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**Discounted Cash Flow Valuation**

Chapter Six Discounted Cash Flow Valuation

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Chapter Outline Future and Present Values of Multiple Cash Flows Valuing Level Cash Flows: Annuities and Perpetuities Comparing Rates: The Effect of Compounding Loan Types and Loan Amortization

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**Multiple Cash Flows 6.1 – FV Example 1**

You currently have $7,000 in a bank account earning 8% interest. You think you will be able to deposit an additional $4,000 at the end of each of the next three years. How much will you have in three years? 7,000 4,000 8,817.98 4,665.60 4,320.00 $21,803.58 1 2 3 4,000.00

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**Multiple Cash Flows – FV Example 2**

Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years? 1 2 9% 500 600 654.00 594.05 Calculator: Year 0 CF: 2 N; -500 PV; 9 I/Y; CPT FV = Year 1 CF: 1 N; -600 PV; 9 I/Y; CPT FV = Total FV = = $1,248.05

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**Multiple Cash Flows – FV Example 2 Continued**

Using the cash flows from the previous page, how much will you have at the end of 5 years, if you were to make no further deposits? 1 2 3 4 5 9% 500 600 846.95 Calculator: First way: Year 0 CF: 5 N; -500 PV; 9 I/Y; CPT FV = Year 1 CF: 4 N; -600 PV; 9 I/Y; CPT FV = Total FV = = Second way – use value at year 2: 3 N; PV; 9 I/Y; CPT FV = 769.31 $ 1,616.26

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**Multiple Cash Flows – FV Example 3**

Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%? 1 2 3 4 5 8% 100 300 349.92 136.05 FV = 100(1.08) (1.08)2 = = Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV = Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV = Total FV = = $

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**Multiple Cash Flows – PV Example 1**

You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the year after, and $800 at the end of the following year. You can earn 12% on similar investments. How much is this investment worth today? 1 2 3 4 12% 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93

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**Uneven Cash Flows – Using the Calculator**

Another way to use the financial calculator for uneven cash flows is to use the cash flow keys HP 10BII Enter the first cash flow, using the +/- keys to indicate outflows Press CF Enter the second cash flow Enter the interest rate as I/Y Use the 2nd function, then the NPV key to compute the present value To clear this function, use 2nd function, Clear All The next example will be worked using the cash flow keys. Note that with the BA-II Plus, the students can double check the numbers they have entered by pressing the up and down arrows. It is similar to entering the cash flows into spreadsheet cells. Other calculators also have cash flow keys. You enter the information by putting in the cash flow and then pressing CF. You have to always start with the year 0 cash flow, even if it is zero. Remind the students that the cash flows have to occur at even intervals, so if you skip a year, you still have to enter a 0 cash flow for that year.

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Decisions, Decisions Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment? Use the CF keys to compute the value of the investment CF 40 CF 75 CF 15 I/Y 2nd NPV -8.51 Note: NPV (Net Present Value) is the addition to, or the destruction of, wealth that occurs from undertaking an investment. If NPV is positive, you create wealth. If NPV is negative, you destroy wealth. Therefore, the decision rule is to accept only projects with an NPV > 0 You can also use this as an introduction to NPV by having the students put –100 in for CF0. When they compute the NPV, they will get – You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV. Reject: NPV < 0

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**Annuities and Perpetuities 6.2**

Annuity – finite series of equal payments that occur at regular intervals If the first payment occurs at the end of the period, it is called an ordinary annuity If the first payment occurs at the beginning of the period, it is called an annuity due Perpetuity – infinite series of equal payments

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**Annuities – Basic Formulas**

Where: PV = Present Value FV = Future Value C = the equal periodic cash flow R = interest or discount rate t = the number of time periods

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**Ordinary Annuities Versus Annuities Due**

Ordinary Annuity – first payment occurs at time period 1 Annuity Due – first payment occurs at time period 0

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**Annuities and the Calculator**

To solve any annuity type problem using the function keys, you must use the PMT key Only use the PMT key when each cash flow is exactly the same size and occurs at regular intervals Ordinary annuity versus annuity due To switch the calculator between an ordinary annuity and an annuity due, enter 2nd BGN (HP 10BII) If you see “BGN” or “Begin” in the display of your calculator, you have it set for an annuity due Most problems are ordinary annuities Other calculators also have a key that allows you to switch between Beg/End.

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Annuity – Example 1 After carefully going over your budget, you have determined that you can afford to pay $632 per month towards a new sports car. Your bank will lend to you at 1% per month for 48 months. How much can you borrow? Formula Approach Calculator Approach 632 PMT 0 FV 48 N 1 I/Y PV $23,999.54

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**Annuity – Sweepstakes Example**

Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333, over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? Formula Approach Calculator Approach 333, PMT 0 FV 30 N 5 I PV $5,124,150.29 Calculator: 30 N; 5 I/Y; 333, PMT; CPT PV = 5,124,150.29

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Finding the Payment Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8%/12 = % per month). If you take a 4 year loan, what is your monthly payment? Formula Approach Calculator Approach 20,000 PV 0 FV 4 x 12 = N 8 ÷ 12 = I PMT $488.26 Note if you do not round the monthly rate and actually use 8/12, then the payment will be Calculator: 4(12) = 48 N; 20,000 PV; I/Y; CPT PMT =

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**Finding the Number of Payments – Example 1**

You ran a little short on your February vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5% per month. How long will you need to pay off the $1,000?

