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Chapter 4 Time Value of Money

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**Time Value Topics Future value Present value Rates of return**

Amortization

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**Determinants of Intrinsic Value: The Present Value Equation**

Net operating profit after taxes Required investments in operating capital − Free cash flow (FCF) = FCF1 FCF2 FCF∞ ... Value = (1 + WACC)1 (1 + WACC)2 (1 + WACC)∞ Weighted average cost of capital (WACC) For value box in Ch 4 time value FM13. Cost of debt Cost of equity

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**Why is timing important?**

You are asked to choose from the following options: 1. Receive $1 million today 2. Receive $1 million 10 years from now Would you choose 1 or 2?

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Money has time value Most people prefer to receive it sooner rather than later because they place a higher value on the cash received earlier.

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**Time value of money: Practical relevance**

Examples Retirement plan Mortgage payment Pricing a financial securities Helping your company to decide which project to undertake

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**Time lines show timing of cash flows.**

CF0 CF1 CF3 CF2 1 2 3 I% Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

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**Time line for a $100 lump sum due at the end of Year 2.**

1 2 Year I%

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**Time line for an ordinary annuity of $100 for 3 years**

1 2 3 I%

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**Time line for uneven CFs**

100 50 75 1 2 3 I% -50

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**Preparing BAII Plus for use**

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 then press ‘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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**FV of an initial $100 after 3 years (I = 10%)**

1 2 3 10% Finding FVs (moving to the right on a time line) is called compounding. 100

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**After 1 year FV1 = PV + INT1 = PV + PV (I) = PV(1 + I) = $100(1.10)**

= $110.00

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**After 2 years FV2 = FV1(1+I) = PV(1 + I)(1+I) = PV(1+I)2 = $100(1.10)2**

= $121.00

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**After 3 years FV3 = FV2(1+I)=PV(1 + I)2(1+I) = PV(1+I)3 = $100(1.10)3**

= $133.10 In general, FVN = PV(1 + I)N

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Four Ways to Find FVs Step-by-step approach using time line (as shown in Slides 12-15). Solve the equation with a regular calculator (formula approach). Use a financial calculator. Use a spreadsheet.

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**Financial Calculator Solution**

Financial calculators solve this equation: FVN = PV (1+I)N There are 5 variables. If 4 are known, the calculator will solve for the 4th.

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**Here’s the setup to find FV**

N I/YR PV PMT FV 133.10 INPUTS OUTPUT

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**Spreadsheet Solution Use the FV function = FV(I, N, PMT, PV)**

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**What’s the PV of $100 due in 3 years if I/YR = 10%?**

Finding PVs is discounting, and it’s the reverse of compounding. 100 1 2 3 PV = ?

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**Solve FVN = PV(1 + I )N for PV**

3 1 PV = $100 1.10 = $100(0.7513) = $75.13

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**Financial Calculator Solution**

N I/YR PV PMT FV -75.13 INPUTS OUTPUT Either PV or FV must be negative. Here PV = Put in $75.13 today, take out $100 after 3 years.

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**Spreadsheet Solution Use the PV function: = PV(I, N, PMT, FV)**

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**Finding the Time to Double**

20% 2 1 ? -1 Q: if deposit $1 today, and i=20%, when will it double?

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**Time to Double (Continued)**

N LN(1.2) = LN(2) N = LN(2)/LN(1.2) N = 0.693/0.182 = 3.8

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**Financial Calculator Solution**

N I/YR PV PMT FV 3.8 INPUTS OUTPUT

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**Spreadsheet Solution Use the NPER function: = NPER(I, PMT, PV, FV)**

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**Finding the interest rate**

?% 2 1 3 -1 FV = PV(1 + I)N $2 = $1(1 + I)3 (2)(1/3) = (1 + I) = (1 + I) I = = 25.99%

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Financial Calculator N I/YR PV PMT FV 25.99 INPUTS OUTPUT

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**Spreadsheet Solution Use the RATE function: = RATE(N, PMT, PV, FV)**

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Exercises Suppose 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? You 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? You 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?

