discounted cash flow valuations -6 (2).ppt

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



FUTURE VALUE WITH MULTIPLE CASH FLOW:

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

M ZAHID KHAN




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

208*1.08=224.64

M ZAHID KHAN




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

1 Compound the accumulated balance forward one year at a time

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

M ZAHID KHAN

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




If you deposit 100$ in one Year.200$ in two year ,and 300$ in three year . How much will you have in three years ? How much of this interest ? How much will you have in five year if you don’t have additional amounts? Assume 7 percent interest rate through out?

M ZAHID KHAN

T6.1 DCFM

Discounted Cash Flow Valuation:


If you deposit 100$ in one Year.200$ in two year ,and 300$ in three year . How much will you have in three years ? How much of this interest ? How much will you have in five year if you don’t have additional amounts? Assume 7 percent interest rate through 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

T6. DCFM



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

628.49Future 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



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 add them up.

M ZAHID KHAN


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




Suppose you need 1,000$ in one year and 2000$ more in two years . 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 at 9%?

M ZAHID KHAN

T6.1 DCFM – PV MULTIPLE CF



Suppose you need 1,000$ in one year and 2000$ more in two years . 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 at 9%?

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


ANNUITIES AND PERPETUTIES :

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

Present value of an annuity of “C” $ per period for “t” period when the 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.

M ZAHID K

HAN


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 of C per period for t periods at r percent per period:

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

III. PV of C per period for t periods at r percent 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 calculations in Corporate Finance. It will be worthwhile to keep them in mind!

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Examples: Annuity Present Value (continued

PRESENT VALUE FOE ANNUITY CASH FLOWS:Suppose an Asset that promised to pay 500 $ at the end of each of

the nest three years. The cash flow from this asset of a three years is Ordinary annuity . If would like to earn 10% on our money , how much we offer for this annuity?

Present Value Factor=

=1/1.1^3 = 1/1.331 =.75131

To calculate the Annuity Present Value Factor== (1 - Present Value Factor) / r= (1 - .75131) / .10= .248685 / .10 = 2.48685 The Present value of our Annuity is then :Annuity Present Value = 500 * 2.48685 = 1,243.43

M ZAHID KHAN

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T6.6 Examples: Annuity Present Value


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

Assume you need the first $20,000 in exactly one year. Suppose you can place your money in a savings account yielding 8% compounded annually. How much do you need to have in the account today?

(Note: Ignore taxes, and keep in mind that you don’t want any funds to be left in the account after the third withdrawal, nor do you want to run short of money.)


T6.6 Examples: Annuity Present Value (continued)

Annuity Present Value - Solution

Here we know the periodic cash flows are $20,000 each. 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

Here’s a shortcut method for solving the problem using the annuity present value factor:

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




Here we know the periodic cash flows are $20,000 each. 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




Let’s continue our tuition problem.

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


T6.6 Examples: Annuity Present Value (concluded)


Again we know the periodic cash flows are $20,000 each. 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

PER AND ANNUITY

FINDING THE PAYMENT:

Finding C;

Suppose you wish to start up a new Business , You need to borrow 100,000. You want to make 5 equal installment and the interest 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 ) / .18 100,000 = C * ( 1 - .4371 ) / .18 100,000 = C * 3.1272

C = 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: Finding 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 monthly payment

be?

A. You will borrow .90 $25,000 = $22,500 . This is the amount today, so it’s 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

PER AND ANNUITY

FINDING THE RATE :

Suppose that Insurance Co offer to pay you 1,000/= per year for 10 yrs if you pay 6,710 up front . What rate is implicit in this for 10 yrs.

Present Value = 6,710/=Cash Flows = 1,000/= per years 6,710 = 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 a

factor of 6.7101 for 8 percent , so we are right away that insurance co is just offering 8 %

Or Just Use TRIAL AND ERROR M ZAHID KHAN

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PER AND ANNUITY

FUTURE VALUE OF ANNUITIES:There are Future Value Factor for annuities as well as present

factor.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 retirement account paying 8 % . If you retire in 30 yrs , how much will you have ?

Annuity FV Factor = ( Future Value Factor – 1) / r= (1.08 ^ 30 – 1) / .08= (10.0627 – 1) / .08= 113.2832Thus the FV of this 30 yrs , 2000 annuity is :Annuity Future Value = 2,000 * 113.2832= 226,566.40

M ZAHID KHAN


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

Example 2: Finding C

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

Now, rather than plunking down $15,091 in one chunk, suppose she would rather invest smaller amounts annually to accumulate the 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 don’t start saving for retirement that early in life. (Many don’t start at all!)

Suppose Bill just turned 40 and has decided it’s time to get serious 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 and make the last one at age 65, how much must each deposit be?

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

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

By waiting, Bill has to set aside over six times as much money each year!



Consider Bill’s retirement plans one more time.