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**Finding the Number of Payments – Example 1**

Formula Approach Start with the basic annuity formula and solve for the unknown exponent t, using logs. Calculator Approach 1,000 PV 0 FV PMT 1.5 I N months You ran a little short on your spring break vacation, so you put $1000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000. This is an excellent opportunity to talk about credit card debt and the problems that can develop if it is not handled properly. Many students don’t understand how it works and it is never discussed. This is something that students can take away from the class, even if they aren’t finance majors. Calculator: 1.5 I/Y 1000 PV -20 PMT CPT N = MONTHS = 7.75 years

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**Finding the Number of Payments – Example 2**

Suppose you borrow $2000 at 5% and you are going to make annual payments of $ How long before you pay off the loan? Calculator Approach 2,000 PV 0 FV PMT 5 I N years Sign convention matters!!! 5 I/Y 2000 PV PMT CPT N = 3 years

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**Finding the Rate On the Financial Calculator**

Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $ per month for 60 months. What is the monthly interest rate? Calculator Approach 10,000 PV 0 FV PMT 60 N I % per month The next slide talks about how to do this without a financial calculator.

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**Annuity – Finding the Rate Without a Financial Calculator**

Trial and Error Process Choose an interest rate and compute the PV of the payments based on this rate Compare the computed PV with the actual loan amount If the computed PV > loan amount, then the interest rate is too low (e.g. r = 2% computed PV= $7,215.48) If the computed PV < loan amount, then the interest rate is too high (e.g. r = 0.5% computed PV= $10,738.11) Adjust the rate and repeat the process until the computed PV and the loan amount are equal

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**Future Values for Annuities – Example 1**

Suppose you begin saving for your retirement by depositing $2000 per year in an RRSP. If the interest rate is 7.5%, how much will you have in 40 years? Formula Approach Calculator Approach 2,000 PMT 0 PV 40 N 7.5 I FV $454,513.04 FV = 2000( – 1)/.075 = 454,513.04 Remember the sign convention!!! 40 N 7.5 I/Y -2000 PMT CPT FV = 454,513.04

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Annuity Due – Example 1 You are saving for a new house and you put $10,000 per year in an account paying 8% compounded annually. The first payment is made today. How much will you have at the end of 3 years? Formula Approach Calculator Approach 2nd BGN 10,000 PMT 0 PV 3 N 8 I FV $35,061.12 Note that the procedure for changing the calculator to an annuity due is similar on other calculators. Calculator 2nd BGN 2nd Set (you should see BGN in the display) 3 N -10,000 PMT 8 I/Y CPT FV = 35,061.12 2nd BGN 2nd Set (be sure to change it back to an ordinary annuity) What if it were an ordinary annuity? FV = 32,464 (so receive an additional by starting to save today.)

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**Perpetuity Perpetuity: A stream of cash flows that continues forever**

Where: PV = Present Value C = the equal periodic cash flow, starting at time period 1 r = discount rate

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Perpetuity – Example 1 The Home Bank of Canada want to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend would the Home Bank have to offer if its preferred stock is going to sell?

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**Perpetuity – Example 1 continued**

First, find the required return for the comparable issue: Then, using the required return found above, find the dividend for new preferred issue: This is a good preview to the valuation issues discussed in future chapters. The price of an investment is just the present value of expected future cash flows. Example statement: Suppose the Fellini Co. wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to offer if the preferred stock is going to sell.

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Growing Perpetuity The perpetuities discussed so far are annuities with constant payments Growing perpetuities have cash flows that grow at a constant rate and continue forever Growing perpetuity formula:

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**Growing Perpetuity – Example 1**

Hoffstein Corporation is expected to pay a dividend of $3 per share next year. Investors anticipate that the annual dividend will rise by 6% per year forever. The required rate of return is 11%. What is the price of the stock today?

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Growing Annuity Growing annuities have a finite number of cash flows, which grow at a constant rate The formula for a growing annuity is:

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**Growing Annuity – Example 1**

Gilles Lebouder has just been offered a job paying $50,000 at the end of his first year. He anticipates his salary will then increase by 5% a year until his retirement in 40 years. Given an interest rate of 8%, what is the present value of his lifetime salary?

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**Effective Annual Rate (EAR) 6.3**

The Effective Annual Interest Rate is the actual rate paid (or received) after accounting for compounding that occurs during the year For example, 10% compounded twice a year has an EAR of 10.25%. This means that 10%, compounded twice a year, is identical to 10.25%, compounded annually. If you want to compare two alternative investments with different compounding periods, you need to compute the EAR for both investments and then compare the EAR’s. Where m is the number of compounding periods per year Using the calculator: The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates. 2nd I Conv NOM is the quoted rate down arrow EFF is the effective rate down arrow C/Y is compounding periods per year. You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.

