ch 7 time value of money - advanced

8/12/2019 CH 7 Time Value of Money - Advanced

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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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ch 7 time value of money - advanced

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