ch 7 time value of money - advanced
TRANSCRIPT
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Chapter 7
Time Value of Money: Advanced
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Learning Objectives
Use a financial calculator to solve TVM problemsinvolving multiple periods and multiple cash flows
General case
Perpetuity
Annuity
Find the rate of return in multi-period (multi-CF) time-
value-of-money problems
The frequency of compounding
Prepare an amortization schedule
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A Special Case: Perpetuity
C C C
|------------|-----------|--------- -----|----- --------> time
0 1 2 t
Cash flows are fixed (same) in each period
N (number of periods) is infinity
What would be the PV of a stream of equal cash flows that occur at the end
of each period and go on forever?
PV = C / r
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Perpetuity Examples
An asset that generates $1,000 per year forever, in otherwords, a perpetuity of $1,000. If the discount rate is 8%, the
present value of this perpetuity will be
PV=1,000/0.08=$12,500.
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A Special Case: Annuity
C C C
|------------|-----------|--------- -----|----- ------------> time
0 1 2 t
Cash flows are fixed (same) in each period
N (number of periods) is fixed
What would be the PV of a stream of equal cash flows that occur at the end
of each period and go on for N periods?
PV = C / r * [ 1 - 1/(1+r)^N ]
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Annuity Examples
What is the PV of a three-year annuity of $700 per year? The
discount rate is 8% p.a.
PV=(700/0.08)*(1-(1/(1+0.08))^3)=$1803.97
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Time Value Calculations with a
Financial Calculator
Texas Instruments BAII PLUS
Basic Setting
Press 2nd and [Format]. The screen will display the number of decimal
places that the calculator will display. If it is not eight, press 8 and thenpress Enter
Press 2nd and then press [P/Y]. If the display does not show one, press
1 and then Enter
Press 2nd and [BGN]. If the display is not END, that is, if it says BGN,press 2nd and then [SET], the display will read END
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Special keys used for TVM problems
N: Number of years (periods)
I/Y: Discount rate per period
PV: Present value
PMT: The periodic fixed cash flow in an annuity
FV: Future value
CPT: Compute
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Practice Example One
How much do you need to deposit today so that you can have $6,000 six yearsfrom now when the discount rate is 14%.
6 and then N
14 and then I/Y
0 and then PMT
6,000 and then FV
Finally, press CPT and then PV
The number -2,733.519286 will be in the displayWhy negative?
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Practice Example TwoSuppose you deposit $150 in an account today and the interest rate is 6 percent p.a..How much will you have in the account at the end of 33 years?
33 then N
6 then I/Y
150 then +/- and then PV
0 then PMT
CPT then FV
The number 1,026.09 will be in the display
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Practice Example ThreeYou deposited $15,000 in an account 22 years ago and now the account has
$50,000 in it. What was the annual rate of return that you received on this
investment?
N = 22, PV = - 15,000, PMT = 0, FV = 50,000,
I/Y = ?
I/Y=5.625%
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Practice Example FourYou currently have $38,000 in an account that has been paying 5.75
percent p.a.. You remember that you had opened this account quite
some years ago with an initial deposit of $19,000.You forget when the
initial deposit was made. How many years (in fractions) ago did you
make the initial deposit?
PV = - 19000, PMT = 0, FV = 38000, I/Y = 5.75,
N = ?
N=12.398
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Practice Example FiveSuppose an investment promises to yield annual cash flows of$13,000 per year for eleven years. If your required rate of return is
13%, what is the maximum price that you would be willing to pay?
N=11, I/Y=13, PMT=13,000, FV=0, PV=?
PV=$73,930.23
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Practice Example SixAn asset promises the following stream of cash flows. It will pay you
$80 per year for twenty years and , in addition, at the end of the
twentieth year, you will be paid $1,000. If your required rate of return
is 9%, what is the maximum price that you would pay for this asset?
N=20, I/Y=9,PMT=80,FV=1,000,PV=?
