1 Chapter 2: Time Value of Money Future value Present value Rates of return Amortization.
6-1 CHAPTER 5 Time Value of Money Read Chapter 6 (Ch. 5 in the 4 th edition) Future value Present...
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Transcript of 6-1 CHAPTER 5 Time Value of Money Read Chapter 6 (Ch. 5 in the 4 th edition) Future value Present...
6-1
CHAPTER 5Time Value of Money
Read Chapter 6 (Ch. 5 in the 4th edition)
Future value Present value Rates of return Amortization
6-2
Time Value of Money Problems
Use a financial calculator Bring your calculator to class Will need on exams We will not use the tables
6-3
Time lines show timing of cash flows.
CF0 CF1 CF3CF2
0 1 2 3i%
Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.
6-4
A. (1) a. Time line for a $100 lump sum due at the
end of Year 2.
100
0 1 2 Year
i %
6-5
A. (1) b. Time line for anordinary annuity of $100 for
3 years.
100 100100
0 1 2 3i%
6-6
A. (1) c. Time line for uneven CFs -$50 at t=0 and$100, $75, and $50 at the end of Years 1 through 3.
100 50 75
0 1 2 3i%
-50
6-7
What’s the FV of an initial$100 after 3 years if i = 10%?
FV = ?
0 1 2 310%
100
Finding FVs is Compounding.
6-8
After 1 year:
FV1 = PV + I1 = PV + PV (i)= PV(1 + i)= $100 (1.10)= $110.00.
After 2 years:
FV2 = PV(1 + i)2
= $100 (1.10)2
= $121.00.
6-9
After 3 years:
FV3 = PV(1 + i)3
= 100 (1.10)3
= $133.10.
In general,
FVn = PV (1 + i)n
6-10
Three ways to find FVs:
1. ‘Solve’ the Equation with aScientific Calculator
2. Use Tables (the book describes this but not for use in this class)
3. Use a Financial Calculator4. Spreadsheet (has built-in formulas) -- won’t work on exams
6-11
3 10 -100 0N I/YR PV PMT FV
133.10
INPUTS
OUTPUT
Here’s the setup to find FV:
Clearing automatically sets everything to0, but for safety enter PMT = 0.
Check your calculator. Set: P/YR = 1 and END (“BEGIN” should not show on the display)
6-12
What’s the PV of $100 duein 3 years if i = 10%?
Finding PVs is discounting,and it’s the reverse of compounding.
100
0 1 2 310%
PV = ?
6-13
Financial Calculator Solution:
3 10 0 100N I/YR PV PMT FV
-75.13
INPUTS
OUTPUT
Either PV or FV must be negative. HerePV = -75.13. Put in $75.13 today, take out $100 after 3 years.
6-14
If sales grow at 20% per year,how long before sales double?
Solve for n:
FVn = 1(1 + i)n; In our case 2 = (1.20)n .Take the log of both sides:ln(2) = n ln(1.2)n = ln(2)/ln(1.2)=.693…/0.1823.. =3.8017
6-15
20 -1 0 2N I/YR PV PMT FV3.8
INPUTS
OUTPUT
Graphical Illustration:
01 2 3 4
1
2
FV
3.8
Year
Financial calculator solution
6-16
What’s the differencebetween an ordinary
annuity and an annuitydue?
6-17
Ordinary vs. Annuity Due
PMT PMTPMT
0 1 2 3i%
PMT PMT
0 1 2 3i%
PMT
6-18
What’s the FV of a 3-yearordinary annuity of $100 at
10%?
100 100100
0 1 2 310%
110
121
FV = 331
6-19
3 10 0 -100
331.00
INPUTS
OUTPUT
N I/YR PV PMT FV
Financial Calculator Solution:
If you enter PMT of 100, you get FV of
-331.
Get used to the fact that you have to figure out the sign.
6-20
What’s the PV of this ordinaryannuity?