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Perpetuity 1 Perpetuity: a stream of equal cash flows ( C ) that occur at the end of each period and go on forever. PV of perpetuity =

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**Perpetuity 2 We use the idea of a perpetuity to determine the value of**

A preferred stock A perpetual debt

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Perpetuity questions Suppose the value of a perpetuity is $38,900 and the discount rate is 12 percent p.a.. What must be the annual cash flow from this perpetuity? Verify that C = $4,668. An asset that generates $890 per year forever is priced at $6,000. What is the required rate of return? Verify that r = %

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Ordinary Annuity Ordinary annuity: a cash flow stream where a fixed amount is received at the end of every period for a fixed number of periods.

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**What’s the FV of a 3-year ordinary annuity of $100 at 10%?**

1 2 3 10% 110 121 FV = 331

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**Financial Calculator Solution**

331.00 N I/YR PMT FV PV INPUTS OUTPUT Have payments but no lump sum PV, so enter 0 for present value.

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**Spreadsheet Solution Use the FV function: = FV(I, N, PMT, PV)**

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**What’s the PV of this ordinary annuity?**

100 1 2 3 10% 90.91 82.64 75.13 = PV

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**Financial Calculator Solution**

INPUTS N I/YR PV PMT FV OUTPUT Have payments but no lump sum FV, so enter 0 for future value.

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**Spreadsheet Solution Use the PV function: = PV(I, N, PMT, FV)**

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Annuity, find FV You open an account today with $20,000 and at the end of each of the next 15 years, you deposit $2,500 in it. At the end of 15 years, what will be the balance in the account if the interest rate is 7 percent p.a.? PV=-20000, PMT=-2500, N=15, I/Y=7, FV=?

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Annuity, find I/Y You lend your friend $100,000. He will pay you $12,000 per year for the ten years and a balloon payment at t = 10 of $50,000. What is the interest rate that you are charging your friend? PV=-100,000, FV=50,000, PMT=12,000, N = 10, I/Y=?

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Annuity, find PMT Next year, you will start to make 35 deposits of $3,000 per year in your Individual Retirement Account (so you will contribute from t=1 to t=35). With the money accumulated at t=35, you will then buy a retirement annuity of 20 years with equal yearly payments from a life insurance company (payments from t=36 to t=55). If the annual rate of return over the entire period is 8%, what will be the annual payment of the annuity?

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Annuity Due Annuity due: a cash flow stream where a fixed amount is received at the beginning of every period for a fixed number of periods.

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**Ordinary Annuity vs. Annuity Due**

PMT 1 2 3 I% Annuity Due

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**Ordinary Annuity vs. Annuity Due**

$300 $300 $300 T = 1 T = 3 T = 0 T = 2 $300 $300 $300 T = 3 T = 0 T = 1 T = 2

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**a relationship between ordinary annuity and annuity due?**

PV of annuity due = (PV of ordinary annuity) x (1 + r) FV of annuity due = (FV of ordinary annuity) x (1 + r)

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**Find the FV and PV if the annuity were an annuity due.**

100 1 2 3 10%

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**PV and FV of Annuity Due vs. Ordinary Annuity**

PV of annuity due: = (PV of ordinary annuity) (1+I) = ($248.69) ( ) = $273.56 FV of annuity due: = (FV of ordinary annuity) (1+I) = ($331.00) ( ) = $364.10

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**PV of Annuity Due: Switch from “End” to “Begin”**

N I/YR PV PMT FV INPUTS OUTPUT BEGIN Mode

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**FV of Annuity Due: Switch from “End” to “Begin”**

N I/YR PV PMT FV INPUTS OUTPUT BEGIN Mode

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**Excel Function for Annuities Due**

Change the formula to: =PV(0.10,3,-100,0,1) The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due: =FV(0.10,3,-100,0,1)

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**What is the PV of this uneven cash flow stream?**

100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 = PV

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**Solving with Calculator**

Input cash flows in the calculator’s “CF” register: Press CF key CF0 = 0, ENTER, C01 = 100, ENTER, F01=1, ENTER, C02 = 300, ENTER, F02=2, ENTER, C03= 50, +/- key, ENTER, F03=1, ENTER, Press NPV key, I = 10, ENTER, press CPT key to get NPV = (Here NPV = PV.)