PV=-908.71
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Practice Example-Retirement ProblemYou plan to retire in 30 years. After that, you need $200,000 per year for 10 years(first withdraw at t=31). At the end of these 10 years, you will enter a retirement
home where you will stay for the rest of your life. As soon as you enter the
retirement home, you will need to make a single payment of 1 million. You want to
start saving in an account that pays you 9% interest p.a. Therefore, beginning from
the end of the first year (t=1), you will make equal yearly deposits into this accountfor 30 years. You expect to receive $500,000 at t=30 from a cash value insurance
policy that you own and you will deposit this money to your retirement account.
What should be the yearly deposits?
Answer: Two annuities.
At t=30: N=10, I/Y=9, PMT=-200,000,FV=-1,000,000, PV(30)=1705942.347
1705942.347-500,000=1205942.347
At t=0: N=30, I/Y=9,PV=0, FV=1205942.347
PMT=-8847.22
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Special topics in Time Value
Compounding period is less than one year
Loan amortization
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Compounding period is less than One year
Suppose that your bank statesthat the interest on your accountis 8% p.a.. However, interest is paid semi annually, that is every
six months or twice a year. The 8% is called the stated interest
rate. (also called the nominal interest rate) But, the bank will
pay you 4% interest every 6 months.In other words, the compounding frequency is two.
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Compounding Frequency exampleSuppose you deposit $100 into the account today.
If the interest had been paid once a year,
100 x1.08=108
If the interest had been paid twice a year,Account balance at end of 6 months:
100 x 1.04 = 104
Account balance at end of 1 year:
104 x 1.04 =108.16
Effective Interest Rate = (108.16100)/100 = 8.16%
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Compounding Frequency exampleSuppose the interest had been paid quarterly, you would have
receive 8/4=2 % interest every quarter.
In this case:
Account balance at end of 1 year:
100 x (1.02)^4 =108.2432
Effective Interest Rate = (108.2432 -100)/100=8.2432%
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Compounding period is less than One year
11rateinterestEffective
1
1
m
nm
nm
m
r
m
rPVFV
m
r
FVPV
20
n = number of years
m = frequency of
compounding per year
r = stated interest
rate
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Example
Suppose you deposit $100 today in your bank account thatstates the interest is 8% p.a.. However, the interest is paidquarterly. Compute your account balance at the end of five yearswith quarterly compounding.
Account balance at end of 5 year:100 x (1.02)^20 =148.59
N =5 x4=20
I/Y=2, PV=-100,PMT=0, CPT FV=148.59
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Loan AmortizationAmortization is the process of separating a payment into two
parts:
The interest payment
The repayment of principal
Note:
Interest payment decreases over time
Principal repayment increases over time
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Example of Loan Amortization
You have borrowed $8,000 from a bank and have promised toreturn it in five equal years payments. The first payment is at the
end of the first year. The interest rate is 10 percent. Draw up the
amortization schedule for this loan.
Amortization schedule is just a table that shows how each
payment is split into principal repayment and interest payment.
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Example of Loan AmortizationStep 1: Compute periodic payment.
PV=8000, N=5, I/Y=10, FV=0, PMT=?
Verify that PMT = 2,110.38
Step 2: Amortization for first year
Interest payment = 8000 x 0.1 = 800
Principal repayment
= 2,110.38800 = 1310.38Immediately after first payment, the principal balance is
= 80001310.38 = 6,689.62
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Example of Loan AmortizationStep 3: Amortization for second year
Interest payment = 6689.62 x 0.1 = 668.96
(using the new balance!)
Principal repayment
= 2,110.38668.96 = 1441.42
Immediately after second payment, the principal balance is = 6,689.621441.42 = 5,248.20
Verify that the entire schedule (on following slide)
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Example of Loan Amortization
Year
Beg.
Balance Payment Interest Principal
End.
Balance
0 8,000.001 8,000.00 2,110.38 800.00 1,310.38 6,689.62
2 6689.62 2,110.38 668.96 1,441.42 5,248.20
3 5248.20 2,110.38 524.82 1,585.56 3,662.64
4 3662.64 2,110.38 366.26 1,744.12 1,918.53
5 1918.53 2,110.38 191.85 1,918.53 0.00
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Summary
TVM problems with multiple periods and multiple cash flows
Solving TVM problems using financial calculator and timelines
Special TopicsCompounding period < One Year
Loan amortization
Practice! Practice! Practice!
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