100 100100
0 1 2 310%
90.91
82.64
75.13
248.69 = PV
6-21
3 10 100 0
-248.69
INPUTS
OUTPUT
N I/YR PV PMT FV
Have payments but no lump sum FV,so enter 0 for future value.
Financial Calculator Solution:
6-22
Technical Aside:
Your calculator really is assuming a NPV equation, with PV as a time zero cash flow as follows:
nn
)i1(FVi
)i1(1PMTPVNPV
When you use the top row of calculator keys, the calculator assumes NPV=0 and solves for one variable.
6-23
Find the FV and PV if theannuity were an annuity due.
100 100
0 1 2 310%
100
6-24
3 10 100 0
-273.55
INPUTS
OUTPUT
N I/YR PV PMT FV
Switch from “End” to “Begin”.
Then enter variables to find PVA3 = $273.55.
Then enter PV = 0 and press FV to findFV = $364.10.
6-25Alternative:
The first payment is in the present and thus has a PV of 100.
The next two payments comprise a two period ordinary annuity -- use the formula with n=2, PMT=100, and i=.10.
Sum the above two for the present value. If you already have the PV, multiply by
To get FV
3)i1(
6-26
Perpetuities A perpetuity is a stream of regular payments that goes on forever
An infinite annuity Future value of a perpetuity
Makes no sense because there is no end point Present value of a perpetuity
A diminishing series of numbers
• Each payment’s present value is smaller than the one before
p
PMTPV
k
6-27
Perpetuities—Example E
xam
ple
p
PMT $5PV $250
k 0.02 You may also work this by inputting a
large n into your calculator (to simulate infinity), as shown below.
PV
N
PMT
I/Y
250
999
5
2
0FV
Answer
Q: The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Money markets are such that investors can earn about 8% compounded quarterly on their money. How much can Longhorn sell this special security for?
A: Convert the k to a quarterly k and plug the values into the equation.
6-28
What is the PV of this uneven cashflow stream?
0
100
1
300
2
300
310%
-50
4
90.91
247.93
225.39
-34.15
530.08 = PV
6-29
Input in “CFLO” register ( CFj ):
CF0 = 0
CF1 = 100
CF2 = 300
CF3 = 300
CF4 = -50 Enter I = 10%, then press NPV button to
get NPV = 530.09. (Here NPV = PV.)
6-30
What’s Project L’s NPV?
10 8060
0 1 2 310%
Project L:
-100.00
9.09
49.59
60.11
18.79 = NPVL
11.1
21.1
31.1
6-31
Calculator Solution:
Enter in CFLO for L:
-100
10
60
80
10
CF0
CF1
NPV
CF2
CF3
i = 18.78 = NPVL
6-32TI Calculators
•BA-35 doesn’t appear to do uneven cash flows (NPV and IRR)
BA II PLUSCF
CF0= -100 Enter
C01= 10 Enter F01= 1.00
C02= 60 Enter F02= 1.00
C03= 80 Enter F03= 1.00 NPV I=10 Enter CPT NPV= 18.78
IRR CPT IRR= 18.13
6-33
The Sinking Fund Problem
Companies borrow money by issuing bonds for lengthy time periods
No repayment of principal is made during the bonds’ lives
• Principal is repaid at maturity in a lump sum– A sinking fund provides cash to pay off a
bond’s principal at maturity• Problem is to determine the periodic
deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem
6-34
The Sinking Fund Problem –Example
Q: The Greenville Company issued bonds totaling $15 million for 30 years. The bond agreement specifies that a sinking fund must be maintained after 10 years, which will retire the bonds at maturity. Although no one can accurately predict interest rates, Greenville’s bank has estimated that a yield of 6% on deposited funds is realistic for long-term planning. How much should Greenville plan to deposit each year to be able to retire the bonds with the money put aside?
A: The time period of the annuity is the last 20 years of the bond issue’s life. Input the following keystrokes into your calculator.
PMT
N
FV
I/Y
407,768.35
20
15,000,000
6
0PV
Answer
Exa
mpl
e
6-35
What interest rate wouldcause $100 to grow to
$125.97 in 3 years?
