Chapter+5+V2+(Cont)+Discounted+Cash+Flow[1]

Chapter 5 (cont.) Discounted Cash Flow Valuation 1. Review Problems 2. Mult Multip iple le Cash ash Fl Flows ows: Com Compu putting ing FV FV, PV, IR, n 3. Valu Valuiing Annu Annuit itie ies s & Perp Perpe etuit tuitie ies s 4. Interes rest Rates: APR and EAR

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First: A Review Problem #1 The company President of Acme Computers estimates he needs to increase sales by 200% before he is profitable. Sales are currently $150,000. His goal is to be profitable 6 years from now. What is the lowest growth rate in sales that is needed to accomplish his goal and be profitable? (Note 200% increase.) Use the 4-button method and check with y x. 2

First: A Review Problem #1 The company President of Acme Computers estimates he needs to increase sales by 200% before he is profitable. Sales are currently $150,000. His goal is to be profitable 6 years from now. What is the lowest growth rate in sales that is needed to accomplish his goal and be profitable? (Note 200% increase.) Use the 4-button method and check with y x. 2

Review Problem #2 • The production manager estimates he will

be producing 500,000 units in year 2020. He bases this on the fact that he estimates his production will increase 7% per year. How many units did he produce in 2002?

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Future Value with Multiple Cash Flows: There are two ways to calculate future values of  multiple cash flows: 1. Compound the accumulated balance forward one period at a time, or 2. Calculate the future value of each cash flow and add them up. **

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Present Value of Multiple Cash Flows: There are two ways to calculate present values of multiple cash flows: 1. Discount the last amount back one period and add them up as you go, or 2. discount each amount to time 0 and then add them all up.**

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Another Way – Annuity Due • I suggest you do not mess with your calculator.

When I ask you to calculate an annuity due: - decrease N by 1 and calculate PV - add 1 payment to your calculated PV An annuity due calls for the first payment up front. So if you buy an annuity that costs $1000 and pays $100 per period you get $100 back right away. We will do a couple examples of this later. 12

Present Value for Annuity Cash Flows Ordinary Annuity - multiple, identical cash flows occurring at the end of each period for a fixed number of periods. Annuity Present Value: APV = C  (1 - [1/(1 + r)t ])/r (Note: this formula is FYI, as you will normally use the calculator for determining PV and FV of annuities).

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Example: If you are willing to make 36 monthly payments of  $100 at 1.5% per period, what size loan (APV) can you obtain? APV = C * [(1 - [1/(1 + r)t ])/r]. C = $100 ; t = 36; r = 1.5% APV = 100 * [(1-[1/(1+.015)36])/.015] = 100*[(1[1/1.70914])/.015] = 100 * (.4149/.015) = 100 * 27.66 = $2766 Using this formula gets a bit hairy and it’s easy to make a mistake. It is much easier to just use the calculator: Using your calculator: N=36; -$100 = PMT; 1.5 = Interest rate; Compute PV  $2766.07 (after you adjust for sign convention)

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Sweepstakes as Annuity Due • $333,333.33 per year for 30 years • First payment is up front. • $10,000,000 in total payments • 5% interest rate • Now what is it worth? Reduce N by 1 and

calculate the PV then add one payment to your PV. 16

Sweepstakes as Annuity Due Answer • N = 30 -1 = 29 because first payment is up • •

• •

front; Interest Rate = 5% ; PMT = -333,333.33 Solve for PV  5,047,024.48 after adjust for sign convention. Now add the first payment of 333,333.33  5,380,357.81 Versus $5,124,150 for Regular Annuity Annuity Due is Always worth more, because each payment is earlier. 17

Annuities – the Calculator Method • 5 possible buttons: you will use only 4 for

any problem and have 3 variables given. • You use either the PV or FV for a problem but not both, so for any problem you will know or solve for N, Interest Rate, PMT (payment), and FV or PV.

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Example: If you borrow $400, promising to repay in 4 monthly installments at 1% a month, how much are your payments? APV = C * [(1 - [1/(1 + r)t ])/r]. $400 = C  (1 - [1/(1.01)4])/.01 $400 = C  3.90196, so C = $400/3.90196 C = $102.51.

Or the easy way on the calculator: PMT = ? ; Interest rate = 1%; PV = 400; t or n = 4; Compute PMT $102.51 19

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Example of calculating n for an annuity: How many $100 payments will pay off a $5,000 loan at 1% per period? Answer 1. $5000 Loan; $100 payments ; 1% per month 2. PV = $5000 ; Interest Rate = 1%; PMT = -$100 3. Compute N  69.66

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Example: A finance company offers to loan you $1,000 today if you will make 48 monthly payments of $32.60. What rate is implicit in the loan? Answer: 1. $1000 today ; 48 payments of $32.60. 2. PV = 1000; N = 48; PMT = -32.60 3. Compute interest rate  2.000%

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Future Values for Annuities: 1. Method 1: Discount the payments, then find the future value: Annuity future value AFV = APV  (1 + r)t. 2. Method 2: AFV = C  ((1 + r)t - 1)/r I would recommend the calculator = Method 3 3. Method 3: Use the calculator: You know n, IR, payment; solve for FV.

