution
Input in “CF” register:CF0 = 0CF1 = 100CF2 = 300CF3 = 300CF4 = -50
Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV)
Spreadsheet Solution
Semiannual and Other Compounding Periods
Annual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added once a year.
Semiannual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added twice a year.
If compounding is more frequent than once a year—for example, semi-annually, quarterly, or daily—interest is earned on interest—that is, compounded—more often.
LARGER!
Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated r constant? Why?
0 1 2 310%
100133.10
Annually: FV3 = 100(1.10)3 = 133.100 1 2 3 4 5 6
5%
134.01100
Semi-annually: FV6/2 = 100(1.05)6 = 134.01
Compounding Annually vs. Semi-Annually
rSIMPLE = Simple (Quoted) RateSimple (Quoted) Rate used to compute the interest paid per period rEAR = Effective Annual RateEffective Annual Ratethe annual rate of interest actually being earned
APR = Annual Percentage RateAnnual Percentage Rate = rSIMPLE periodic rate X the number of periods per year
Distinguishing Between Different Interest Rates
Comparison of Different Types of Interest Rates
rSIMPLE: Written into contracts, quoted by
banks and brokers. Not used in calculations or shown on time lines.
rPER: Used in calculations, shown on time
lines.
rEAR: Used to compare returns on
investments with different payments per year.
Simple (Quoted) Rate
rSIMPLE is stated in contracts Periods per year (m) must also be given
Examples:8%, compounded quarterly8%, compounded daily (365 days)
Periodic Rate
Periodic rate = rPER = rSIMPLE/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.
Examples:8% quarterly: rPER = 8/4 = 2%
8% daily (365): rPER = 8/365 = 0.021918%
The annual rate that causes PV to grow to the same FV as under multi-period compounding.
Example: 10%, compounded semiannually:
rEAR = (1 + rSIMPLE/m)m - 1.0
= (1.05)2 - 1.0 = 0.1025 = 10.25%
Effective Annual Rate
1 - m
r + 1 = r
mSIMPLE
EAR
10.25% = 0.1025 =1.0 - 1.05 =
1.0 - 2
0.10 +1 =
2
2
How do we find rEAR for a simple rate of 10%, compounded semi-annually?
FVn = PV 1 + rSIMPLE
m
mn
$134.0110)$100(1.3402
0.10 + 1$100 = FV
32
23
$134.4989)$100(1.3444
0.10 + 1$100 = FV
34
43
FV of $100 after 3 years if interest is 10% compounded semi-annual? Quarterly?
0 0.25 0.50 0.7510%
- 100
1.00
FV = ?
Example: $100 deposited in a bank at EAR = 10% for 0.75 of the year
INPUTS
OUTPUT
0.75 10 -100 0 ?
107.41
N I/YR PV PMT FV
Fractional Time Periods
Spreadsheet Solution
Amortized Loans
Amortized Loan: A loan that is repaid in equal payments over its life
Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, and so forth to determine how much of each payment represents principal repayment and how much represents interest
PMT PMTPMT
0 1 2 310%
-1,000
Construct an amortization schedule for a $1,000, 10 percent loan that requiresthree equal annual payments.
PMT PMTPMT
0 1 2 310%
-1000
Step 1: Determine the required payments
INPUTS
OUTPUT
3 10 -1000 ? 0 N I/YR PV PMT FV
402.11
INTt = Beginning balancet (r)
INT1 = 1,000(0.10) = $100.00
Step 2: Find interest charge for Year 1
Repayment = PMT - INT= $402.11 - $100.00= $302.11.
Step 3: Find repayment of principal in Year 1
Ending bal. = Beginning bal. - Repayment
= $1,000 - $302.11 = $697.89.
Repeat these steps for the remainder of the payments (Years 2 and 3 in this case) to complete the amortization table.
Step 4: Find ending balance after Year 1
Spreadsheet Solution
Loan Amortization Table10 Percent Interest Rate
YR Beg Bal PMT INT Prin PMT End Bal
1 $1000.00 $402.11 $100.00 $302.11 $697.89
2 697.89 402.11 69.79 332.32 365.57
3 365.57 402.11 36.56 365.55 0.02
Total 1,206.33 206.35 999.98 *
* Rounding difference
Chapter 4 EssentialsWhy is it important to understand and apply time value to money concepts?To be able to compare various investments
What is the difference between a present value amount and a future value amount?Future value adds interest - present value subtracts
interest
What is an annuity?A series of equal payments that occur at equal time
intervals
Chapter 4 EssentialsWhat is the difference between the Annual Percentage Rate and the Effective Annual Rate?APR is a simple interest rate quoted on loans. EAR is the
actual interest rate or rate of return.
What is an amortized loan?A loan paid off in equal payments over a specified period
How is the return on an investment determined?The amount to which the investment will grow in the
future minus the cost of the investment