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**Excel Formula in cell A3: =NPV(10%,B2:E2)**

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**In-class group project**

You will need to pay for your son’s private school tuition (first grade through 12th grade) a sum of $8,000 per year for Years 1 through 5, $10,000 per year for Years 6 through 8, and $12,500 per year for Years 9 through 12. Assume that all payments are made at the beginning of the year, that is, tuition for Year 1 is paid now (i.e., at t = 0), tuition for Year 2 is paid one year from now, and so on. In addition to the tuition payments you expect to incur graduation expenses of $2,500 at the end of Year 12. If a bank account can provide a certain 10 percent p.a. rate of return, how much money do you need to deposit today to be able to pay for the above expenses?

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Nominal rate (INOM) Stated in contracts, and quoted by banks and brokers. Not used in calculations or shown on time lines Periods per year (M) must be given. Examples: 8%; Quarterly 8%, Daily interest (365 days)

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Periodic rate (IPER ) IPER = INOM/M, where M is number of compounding periods per year. M = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Used in calculations, shown on time lines. Examples: 8% quarterly: IPER = 8%/4 = 2%. 8% daily (365): IPER = 8%/365 = %.

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**The Impact of Compounding**

Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why?

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**The Impact of Compounding (Answer)**

LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

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**Six-months / semiannual**

Common examples Compounding period Compounding frequency Six-months / semiannual 2 Quarter 4 Month 12 Day 365

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**When frequency of compounding is more than once a year**

‘n’ = number of years ‘m’ = frequency of compounding per year ‘r’ = nominal rate

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**Effective Annual Rate (EAR = EFF%)**

The EAR is the annual rate that causes PV to grow to the same FV as under multi-period compounding.

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**Effective Annual Rate Example**

Example: Invest $1 for one year at 12%, semiannual: FV = PV(1 + INOM/M)M FV = $1 (1.06)2 = $ EFF% = 12.36%, because $1 invested for one year at 12% semiannual compounding would grow to the same value as $1 invested for one year at 12.36% annual compounding.

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**$100 at a 12% nominal rate with semiannual compounding for 5 years**

INOM FVN = PV M M N 0.12 FV5S = $ 2 2x5 = $100(1.06)10 = $179.08

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**FV of $100 at a 12% nominal rate for 5 years with different compounding**

FV(Ann.) = $100(1.12)5 = $176.23 FV(Semi.) = $100(1.06)10 = $179.08 FV(Quar.) = $100(1.03)20 = $180.61 FV(Daily) = $100(1+(0.12/365))(5x365) = $182.19 How to solve with financial calculator?

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Comparing Rates An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons. Banks say “interest paid daily.” Same as compounded daily.

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**EFF% for a nominal rate of 12%, compounded semiannually**

INOM M = − 1 0.12 2 = (1.06) = = 12.36%.

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**EAR (or EFF%) for a Nominal Rate of of 12%**

EARAnnual = 12%. EARQ = ( /4)4 - 1 = %. EARM = ( /12) = %. EARD(365) = ( /365) = %.

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**Can the effective rate ever be equal to the nominal rate?**

Yes, but only if annual compounding is used, i.e., if M = 1. If M > 1, EFF% will always be greater than the nominal rate.

