1 Chapter 5 The Time Value of Money Some Important Concepts.
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Transcript of 1 Chapter 5 The Time Value of Money Some Important Concepts.
1
Chapter 5The Time Value of Money
Some Important Concepts
2
Topics Covered
Future Values and Compound Interest Present ValuesMultiple Cash FlowsPerpetuities and AnnuitiesEffective Annual Interest Rates.Inflation
3
Future Values
Future Value - Amount to which an investment will grow after earning interest.
Compound Interest - Interest earned on interest.
Simple Interest - Interest earned only on the original investment.
4
Future Values
Example - Simple InterestInterest earned at a rate of 6% for five years on a principal balance of $100.
Today Future Years 0 1 2 3 4 5
Interest EarnedValue 100
6106
6112
6118
6124
6130
Value at the end of Year 5 = $130
5
Future Values
Example - Compound InterestInterest earned at a rate of 6% for five years on the previous year’s balance.
Today Future Years 0 1 2 3 4 5
Interest EarnedValue 100
6106
6.36112.36
6.74119.10
7.15126.25
7.57133.82
Value at the end of Year 5 = $133.82
6
Future Values
Future Value of $100 = FV
FV r t $100 ( )1
7
Future Values
0
1000
2000
3000
4000
5000
6000
7000
Number of Years
FV
of
$100
0%
5%
10%
15%
8
Future Values
Example – Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal?
trillion
FV
57.120$
)08.1(24$ 380
To answer, determine $24 is worth in the year 2006, compounded at 8%.
Financial calculator: n=380, i=8, PV=-24, PMT=0 FV= $120.57 trillion
Set up the Texas instrument
2nd, “FORMAT”, set “DEC=9”, ENTER2nd, “FORMAT”, move “↓” several times,
make sure you see “AOS”, not “Chn”.2nd, “P/Y”, set to “P/Y=1”2nd, “BGN”, set to “END” P/Y=periods per year, END=cashflow happens end of periods
10
Present Values
If the interest rate is at 6 percent per year, how much do we need to invest now in order to produce $106 at the end of the year?
How much would we need to invest now to produce $112.36 after two years?
100$06.1
106$
1.06
valuefuture(PV) luePresent va
100$06.1
36.112$
1.06
valuefuture(PV) luePresent va
22
11
Present Values
tr)+(1
periodsafter t Value Future=PV
Discount Factor (DF) = PV of $1
Discount Factors can be used to compute the present value of any cash flow.
PV: Discounted Cash-Flow
(DCF)
r: Discount Rate
DFr t
1
1( )
12
Present Values
0
20
40
60
80
100
120
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of Years
PV
of
$100
5%
10%
15%
Interest Rates
13
Present Values
Example
In 2007, Puerto Rico needed to borrow about $2.6 billion for up to 47 years. It did so by selling IOUs, each of which simply promised to pay the holder $1,000 at the end of that time. The market interest rate at the time was 5.15%. How much would you have been prepared to pay for one of these IOUs?
4.94$0944.000,1$)0515.1(
1000,1$
47PV
14
Present Values
Example
Kangaroo Autos is offering free credit on a $20,000 car. You pay $8,000 down and then the balance at the end of 2 years. Turtle Motors next door does not offer free credit but will give you $1,000 off the list price. If the interest rate is 10%, which company is offering the better deal?
15
Present Values
$8,000
$12,000
0 1 2
$8,000
$9,917.36
$17,917.36
PV at Time=0
2)10.1(
1000,12$
Total
16
Present Values
Example – Finding the Interest Rate
Each Puerto Rico IOU promised to pay holder $1,000 at the end of 47 years. It is sold for $94.4. What is the interest rate?
5.15%or 0.0515,r
0515.110.593 r)(1
593.10$94.4
$1,000r)(1
$1,000r)(1$94.4
1/47
47
47
Financial calculator: n=47, FV=1,000 PV=-94.4, PMT=0 i=5.15
17
Present Values
Example – Double your money
Suppose your investment advisor promises to double your money in 8 years. What interest rate is implicitly being promised?
