discounted cash flow valuations -6 (2)

7/28/2019 Discounted Cash Flow Valuations -6 (2)

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T6.1 Chapter Outline

Discounted Cash Flow Valuation

Chapter Organization

6.1 Future and Present Values of Multiple Cash Flows

6.2 Valuing Level Cash Flows: Annuities and Perpetuities

6.3 Comparing Rates: The Effect of Compounding

6.4 Loan Types and Loan Amortization

6.5 Summary and Conclusions

M ZAHID KHAN


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FUTURE VALUE WITH MULTIPLE CASH FLOW:

So far we have restricted our attention to either the cashflow of a lump-sum present amount or the present value ofsome single future cash flows. In this section , we begin to

study ways to value multiple cash flows.

M ZAHID KHAN


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Suppose that you have deposited 100 today in accountpaying 8 % . In one Year , you will be deposited another100. How much you will paid in two years?

208*1.08=224.64

M ZAHID KHAN


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There are two ways to calculate the future values for Multiple cashFlows:

1 Compound the accumulated balance forward one year at atime

2 Calculate the future value of each cash flow first and thenadd these up.

M ZAHID KHAN


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Irwin/McGraw-Hill copyright 2002 McGraw-Hill Ryerson, Ltd Slide 5

T6.2 Future Value Calculated (Fig. 6.3-6.4)

Future value calculated by compounding forward one period at a time

Future value calculated by compounding each cash flow separately


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If you deposit 100$ in one Year.200$ in two year ,and 300$ inthree year . How much will you have in three years ? How muchof this interest ? How much will you have in five year if you dont

have additional amounts? Assume 7 percent interest ratethrough out?

M ZAHID KHAN


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T6.1 DCFM

Discounted Cash Flow Valuation:


If you deposit 100$ in one Year.200$ in two year ,and 300$ inthree year . How much will you have in three years ? How much

of this interest ? How much will you have in five year if you donthave additional amounts? Assume 7 percent interest ratethrough out?

Ans: 100$ * 1.07^2 = 114.49$200$ * 1.07 = 214.00300$ * = 300.00

628.49

Future Value = 628.49-( 100+200+300)= 28.49

M ZAHID KHAN


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T6. DCFM



Ans: 100$ * 1.07 = 114.49$200$ * 1.07 = 214.00

300$ * = 300.00628.49

Future Value = 628.49-( 100+200+300)= 28.49

5 Years amount ?We know we have 628.49*1.07^2 = 628.49*1.1449=719.56

M ZAHID KHAN


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PRESENT VALUE WITH MULTIPLE CASH FLOW:

There are 2 ways :

We can either discount back one period at a time

We can calculate the present values individually and then addthem up.

M ZAHID KHAN


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T6.3 Present Value Calculated (Fig 6.5-6.6)Present value

calculated by

discounting each

cash flow separately

Present value

calculated by

discounting back one

period at a time


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Suppose you need 1,000$ in one year and 2000$ more in twoyears . If we can earn 9 % on your money , how much do you

have to put up today to exactly cover these amount in the future? In other words , what is the present value of the cash flows at9%?

M ZAHID KHAN


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T6.1 DCFM PV MULTIPLE CF



Suppose you need 1,000$ in one year and 2000$ more in twoyears . If we can earn 9 % on your money , how much do youhave to put up today to exactly cover these amount in the future

? In other words , what is the present value of the cash flows at9%?

The PV of 2000 in two yrs at 9% is :2000/1.09^2 = 1,683.36The PV of 1000 in one yrs at 9% is :1000/1.09=917.43Total 1683.36+917.43=2600.79To checking :2600.79*1.09=2834.86 almost 3000

M ZAHID KHAN


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ANNUITIES AND PERPETUTIES :

A series of constant , or level , cash flows that occur at the end ofeach period for some fixed number of years, is called ordinaryannuity or more correctly , the cash flows are said to in ordinaryannuity form.