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Summary of Annuities • Fixed Payment for a set time period, not forever

like a perpetuity. • You know 3 of 4 things and have to figure out the fourth. You also have to decide if you are dealing with PV or FV. • Use your calculator to solve. No good method to check other that testing by plugging in your answer and 2 of the other variables to confirm that you get the other one. 28

Perpetuities = Series of level cash flows forever.

Perpetuity present value: PPV= C/r, since PPV  r must give payment, C.

Preferred stock is an important example of a perpetuity.

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Sample Perpetuity Problem #1 Preferred stock pays a perpetual dividend. If the current market rate is 8% for an investment in the risk category of ABC Company’s preferred stock,

the price of its common stock is $1000, and the annual dividend on the preferred stock is $100 with the next dividend payable a year from now, what is the appropriate market price for ABC’s preferred

stock?

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Answer #1 • Price of common stock has no impact on

preferred stock; I often will give you extraneous info, which you should ignore. • PPV = C/r • PPV = 100/.08 =1250

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Sample Perpetuity Problem #2 • Preferred stock pays a perpetual dividend.

If the current market price of the preferred stock is $95, the price of its common stock is $75, and the annual dividend (coupon) on the preferred stock is $6.75 with the next dividend payable a year from now, what is the interest rate for ABC’s

preferred stock? 33

Answer #2 • Price of common stock has no impact on

preferred stock; I often will give you extraneous info, which you should ignore. • PPV = C/r • 95 = 6.75/r  r = 6.75/95 = .071 = 7.1%

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Annuity Due: The difference between Annuity Due cash flows and

Ordinary Annuity cash flows is that the Annuity Due cash flows are always made or received at the beginning of each period. 3 ways to solve for the PV or FV of an annuity due. 1. The first method is to use a special annuity key on a financial calculator, and then provide the same inputs as for an ordinary annuity. This is the method I do not recommend as it involves changing calculator settings. 2. The second method called multiplying by the base. To multiply by the base, first solve the entire problem as if it were an ordinary annuity, and then multiply the solution by (1+r). 3. The third method** recognizes that an Annuity Due is somewhat of a silly concept in that you pay PV for the annuity and then immediately get the first payment back; so its value = the value of Ordinary Annuity (in arrears) for n-1 payments + the value of the payment up front. In other words, the value of 10 payments of $50 Annuity Due = the value of 9 payments of $50 Ordinary Annuity + $50 35

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Example: You are leasing an SUV for 36 months at $400 per month, payable on the first day of each month. If the appropriate interest rate is 0.5% per month, what is the present value of this lease to the finance company? Let’s try this 2 ways. First, do it by solving for a regular annuity and then multiplying by (1+r). Second, do it by reducing N by 1, solving for PV and then adding 1 payment to the Calculated PV. 37

Multiplication by the base, first set up a financial calculator solution for an ordinary annuity solution: 36 N, -400 PMT, .5 I/Y. Then compute PV $13,148.41. Multiply this by 1.005 (which is 1+r) 

$13,214.15

Reduce N by 1: 35 = N; -400 = PMT; .5 = interest rate; Compute PV  $12,814.15; add $400  $13,214.15 THIS IS THE WAY I LIKE. 38

Stated or quoted interest rate - rate before considering any compounding effects, such as 10% compounded quarterly . This is the APR (Annual  Percentage Rate). It is also called the nominal  interest rate. Effective annual interest rate (EAR) - rate, on an annual basis, that reflects compounding effects, e.g., 10% compounded quarterly is effective rate of 10.38% = (1.025)4 - 1.

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T5.13 Compounding Periods, EARs, and APRs

Compounding period

Number of times

Effective

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

 Irwin/McGraw-Hill 

 © The

McGraw-Hill Companies, Inc. 1999

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Calculating and Comparing Effective Annual Rates (EAR) To get the effective rate, divide the quoted annual rate by number of periods in a year (semi-annual = 2, quarterly = 4, monthly = 12, etc.), add 1, raise to the number of periods power, then subtract 1. That is,

EAR = [1 + (quoted rate or APR)/m]m – 1 where m = number of periods per year This formula is in the Table. (This is one reason that it was helpful to learn the y x key.) Example: 18% compounded monthly = 1 + (.18/12)]12  – 1 = 19.56% 42