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**When is each rate used? INOM:**

Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

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**When is each rate used? (Continued)**

IPER: Used in calculations, shown on time lines. If INOM has annual compounding, then IPER = INOM/1 = INOM.

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**When is each rate used? (Continued)**

EAR (or EFF%): Used to compare returns on investments with different payments per year. Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.

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**Annuity with semiannual compounding**

You would like to accumulate $16,500 over the next 8 years. How much must you deposit every six months, starting six months from now, given a 4 percent per annum rate with semiannual compounding?

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Loan Amortization Amortization is the process of separating a payment into interest payment and repayment of principal. Amortization schedule is a table that shows how each payment is split into principal repayment and interest payment.

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Amortization Example 1 Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

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**Step 1: Find the required payments.**

PMT 1 2 3 10% -1,000 INPUTS OUTPUT N I/YR PV FV 402.11

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**Step 2: Find interest charge for Year 1.**

INTt = Beg balt (I) INT1 = $1,000(0.10) = $100

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**Step 3: Find repayment of principal in Year 1.**

Repmt = PMT - INT = $ $100 = $302.11

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**Step 4: Find ending balance after Year 1.**

End bal = Beg bal - Repmt = $1,000 - $ = $697.89 Repeat these steps for Years 2 and 3 to complete the amortization table.

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**Amortization Table YEAR BEG BAL PMT INT PRIN PMT END BAL 1 $1,000 $402**

$100 $302 $698 2 698 402 70 332 366 3 37 TOT 1,206.34 206.34 1,000

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**Interest declines because outstanding balance declines.**

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Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, and more. They are very important!

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**Example of loan amortization 1**

You have borrowed $8,000 from a bank and have promised to repay the loan in five equal yearly 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.

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**Example of loan amortization 2**

1) Compute periodic payment. PV=8000, N=5, I/Y=10, FV=0, PMT=? Verify that PMT = -2,110.38

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**Example of loan amortization 3**

Suppose we want to work out the remaining balance immediately after the 2nd payment. Press [2ND], [AMORT] to activate the Amortization worksheet in BA II Plus. Press P1=2, [ENTER], , Press P2=2, [ENTER], , You will see BAL=5,248.20 P1 = starting point in a range of payments, the first payment of interest P2 = ending point in a range of payments, the last payment of interest

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**Example of loan amortization 4**

Press again and you see the portion of the year 2 payment going towards repaying principal, PRN = -1,441.42 Press again and you see the portion of year 2 payment going towards interest, INT = To get out of the Amortization schedule, press [2ND], Quit.

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**Verify the amortization schedule**

Year Beg. Balance Payment Interest Principal End. 8,000.00 1 2,110.38 800.00 1,310.38 6,689.62 2 668.96 1,441.42 5,248.20 3 524.82 1,585.56 3,662.64 4 366.26 1,744.12 1,918.53 5 191.85 0.00

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**Non-matching rates and periods**

What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually?

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**Time line for non-matching rates and periods**

1 100 2 3 5% 4 5 6 6-mos. periods

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**Non-matching rates and periods**

Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

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**1st Method: Compound Each CF**

1 100 2 3 5% 4 5 6 100.00 110.25 121.55 331.80 FVA3 = $100(1.05)4 + $100(1.05)2 + $100 = $331.80

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**2nd Method: Treat as an annuity, use financial calculator**

Find the EFF% (EAR) for the quoted rate: EFF% = − 1 = 10.25% 0.10 2

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**Use EAR = 10.25% as the annual rate in calculator.**

INPUTS N I/YR PV PMT FV OUTPUT 331.80

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**What’s the PV of this stream?**

100 1 5% 2 3 90.70 82.27 74.62 247.59

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**What’s the PV of this stream? EAR = 10.25%**

INPUTS N I/YR PV PMT FV OUTPUT 247.59

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**After Chapter Homework**

Problems: 1- 25,

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CHAPTER 6 Time Value of Money

CHAPTER 6 Time Value of Money

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