Present value of an annuity of C $ per period for t period whenthe rate of return , or the interest rate , is r is given by:

Annuity present Value = C * ( 1- Present Value Factor / r)

= C * ( 1- (1-(1/1+r) ^ t ) / r

Notice that 1 / 1+r ^ t is the same present value Interest factor.


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T6.10 Summary of Annuity and Perpetuity Calculations (Table 6.2)

I. Symbols

PV = Present value, what future cash flows bring todayFVt = Future value, what cash flows are worth in the future

r = Interest rate, rate of return, or discount rate per period

t = Number of time periods

C = Cash amount

II. FV ofCper period fortperiods at rpercent per period:

FVt = C {[(1 + r)t- 1]/r}

III. PV ofCper period fortperiods at rpercent per period:

PV = C {1 - [1/(1 + r)t]}/r

IV. PV of a perpetuity of C per period:

PV = C/r

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T6.5 Annuities and Perpetuities -- Basic Formulas

Annuity Present Value

PV = C {1 - [1/(1 + r)t]}/r

Annuity Future Value

FVt = C {[(1 + r)t- 1]/r}

Perpetuity Present Value

PV = C/r

The formulas above are the basis of many of the calculationsin Corporate Finance. It will be worthwhile to keep them in

mind!

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T6.6 Examples: Annuity Present Value


Suppose you need $20,000 each year for the next threeyears to make your tuition payments.

Assume you need the first $20,000 in exactly one year.Suppose you can place your money in a savingsaccount yielding 8% compounded annually. How much

do you need to have in the account today?(Note: Ignore taxes, and keep in mind that you dontwant any funds to be left in the account after the thirdwithdrawal, nor do you want to run short of money.)


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T6.6 Examples: Annuity Present Value (continued)

Annuity Present Value - Solution

Here we know the periodic cash flows are $20,000each. Using the most basic approach:

PV = $20,000/1.08 + $20,000/1.082 + $20,000/1.083

= $18,518.52 + $_______ + $15,876.65= $51,541.94

Heres a shortcut method for solving the problem using theannuity p resent value factor:

PV = $20,000 [____________]/__________= $20,000 2.577097= $________________


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Here we know the periodic cash flows are $20,000each. Using the most basic approach:

PV = $20,000/1.08 + $20,000/1.082 + $20,000/1.083

= $18,518.52 + $17,146.77 + $15,876.65= $51,541.94


PV = $20,000 [1 - 1/(1.08)3]/.08= $20,000 2.577097= $51,541.94


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Lets continue our tuition problem.

Assume the same facts apply, but that you can onlyearn 4% compounded annually. Now how much do youneed to have in the account today?


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T6.6 Examples: Annuity Present Value (concluded)


Again we know the periodic cash flows are $20,000each. Using the basic approach:

PV = $20,000/1.04 + $20,000/1.042 + $20,000/1.043

= $19,230.77 + $18,491.12 + $17,779.93= $55,501.82


PV = $20,000 [1 - 1/(1.04)3]/.04= $20,000 2.775091= $55,501.82


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PER AND ANNUITY

FINDING THE PAYMENT:

Finding C;

Suppose you wish to start up a new Business , You need toborrow 100,000. You want to make 5 equal installment and theinterest rate is 5 percent.

Present Value = 100,000/=Annuity Present Value =

100,000 = C * ( 1 Present Value Factor) / r100,000 = c * (1 1/ 1.18^ 5 ) / .18100,000 = C * ( 1 - .4371 ) / .18

100,000 = C * 3.1272C = 100,000 / 3.1272 = 31,978

Just under 32,000/=M ZAHID KHAN

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T6.4 Chapter 6 Quick Quiz: Part 1 of 5 (concluded)

Example: Findin g C

Q. You want to buy a Mazda Miata to go cruising. It costs $25,000.With a 10% down payment, the bank will loan you the rest at

12% per year (1% per month) for 60 months. What will your monthlypayment be?