Example: What is the present value of $100 in two years at 10% compounded quarterly? What is the EAR? We need to figure out the rate at each period being compounded. 10% compounded quarterly = 10%/4 each quarter = 2.5% each quarter. There are 8 quarters in 2 years, so we are compounding 8 times. PV = FV/(1+r)t PV = 100/(1+.025)8 PV = $82.07. EAR is an annual rate. EAR = (1.025)4 -1 = .1038 or 10.38% Remember EAR is an annual rate. 43

Another Example • 15% interest rate compounded monthly for

3 years. If we start with $200, what do we end with? What is the EAR? For problems: • Use the EAR rate & the # of years; or • Use the APR/m & the appropriate # of periods (m * # of years); • Use the yx key. 44

Another Example • 15% APR interest rate compounded monthly

• • • •

for 3 years. If we start with $200, what do we end with? What is the EAR? We are compounding monthly, so the monthly rate is 15%/12 = 1.25%. FV = PV (1+r)t FV = $200 (1.0125)36 = 200(1.5639) = $312.79 EAR = 1.012512 -1 = 16.1%

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Annual Percentage Rate (APR) = simply the rate per period  number of periods per year, making it a quoted  or stated  rate. APR is the rate quoted in most leases, mortgages, etc. Why is that? (EAR is normally not quoted on these.) However, it understates the effective rate if there is more than 1 period per year. Conversely, most investments quote both EAR and APR. Why? 46

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EAR vs. APR Formula • EAR = (1+APR/m)m – 1; m = # of times

compounded in a year. • Essentially APR is a quoted annual rate. To convert it to EAR, you have to know the type of compounding. A 10% APR can be compounded daily, monthly, quarterly, semi-annually, etc.

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EAR vs. APR • APR is a legal term used in contracts. You

normally have to convert it to the rate and time period being compounded. Comparing APRs is inexact. Example: 15% APR compounded daily = 16.18% [calculated as (1+.15/365) 365] so it is a higher rate than 16% compounded annually. • EAR is the “real” financial rate. Comparing

EARs works well.

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Loan Types and Loan Amortization Pure Discount Loans: Borrower pays a single lump sum

(principal and interest) at maturity. These are types of  problems we solved for already (4 button method and checked with yx method.) Example: A U.S. Treasury bill Interest-Only Loans: Borrower pays interest only each period

and entire principal at maturity. Example: A typical corporate bond Amortized Loans: Principal and interest are paid together in

payments. Example: A typical/conventional mortgage on a house

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Amortized Loans Borrower repays part or all of the principal over the life of the loan. Two methods are: 1) fixed amount of principal to be repaid each period, which results in uneven payments, and 2) fixed payment (i.e., an annuity), which results in uneven principal reduction. (Interest decreases and principal increases as the loan “amortizes” or is paid down with each payment.) A traditional automobile loan or fixed-rate home mortgage. These normally have a fixed payment per month (with increasing principal and decreasing interest but the same total payment.) Solving for these is solving for an annuity. 56

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Problem: 30-year mortgage • $100,000 loan • Fixed rate of APR 6%, compounded

monthly. • 30 year mortgage • What is the EAR? What is the monthly payment?

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More Loan Problems Problem #1: You are considering buying a car. The car’s total cost is $17,500. The salesperson quotes a monthly interest rate of 1% based on 36 monthly payments with the first payment due in 30 days. Excluding taxes and any other fees, what should your monthly payment be? Problem #2: You are evaluating a car loan. The car’s cost is $20,500. On a 60-month loan, the quoted payment is $396.32, what is the monthly interest rate? What is the APR? What is the EAR?

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Chapter 5 – Topics to Remember • Present Value of Multiple Cash Flows (take PV of each one and add up the PVs) • Annuity vs. Perpetuity • Annuities – how to find the 4th variable if know 3 • Annuity Due (in advance) vs. Regular Annuity  – how to find the PV of Annuity Due • What makes an PV or FV of an annuity worth more or less? Higher or lower N, interest rate, regular vs. due?

• Perpetuity – 3 variables,

find the answer if you know 2; PPV = C/r • EAR vs. APR • Be able to compute EARs and APRs • Types of Loans - Lump Sum - Interest Only -Amortized Loans – use annuity method for fixed payments

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Answer to Review Problem #1 • N=6; PV=150,000; • FV= PV(1+200%) =150000*(3.00) =

450000 • Compute Interest Rate = 20.09% • Check using yx method: (1.2009)6 = 3 and 3*150,000 = 450,000

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Answer to Review Problem #2 • N= 2020 – 2002 = 18; PV= ?; Interest rate

= 7%; FV = 500000 • Compute PV = 147932 • Check using yx method: (1.07)18 = 3.380 and 3.380*147932 = 500,000

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Solution to 30-year Mortgage • $100,000 loan for 30 years, 6% APR,

compounded monthly, what is the monthly payment? • EAR = (1+.06/12)12 = 1.00512 = 6.17% • PV = -100,000 ; N = 360 ; 0.5% = interest rate ; Compute PMT  $599.55

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