3 -100 0 125.97
INPUTS
OUTPUT
N I/YR PV FVPMT
8%
$100 (1 + i )3 = $125.97.
6-36Will the FV of a lump sum belarger or smaller if we
compound more often, holdingthe stated i% constant? Why?
LARGER! If compounding is morefrequent than once a year--forexample, semi-annually, quarterly,or daily--interest is earned on interestmore often.
6-37
0 1 2 310%
0 1 2 35%
4 5 6
134.01
100133.10
1 2 30
100
Semi-annually:
Annually: FV3 = 100(1.10)3 = 133.10.
FV6/2 = 100(1.05)6 = 134.01.
6-38
We will deal with 3different rates:
iNom = nominal, or stated, or quoted, rate per year.
iPer = periodic rate. The literal rate applied each period
EAR= EFF% = effective annual rate.
6-39
iNom is stated in contracts. Periods per year (m) must also be given. Sometimes (incorrectly) referred to as the “simple” interest rate.
Examples:• 8%, Daily interest (365 days)• 8%; Quarterly
6-40
Periodic rate = iPer = iNom/m, where m is periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.
Examples:8% quarterly: iper = 8/4 = 2%
8% daily (365): iper = 8/365 = 0.021918%
6-41 Effective Annual Rate (EAR = EFF%):
The annual rate which cause PV to grow to the same FV as under multiperiod compounding.
Example: EFF% for 10%, semiannual:
FV = (1 + inom/m)m
= (1.05)2 = 1.1025.
Any PV would grow to same FV at 10.25% annually or 10% semiannually:
(1.1025)1 = 1.1025
(1.05)2 = 1.1025
6-42
Comparing Financial Investments
An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.
Banks say “interest paid daily.” Same as compounded daily.
6-43
How do we find EFF% for a nominal rate of 10%, compounded
semi-annually?
EFF% = 1 + im
- 1nomm
= 1+ 0.10
2 - 1.0
= 1.05 - 1.0= 0.1025 = 10.25%.
2
2
6-44
EAR = EFF% of 10%
105170918.1e:Continuous 10.
EARAnnual = 10%.
EARQ = (1 + 0.10/4)4 - 1 = 10.38%.
EARM = (1 + 0.10/12)12 - 1 = 10.47%.
EARD = (1 + 0.10/360)360 - 1= 10.5155572%.
6-45
Can the effective rate ever beequal to the nominal rate?
Yes, but only if annual compounding is used, i.e., if m = 1.
If m > 1, EFF% will always be greater than the nominal rate.
6-46
When is each rate used?
inom: Written into contracts,quoted by banks andbrokers. Not used incalculations or shownon time lines.
6-47
iper: Used in calculations,shown on time lines.
If inom has annual compounding,then iper = inom/1 = inom.
6-48
EAR = EFF%: Used to compare returnson investments with different paymentsper year and in advertising of deposit interestrates.
(Used for calculations if and only ifdealing with annuities where paymentsdon’t match interest compounding periods.)
6-49
FV of $100 after 3 yearsunder 10% semi-annual
compounding? Quarterly?
FV = PV 1 + imnnom
mn
FV = $100 1 + 0.10
23s
2x3
= $100(1.05)6 = $134.01
FV3Q = $100(1.025)12 = $134.49
6-50What’s the value at the endof Year 3 of the following CF stream if the quoted interest
rate is 10%, compoundedsemi-annually?
0 1
100
2 35%
4 5 6
100 100
6-month periods
6-51
Payments occur annually, but compounding occurs each 6 months.
So we can’t use normal annuity valuation techniques.
6-52
1st method: Compound each CF
0 1
100
2 35%
4 5 6
100 100.00
110.25
121.55
331.80
FVA3 = 100(1.05)4 + 100(1.05)2 + 100= 331.80
2)05.1(100
4)05.1(100
6-53
What’s the PV of this stream?