A. You will borrow .90

$25,000 = $22,500 . This is the amount today, soits the present value. The rate is 1%, and there are 60 periods:

$ 22,500 = C {1 - (1/(1.01)60}/.01= C {1 - .55045}/.01= C 44.955

C = $22,500/44.955

C = $500.50 per month


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PER AND ANNUITY

FINDING THE RATE :

Suppose that Insurance Co offer to pay you 1,000/= per year for10 yrs if you pay 6,710 up front . What rate is implicit in this for10 yrs.

Present Value = 6,710/=Cash Flows = 1,000/= per years6,710 = 1,000 * ( 1 Present Value Factor) / r(6,710 / 1000 ) =6.71 = 1- Present Value Factor / r

If you look across the row 10 periods in Table A.3 . You will see afactor of 6.7101 for 8 percent , so we are right away thatinsurance co is just offering 8 %

Or Just Use TRIAL AND ERRORM ZAHID KHAN

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PER AND ANNUITY

FUTURE VALUE OF ANNUITIES:

There are Future Value Factor for annuities as well as presentfactor.The Future Value Factor of Annuity is :

Annuity FV Factor = ( Future Value Factor 1) / r= ( ( 1 + r ) ^ t - 1) / r

Suppose you plan to Contribute 2,000 per yr into the retirementaccount paying 8 % . If you retire in 30 yrs , how much will youhave ?

Annuity FV Factor = ( Future Value Factor 1) / r= (1.08 ^ 30 1) / .08= (10.0627 1) / .08= 113.2832

Thus the FV of this 30 yrs , 2000 annuity is :Annuity Future Value = 2,000 * 113.2832= 226,566.40

M ZAHID KHAN


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T6.7 Chapter 6 Quick Quiz -- Part 2 of 5

Example 2: Findin g C

21-year old could accumulate $1 million by age 65 by investing$15,091 today and letting it earn interest (at 10%compoundedannually) for 44 years.

Now, rather than plunking down $15,091 in one chunk, supposeshe would rather invest smaller amounts annually to accumulatethe million. If the first deposit is made in one year, and deposits

will continue through age 65, how large must they be? Set this up as a FV problem:

$1,000,000 = C [(1.10)44 - 1]/.10

C= $1,000,000/652.6408 = $1,532.24

Becoming a millionaire just got easier!

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T6.8 Example: Annuity Future Value

Previously we found that, if one begins saving at age 21,accumulating $1 million by age 65 requires saving only$1,532.24 per year.

Unfortunately, most people dont start saving for retirementthat early in life. (Many dont start at all!)

Suppose Bill just turned 40 and has decided its time to getserious about saving. Assuming that he wishes to accumulate

$1 million by age 65, he can earn 10% compounded annually,and will begin making equal annual deposits in one year andmake the last one at age 65, how much must each deposit be?

Setup: $1 million = C [(1.10)25 - 1]/.10

Solve forC: C= $1 million/98.34706 = $10,168.07

By waiting, Bill has to set aside oversix t imesas much moneyeach year!


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Consider Bills retirement plans one more time.

Again assume he just turned 40, but, recognizing that he hasa lot of time to make up for, he decides to invest in somehigh-risk ventures that may yield 20% annually. (Or he maylose his money completely!) Anyway, assuming that Bill stillwishes to accumulate $1 million by age 65, and will beginmaking equal annual deposits in one year and make the lastone at age 65, nowhow much must each deposit be?

Setup: $1 million = C [(1.20)25 - 1]/.20

Solve forC: C= $1 million/471.98108 = $2,118.73

So Bill can catch up, but only if he can earn a much h igherreturn(which will probably require taking a lot more risk!).


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T6.10 Summary of Annuity and Perpetuity Calculations (Table 6.2)

I. Symbols

PV = Present value, what future cash flows bring today

FVt = Future value, what cash flows are worth in the future

r = Interest rate, rate of return, or discount rate per period

t = Number of time periods

C = Cash amount

II. FV ofCper period fortperiods at rpercent per period:

FVt = C {[(1 + r)t- 1]/r}

III. PV ofCper period fortperiods at rpercent per period:

PV = C {1 - [1/(1 + r)t]}/r

IV. PV of a perpetuity of C per period:

PV = C/r

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T6.11 Perpetuity:

An important special case of an annuity arises when the levelstream of cash flows continuous forever, since the cash

flows are perpetual. Perpetuities are also called console.