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**Annual Percentage Rate (Nominal Rate)**

This is the annual rate that is quoted by law By definition, the nominal rate or APR = period rate times the number of periods per year Consequently, to get the period rate we rearrange the APR equation: Period rate = APR / number of periods per year You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate

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**Computing APRs (Nominal Interest Rate)**

What is the APR if the monthly rate is .5%? .5(12) = 6% What is the APR if the semiannual rate is .5%? .5(2) = 1% What is the monthly rate if the APR is 12% with monthly compounding? 12 / 12 = 1%

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Things to Remember You ALWAYS need to make sure that the compounding period and the payment period match. If payments are annual, then you use annual compounding. If payments are monthly, then you use monthly compounding

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**Calculating EARs – Example 1**

To calculate the EAR, use the following formula: For example, the EAR for 10% is: Compounding Periods EAR % % % % % Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.

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**Decisions, Decisions II**

You are comparing two savings accounts. One pays 5.25%, compounded daily. The other pays 5.3%, compounded semi-annually. Which account would you use? First account: EAR = ( /365)365 – 1 = 5.39% Second account: EAR = ( /2)2 – 1 = 5.37% You should choose the first account (the account that compounds daily), because you are earning a higher effective interest rate. Remind students that rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates. Calculator: 2nd I conv 5.25 NOM up arrow 365 C/Y up arrow CPT EFF = 5.39% 5.3 NOM up arrow 2 C/Y up arrow CPT EFF = 5.37%

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**Decisions, Decisions II Continued**

Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year? First Account: Daily rate = / 365 = FV = 100( )365 = , OR, 365 N; 5.25 / 365 = I/Y; 100 PV; FV = Second Account: Semiannual rate = .053 / 2 = .0265 FV = 100(1.0265)2 = , OR, 2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV = You have more money in the first account. It is important to point out that the daily rate is NOT .014, it is First Account: 365 N; 5.25 / 365 = I/Y; 100 PV; CPT FV = Second Account: 2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV =

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**Computing APRs from EARs**

If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:

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APR – Example 1 Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? On the calculator: 2nd I conv down arrow 12 EFF down arrow 12 C/Y down arrow CPT NOM

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**Computing Payments with APRs – Example 1**

Suppose you want to buy a new computer system and the store is willing to allow you to make monthly payments. The entire computer system costs $3500. The loan period is for 2 years and the interest rate is 16.9% with monthly compounding. What is your monthly payment? Formula Approach Calculator Approach 3,500 PV 0 FV 2 x 12 = N 16.9 ÷ 12 = I PMT $172.88 2(12) = 24 N; 16.9 / 12 = I/Y; 3500 PV; CPT PMT =

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**Future Values with Monthly Compounding**

Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years? Formula Approach Calculator Approach 50 PMT 0 PV 35 x 12 = N 9 ÷ 12 = I FV $147,089.22 35(12) = 420 N 9 / 12 = .75 I/Y 50 PMT CPT FV = 147,089.22

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Mortgages In Canada, financial institutions are required by law to quote mortgage rates with semi-annual compounding Since most people pay their mortgage either monthly (12 payments per year), semi-monthly (24 payments), bi-weekly (26 payments), or weekly (52 payments), you need to remember to convert the interest rate before calculating the mortgage payment! Remember, the payment period and the compounding period must always be the same.

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The Three Step Process Whenever you are faced with a situation when the payment period and the compounding period are not equal, you can apply the Three Step Process to solve the problem. Step #1: Calculate the Effective Annual Interest Rate, using the number of compounding periods per year Step #2: Calculate a new notional nominal rate (APR), using the number of payment periods per year Step #3: Calculate the per period interest rate by dividing the answer from Step #2 by the number of payments per year

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Mortgages – Example 1 Theodore D. Kat is applying to his friendly, neighbourhood bank for a mortgage of $200,000. The bank is quoting 6%. He would like to have a 25-year amortization period and wants to make payments monthly. How much are Theodore’s monthly payments?

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**Mortgage – Example 1 Step #1: Calculate EAR**

Step #2: Calculate the new notional nominal rate

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Mortgage – Example 1 Step #3: Calculate the monthly interest rate by dividing the answer from Step #2 by the number of payments per year

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Mortgage – Example 1 Complete the problem by solving for the monthly payment, using the annuity formula Formula Approach Calculator Approach 200,000 PV 0 FV 25 x 12 = N I PMT $1,279.61

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**Continuous Compounding**

Sometimes investments or loans are calculated based on continuous compounding EAR = er – 1 e is the base of the natural logarithms, equal to (although it is actually a non-terminating, non-repeating decimal) e is shown on most calculators ex Example: What is the effective annual rate of 7% compounded continuously? EAR = e.07 – 1 = or 7.25%

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