9.05%or 0.0905,r
r)(1$1$2
r)(1PVFV8
t
Financial calculator: n=8, FV=2 PV=-1, PMT=0 i=9.05
18
Present Value of Multiple Cash Flows
Example
Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?
$15,133.06 PVTotal
36.429,3
70.703,3
8,000.00
2
1
)08.1(
000,42
)08.1(
000,41
payment Immediate
PV
PV
What is the PV of this uneven cash flow stream?
0
100
1
300
2
300
310%
-50
4
90.91247.93225.39 -34.15530.08 = PV
Detailed steps (Texas Instrument calculator)
To clear historical data: CF, 2nd ,CE/CTo get PV:CF , ↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2,
Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter, ↓,CPT
“NPV=530.0867427”
21
Future Value of Multiple Cash Flows
Example
You plan to save some amount of money each year. You put $1,200 in the bank now, another $1,400 at the end of the first year, and a third deposit of $1,000 at the end of the second year. If the interest rate is 8% per year, how much will be available to spend at the end of the 3 year?
22
Future Value of Multiple Cash Flows
$1,200
$1,400
$1,000
Year
0 1 2 3
$1,080.00
$1,632.96
$1,511.65
$4,224.61
FV in year 3
08.1000,1$ 2)08.1(400,1$ 3)08.1(200,1$
23
Perpetuities and Annuities
AnnuityEqually spaced level stream of cash flows
for a limited period of time.
PerpetuityA stream of level cash payments that never
ends.
24
Perpetuities and Annuities
rateinterest
paymentcash rCyPerpertuit of PV
25
Perpetuities and Annuities
Example - Perpetuity
In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?
000,000,1$10.0000,100 PV
26
Perpetuities and Annuities
Example - continued
If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?
446,826$2)10.01(
000,000,1
PV
27
Perpetuities and Annuities
Cash FlowYear 0 1 2 3 4 5 6 … Present
Value
1. Perpetuity A $1 $1 $1 $1 $1 $1 …
2. Perpetuity B $1 $1 $1 …
3. Three-year annuity $1 $1 $1
r
1
3)1(
1
rr
3)1(
11
rrr
28
Perpetuities and Annuities
C = cash payment r = interest rate t = Number of years cash payment is received
trrrCPV
)1(
11annuityyear - tof
29
Perpetuities and Annuities
PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.
trrrPVAF
)1(
11
30
Perpetuities and Annuities
Example – LotterySuppose you bought lottery tickets and won $295.7 million. This sum was scheduled to be paid in 25 equal annual installments of $11.828 million each. Assuming that the first payment occurred at the end of 1 year, what was the present value of the prize? The interest rate was 5.9 percent.
6.152$
])059.1(059.0
1
059.0
1[828.11$
25
PV
PV
Financial calculator: n=25, i=5.9, PMT=11.828, FV=0, PV=-152.6
31
Perpetuities and Annuities
Annuity Due Level stream of cash flows starting
immediately.
annuityan of PVr)(1 dueannuity an of PV
32
Perpetuities and Annuities
Example – Lottery (Continued)What is the value of the prize if you receive the first installment of $11.828 million up front and the remaining payments over the following 24 years?
7.161$059.16.152$
059.1])059.1(059.0
1
059.0
1[828.11$
25
PV
PV
33
Perpetuities and Annuities
Future Value of annual payments
r
rC
rtrrr
CFV
t
t
1)1(
)1()1(
11
34
Perpetuities and Annuities
Example - Future Value of annual payments
You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?
100,229$
20)10.01(20)10.01(10.0
110.01000,4
FV
FV
Financial calculator: n=20, i=10, PMT=-4,000, PV=0, FV=229,100
35
Effective Annual Interest Rate
Effective Annual Interest Rate - Interest rate that is annualized using compound interest.
Annual Percentage Rate (APR) - Interest rate that is annualized using simple interest.