Since a perpetuity has a infinite number of cash flows , weobviously cant compute its value by discounting each one.Fortunately , valuing a perpetuity turn out be the easiest possiblecase.

The PV of Perpetuity is simply = C/r

Example :


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T6.11 Example: Perpetuity Calculations

Suppose we expect to receive $1000 at the end of each of thenext 5 years. Our opportunity rate is 6%. What is the value

today of this set of cash flows?

PV = $1000 {1 - 1/(1.06)5}/.06

= $1000 {1 - .74726}/.06

= $1000 4.212364

= $4212.36

Now suppose the cash flow is $1000 per yearforever. This iscalled a perpetui ty. And the PV is easy to calculate:

PV = C/r= $1000/.06 = $16,666.66 So, payments in years 6 thru have a total PV of $12,454.30!


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Consider the following questions.

The present value of a perpetual cash flow stream has a finite value(as long as the discount rate, r, is greater than 0). Heres a questionfor you: How can an infinite number of cash payments have a f in i tevalue?

Heres an example related to the question above. Suppose you areconsidering the purchase of a perpetual bond. The issuer of thebond promises to pay the holder $100 per year forever. If youropportunity rate is 10%, what is the most you would pay for the bondtoday?

One more question: Assume you are offered a bond identical to theone described above, but with a life of 50 years. What is thedifference in value between the 50-year bond and the perpetualbond?


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T6.12 Solution to Chapter 6 Quick Quiz -- Part 4 of 5

An infinite number of cash payments has a finite presentvalue is because the present values of the cash flows in the

distant future become infinitesimally small.

The value today of the perpetual bond = $100/.10 = $1,000.

Using Table A.3, the value of the 50-year bond equals

$100 9.9148 = $991.48

So what is the present value of payments 51 through infinity(also an infinite stream)?

Since the perpetual bond has a PV of $1,000 and theotherwise identical 50-year bond has a PV of $991.48, thevalue today of payments 51 through infinity must be

$1,000 - 991.48 = $8.52 (!)


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T6.13 Compounding Periods, EARs, and APRs

Compounding Number of times Effective

period compounded annual rate

Year 1 10.00000%

Quarter 4 10.38129

Month 12 10.47131

Week 52 10.50648

Day 365 10.51558

Hour 8,760 10.51703

Minute 525,600 10.51709


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T6.13 Compounding Periods, EARs, and APRs

Compounding Number of times Effective

period compounded annual rate

Quarter 4 10.38129

10 PERCENT COMPOUND QUARTERLY

10 % OR .10 /4 = .025 OR 2.5 % PER QUARTER

1 $ Investment for 4 qtr

1*1.025^ 4= 1.103812891 $

The EAR is 10.38122 PERCENT


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T6.13 Compounding Periods, EARs, and APRs (continued)

EARs and APRs

Q. If a rate is quoted at 16%, compounded semiannually,then the actual rate is 8% per six months. Is 8% per sixmonths the same as 16% per year?

A. If you invest $1000 for one year at 16%, then youllhave $1160 at the end of the year. If you invest at

8% per period for two periods, youll have

FV = $1000 (1.08)2

= $1000 1.1664

= $1166.40,

or $6.40 more. Why? What rate per year is thesame as 8% per six months?


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T6.13 Compounding Periods, EARs, and APRs (concluded)

The Effect ive Annual Rate(EAR) is _____%. The 16%compounded semiannually is the quoted or stated rate,not the effective rate.

By law, in consumer lending, the rate that must be quotedon a loan agreement is equal to the rate per periodmultiplied by the number of periods. This rate is called the

_________________(____).

Q. A bank charges 1% per month on car loans. What is theAPR? What is the EAR?

A. The APR is __ __ = ___%. The EAR is:

EAR = _________ - 1 = 1.126825 - 1 = 12.6825%


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T6.13 Compounding Periods, EARs, and APRs (concluded)

The Effect ive Annual Rate(EAR) is 16.64%. The 16%compounded semiannually is the quoted or stated rate,not the effective rate.