0
100
15%
2 3
100 100
90.70
82.27
74.62
247.59
Years
2)05.1(100
4)05.1(100
6)05.1(100
6-54
Second Method: use yourfinancial calculator!
Follow these two steps:
a. Find the EAR for the quoted rate:
EAR = 1 + 0.10
2 - 1 = 10.25%.
2
This is the iper for a period of one year. Use in formula (or calculator) with the period equal to a year.
6-55
0
100
110.25%
2 3
100 100
Time line
6-56
3 10.25 0 -100
INPUTS
OUTPUT
N I/YR PV FVPMT
331.80
b. Calculator inputs
6-57
N I PV PMT FV
10 10 100 0
5 8 0 100
7 -500 100 0
15 -750 100 1000
240 8/12 -100,000 0
50 10 100 10
Calculator Workout: fill in the blanks
6-58
0.75 10 - 100 0 ?
=107.41
INPUTS
OUTPUTN I/YR PV PMT FV
Fractional Time Periods
0 0.25 0.50 0.7510%
- 100
1.00
FV = ?
Example: $100 deposited in a bank at 10% interest for 0.75 of the year
6-59
AMORTIZATION
Construct an amortization schedulefor a $1,000, 10% annual rate loanwith 3 equal payments.
6-60
This is what an amortization schedule looks like.
Amortization Table
Beginning Ending
Principal Total Interest Principal Principal
Period Balance Payment Payment Payment Balance
1 $1,000.00 $402.11 $100.00 $302.11 $697.89
2 $697.89 $402.11 $69.79 $332.33 $365.56
3 $365.56 $402.11 $36.56 $365.56 $0.00
6-61
Step 1: Find the required payment.
PMT PMTPMT
0 1 2 310%
-1000
3 10 -1000 0
INPUTS
OUTPUT
N I/YR PV FVPMT
402.11
6-62
Step 2: Find interest chargefor Year 1.
INTt = Beg balt (i)INT1 = 1000(0.10) = $100.
Step 3: Find repayment of principal in Year 1.
Repmt. = PMT - INT= 402.11 - 100= $302.11.
6-63
Step 4: Find ending balanceafter Year 1.
End bal = Beg bal - Repmt = 1000 - 302.11 = $697.89.
Repeat these steps for Years 2 and 3to complete the amortization table.
6-64
Amortization Table
Beginning Ending
Principal Total Interest Principal Principal
Period Balance Payment Payment Payment Balance
1 $1,000.00 $402.11 $100.00 $302.11 $697.89
2 $697.89 $402.11 $69.79 $332.33 $365.56
3 $365.56 $402.11 $36.56 $365.56 $0.00
Interest declines. Tax Implications.
6-65
Amortization tables are widely used-- for home mortgages, auto loans, business loans, retirement plans, etc. They are very important!
Financial calculators (and spreadsheets) are great for setting up amortization tables.
6-66
Amortized Loans—Example E
xam
ple
PMT
N
PV
I/Y
293.75
48
10,000
1.5
0FV
Answer
This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 48 and a k of 1.5%,
resulting in $10,000 = PMT[34.0426] = $293.75.
Q: Suppose you borrow $10,000 over four years at 18% compounded monthly repayable in monthly installments. How much is your loan payment?
A: Adjust your interest rate and number of periods for monthly compounding and input the following keystrokes into your calculator.
6-67
Amortized Loans—Example
PV
N
FV
I/Y
15,053.75
36
0
1
500PMT
Answer
Exa
mpl
e
This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 36 and a k of 1%,
resulting in PVA = $500[30.1075] = $15,053.75.
Q: Suppose you want to buy a car and can afford to make payments of $500 a month. The bank makes three-year car loans at 12% compounded monthly. How much can you borrow toward a new car?
A: Adjust your k and n for monthly compounding and input the following calculator keystrokes.
6-68
Loan Amortization Schedules—Example E
xam
ple
Q: Develop an amortization schedule for the loan demonstrated in Example 5.12.
Note that the Interest portion of the payment is decreasing
while the Principal portion is increasing.