36
Effective Annual Interest Rate
Example
Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?
12.00%or 0.12=12 x 0.01=APR
12.68%or 0.1268=1-0.01)+(1=EAR 12
Classifications of interest rates
1. Nominal rate (iNOM) – also called the APR, quoted rate, or stated rate. An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly.
2. Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly. iPER = iNOM / m, where m is the number of
compounding periods per year. e.g., m = 12 for monthly compounding.
Classifications of interest rates
3. Effective (or equivalent) annual rate (EAR, also called EFF, APY) : the annual rate of interest actually being earned, taking into account compounding.
If the interest rate is compounded m times in a year, the effective annual interest rate is
1 1m
nomi
m
Example, EAR for 10% semiannual investment
EAR= ( 1 + 0.10 / 2 )2 – 1 = 10.25%
An investor would be indifferent between an investment offering a 10.25% annual return, and one offering a 10% return compounded semiannually.
EAR on a Financial Calculator
keys: description:
[2nd] [ICONV] Opens interest rate conversion menu
[↓] [EFF=] [CPT] 10.25
Texas Instruments BAII Plus
[↓][NOM=] 10 [ENTER] Sets 10 APR.
[↑] [C/Y=] 2 [ENTER] Sets 2 payments per year
Why is it important to consider effective rates of return?
An investment with monthly payments is different from one with quarterly payments.
Must use EAR for comparisons. If iNOM=10%, then EAR for different
compounding frequency:Annual 10.00%Quarterly 10.38%Monthly 10.47%Daily 10.52%
What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?
Payments occur annually, but compounding occurs every 6 months.
Cannot use normal annuity valuation techniques.
0 1
100
2 35%
4 5
100 100
61 2 3
Method 1:Compound each cash flow
110.25121.55331.80
FV3 = $100(1.05)4 + $100(1.05)2 + $100
FV3 = $331.80
0 1
100
2 35%
4 5
100
61 2 3
100
Method 2:Financial calculator
Find the EAR and treat as an annuity.EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%.
INPUTS
OUTPUT
N I/YR PMTPV FV
3 10.25 -100
331.80
0
When is periodic rate used?
iPER is often useful if cash flows occur several times in a year.
48
Inflation
Inflation - Rate at which prices as a whole are increasing.
Nominal Interest Rate - Rate at which money invested grows.
Real Interest Rate - Rate at which the purchasing power of an investment increases.
49
Inflation
1 real interest rate = 1+nominal interest rate1+inflation rate
approximation formula
Real int. rate nominal int. rate - inflation rate
50
Inflation
Example
If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate?
2.8%or .028=.022-.050=ionApproximat
2.7%or .027 = rateinterest real
1.027 =rateinterest real1.022+1.050+1=rateinterest real1
51
Inflation
Be consistent in how you handle inflation!!
Use nominal interest rates to discount nominal cash flows.
Use real interest rates to discount real cash flows.
You will get the same results, whether you use nominal or real figures
52
Inflation
ExampleYou own a lease that will cost you $8,000 next year, increasing at 3% a year (the forecasted inflation rate) for 3 additional years (4 years total). If discount rates are 10% what is the present value cost of the lease?
53
Inflation
Answer – Nominal figures
99.429,26$
78.59708741.82=8000x1.034
56.63768487.20=8000x1.033
92.68098240=8000x1.032
73.727280001
10% @ PVFlowCash Year
4
3
2
10.182.87413
10.120.84872
10.18240
1.108000
54
Inflation
Answer – Real Figures
Year Cash Flow [email protected]%
1 = 7766.99
2 = 7766.99
= 7766.99
= 7766.99
80001.03
7766.991.068
82401.03
8487.201.03
8741.821.03
2
3
4
7272 73
6809 92
3 6376 56
4 5970 78
26 429 99
7766 991 068
7766 991 068
7766 991 068
2
3
4
.
.
.
.
..
..
..
= $ , .