By law, in consumer lending, the rate that must be quotedon a loan agreement is equal to the rate per periodmultiplied by the number of periods. This rate is called theAnnual Percentage Rate(APR).

Q. A bank charges 1% per month on car loans. What is theAPR? What is the EAR?

A. The APR is 1% 12 = 12%. The EAR is:

EAR = (1.01)12 - 1 = 1.126825 - 1 = 12.6825%

The APR is thu s a qu oted rate, no t an effect ive rate!


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T6.13 AMORTIZED LOAN

The process of paying off a loan by m aking regularprinc ipal reductio ns is cal led amor t izing loan.

Suppose you take ou t a loan of 5000/= 5 yrs lo an at 9% .The loan agreement cal l for a bo rrow er to pay interest onthe loan balance each year and to reduce the loan balance

each year by 1000 . Sinc e the loan is d ecl ined b y 1000 eachyear i t wi l l be paid in 5 yrs com pletely ?

Make the Amo rt izat ion Schedule ?


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T6.13 AMORTIZED LOAN

Suppose you take ou t a loan of 5000/= 5 yrs lo an at 9% .The loan agreement cal l for a bo rrow er to pay interest onthe loan balance each year and to reduce the loan balanceeach year by 1000 . Sinc e the loan is d ecl ined b y 1000 eachyear i t wi l l be paid in 5 yrs com pletely ?

Make the Amo rt izat ion Schedule ?

1s t

Year Interes t = 5000*.09 = 450

Total Payment = 1000+450=1450

2ndYear Interes t = 4000*.09=360

Total Payment 2ndYear = 1000+360=1360

Since the Princip al amount is d ecl ining the InterestCharges are decl ining each year.


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T6.14 Example: Amortization Schedule - Fixed Principal

Beginning Total Interest Principal EndingYear Balance Payment Paid Paid Balance

1 $5,000 $1,450 $450 $1,000 $4,000

2 4,000 1,360 360 1,000 3,000

3 3,000 1,270 270 1,000 2,000

4 2,000 1,180 180 1,000 1,000

5 1,000 1,090 90 1,000 0

Totals $6,350 $1,350 $5,000


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T6.13 AMORTIZED LOAN FIXED PAYMENT

The Most comm on w ay of amort izating a loan is to havethe borr ower make a sing le , f ixed payment every period .

Suppose ou r 5 yrs , 9 % , 5000 loan was amo rt ized th is way

5000 = C*(1 -1/1.09^ 5) / .09

= c* (1 1-.6499) / .09

C= 5000 / 3.8897

= 1285.46

The Interest is = 450 ( from the previous sheet)

1285-450= 835.46 ( The Prin cip al amoun t )

450+835.46=1285.46


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T6.15 Example: Amortization Schedule - Fixed Payments

Beginning Total Interest Principal Ending

Year Balance Payment Paid Paid Balance

1 $5,000.00 $1,285.46 $ 450.00 $ 835.46 $4,164.54

2 4,164.54 1,285.46 374.81 910.65 3,253.88

3 3,253.88 1,285.46 292.85 992.61 2,261.274 2,261.27 1,285.46 203.51 1,081.95 1,179.32

5 1,179.32 1,285.46 106.14 1,179.32 0.00

Totals $6,427.30 $1,427.31 $5,000.00


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How to lie, cheat, and steal with interest rates:

RIPOV RETAILINGGoing out fo rbusiness sale!

$1,000 instant credit!

12% simple interest!

Three years to pay!

Low, low monthly payments!


Assume you buy $1,000 worth of furniture

from this store and agree to the above creditterms. What is the APR of this loan? TheEAR?