Again assume he just turned 40, but, recognizing that he has a lot of time to make up for, he decides to invest in some high-risk ventures that may yield 20% annually. (Or he may lose his money completely!) Anyway, assuming that Bill still wishes to accumulate $1 million by age 65, and will begin making equal annual deposits in one year and make the last one at age 65, now how much must each deposit be?

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

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

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


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 of C per period for t periods at r percent per period:

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

III. PV of C per period for t periods at r percent 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 level stream 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 , we obviously can’t compute its value by discounting each one. Fortunately , valuing a perpetuity turn out be the easiest possible case.

The PV of Perpetuity is simply = C/r

Example :


T6.11 Example: Perpetuity Calculations

Suppose we expect to receive $1000 at the end of each of the next 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 year forever. This is called a perpetuity. 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!



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). Here’s a question for you: How can an infinite number of cash payments have a finite value?

Here’s an example related to the question above. Suppose you are considering the purchase of a perpetual bond. The issuer of the bond promises to pay the holder $100 per year forever. If your opportunity rate is 10%, what is the most you would pay for the bond today?

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


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

An infinite number of cash payments has a finite present value 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 the otherwise identical 50-year bond has a PV of $991.48, the value today of payments 51 through infinity must be

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


T6.13 Compounding Periods, EARs, and APRs

Compounding Number of times Effective

periodcompounded annual rate

Year 1 10.00000%

Quarter 4 10.38129

Month 12 10.47131

Week 5210.50648

Day 36510.51558

Hour 8,76010.51703

Minute 525,600 10.51709


T6.13 Compounding Periods, EARs, and APRs

Compounding Number of times Effective

periodcompounded 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


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 six months the same as 16% per year?

A. If you invest $1000 for one year at 16%, then you’ll have $1160 at the end of the year. If you invest at 8% per period for two periods, you’ll have

FV = $1000 (1.08)2

= $1000 1.1664

= $1166.40,

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


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

The Effective 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 quoted on a loan agreement is equal to the rate per period multiplied by the number of periods. This rate is called the _________________ (____).

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

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

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


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

The Effective 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 quoted on a loan agreement is equal to the rate per period multiplied by the number of periods. This rate is called the Annual Percentage Rate (APR).

Q. A bank charges 1% per month on car loans. What is the APR? 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 thus a quoted rate, not an effective rate!


T6.13 AMORTIZED LOAN

The process of paying off a loan by making regular principal reductions is called amortizing loan.

Suppose you take out a loan of 5000/= 5 yrs loan at 9% . The loan agreement call for a borrower to pay interest on the loan balance each year and to reduce the loan balance each year by 1000 . Since the loan is declined by 1000 each year it will be paid in 5 yrs completely ?

Make the Amortization Schedule ?


T6.13 AMORTIZED LOAN

Suppose you take out a loan of 5000/= 5 yrs loan at 9% . The loan agreement call for a borrower to pay interest on the loan balance each year and to reduce the loan balance each year by 1000 . Since the loan is declined by 1000 each year it will be paid in 5 yrs completely ?

Make the Amortization Schedule ?

1st Year Interest = 5000*.09 = 450

Total Payment = 1000+450=1450

2nd Year Interest = 4000*.09=360

Total Payment 2nd Year = 1000+360=1360

Since the Principal amount is declining the Interest Charges are declining each year.


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


T6.13 AMORTIZED LOAN – FIXED PAYMENT

The Most common way of amortizating a loan is to have the borrower make a single , fixed payment every period.

Suppose our 5 yrs , 9 % , 5000 loan was amortized this 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 Principal amount )

450+835.46=1285.46


T6.15 Example: Amortization Schedule - Fixed Payments

Beginning Total Interest Principal EndingYear 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 2,261.27

4 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


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

RIPOV RETAILINGGoing out for business 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 credit terms. What is the APR of this loan? The EAR?


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, you will 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 Solution to Problem 6.10

Seinfeld’s 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 most a 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.)



In the previous problem, Seinfeld’s 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 you the 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!



Congratulations! You’ve just won the $20 million first prize in the Subscriptions R Us Sweepstakes. Unfortunately, the sweepstakes will actually give you the $20 million in $500,000 annual installments over the next 40 years, beginning next year. 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 value problem.

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

FINDING THE NUMBER OF PAYMENTS:

Suppose you PUT 1,000/= on your credit card . You can only make payment of 20 per month . Interest Rate is 1.5 % per month . 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 / . 015 Present Value Factor = 0.25 = 1 ( 1+ r ) ^ t1.015^ t = 1/.25 = 4 The 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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Q. Suppose you owe $2000 on a Visa card, and the interest rate is 2% per month. If you make the minimum monthly payments of $50, how long will it take you to pay off the debt? (Assume you quit charging stuff immediately!)

Example 1: Finding t



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

Example 1: Finding t

A. A long time:

$2000 = $50 {1 - 1/(1.02)t}/.02 .80 = 1 - 1/1.02t

1.02t = 5.0 t = 81.3 months, or about 6.78 years!

discounted cash flow valuations -6 (2).ppt

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