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T6.16 Solution to Chapter 6 Quick Quiz -- Part 5 of 5 (concluded)

Your payment is calculated as:

1. Borrow $1,000 today at 12% per year for three years, youwill owe $1,000 + $1000(.12)(3)= $1,360.

2. To make it easy on you, make 36 low, low payments of$1,360/36 = $37.78.

3. Is this a 12% loan?

$1,000 = $37.78 x (1 - 1/(1 + r)36)/r

r = 1.767% per month

APR = 12(1.767%) = 21.204%

EAR = 1.0176712 - 1 = 23.39% (!)

T6 17 S l ti t P bl 6 10


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T6.17 Solution to Problem 6.10

Seinfelds Life Insurance Co. is trying to sell you an

investment policy that will pay you and your heirs $1,000

per year forever. If the required return on this investment is

12 percent, how much will you pay for the policy?

The present value of a perpetuity equals C/r. So, the mos ta

rational buyer would pay for the promised cash flows is

C/r= $1,000/.12 = $8,333.33

Notice: $8,333.33 is the amount which, invested at 12%,

would throw off cash flows of $1,000 per year forever.

(That is, $8,333.33 .12 = $1,000.)

T6 18 S l ti t P bl 6 11


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In the previous problem, Seinfelds Life Insurance Co. is

trying to sell you an investment policy that will pay you

and your heirs $1,000 per year forever. Seinfeld told youthe policy costs $10,000. At what interest rate would this

be a fair deal?

Again, the present value of a perpetuity equals C/r. Now

solve the following equation:

$10,000 = C/r= $1,000/r

r= .10 = 10.00%

Notice: If your opportunity rate is less than 10.00%, this is

a good deal for you; but if you can earn more than 10.00%,you can do better by investing the $10,000 yourself!

T6 18 Solution to Problem 6 11


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Congratulations! Youve just won the $20 million first prize inthe Subscriptions R Us Sweepstakes. Unfortunately, thesweepstakes will actually give you the $20 million in $500,000annual installments over the next 40 years, beginning nextyear. If your appropriate discount rate is 12 percent per year,how much money did you really win?

How much money did you really win? translates to, What is

the value today of your winnings? So, this is a present valueproblem.

PV = $ 500,000 [1 - 1/(1.12)40]/.12

= $ 500,000 [1 - .0107468]/.12

= $ 500,000 8.243776= $4,121,888.34 (Not quite $20 million, eh?)

PER AND ANNUITY


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FINDING THE NUMBER OF PAYMENTS:

Suppose you PUT 1,000/= on your credit card . You can onlymake payment of 20 per month . Interest Rate is 1.5 % permonth . How long it would take to pay of 1000.?

Present Value = 1000/=

1000 = 20 * ( 1 Present Value Factor) / .015(1000 / 20 ) / 0.015 = 1- Present Value Factor / . 015Present Value Factor = 0.25 = 1 ( 1+ r ) ^ t

1.015^ t = 1/.25 = 4The question is How long does it take for your money to quadruple

at 1.5 % per month ? The answer is about ( Use F Calculator)1.015 ^ 93 = 3.99 = 4It will take you about 93 / 12 = 7.75 years at this rate.

M ZAHID KHAN

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T6 7 Chapter 6 Quick Quiz Part 2 of 5


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Q. Suppose you owe $2000 on a Visa card, and theinterest rate is 2% per month. If you make theminimum monthly payments of $50, how long will ittake you to pay off the debt? (Assume you quitcharging stuff immediately!)

Example 1: Findin g t

T6 7 Chapter 6 Quick Quiz -- Part 2 of 5


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Q. Suppose you owe $2000 on a Visa card, and theinterest rate is 2% per month. If you make theminimum monthly payments of $50, how long will ittake you to pay off the debt? (Assume you quitcharging stuff immediately!)

Example 1: Findin g t

A. A longtime:

$2000 = $50 {1 - 1/(1.02)t}/.02.80 = 1 - 1/1.02t

1.02t

= 5.0t = 81.3 months, or about 6.78 years!

discounted cash flow valuations -6 (2)

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