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1 Chapter 2 Time Value of Money

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2 Time Value Topics Future value Present value Rates of return Amortization

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3 Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF 2 0123 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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4 Time line for a $100 lump sum due at the end of Year 2. 100 012 Year I%

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5 Time line for an ordinary annuity of $100 for 3 years 100 0123 I%

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6 Time line for uneven CFs 100 50 75 0123 I% -50

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7 FV of an initial $100 after 3 years (i = 10%) FV = ? 0123 10% Finding FVs (moving to the right on a time line) is called compounding. 100

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8 After 3 years FV 3 = FV2(1+I)=PV(1 + I) 2 (1+I) = PV(1+I) 3 = $100(1.10) 3 = $133.10 In general, FV N = PV(1 + I) N.

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9 One Way to Find FVs Use a financial calculator.

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10 310-100 0 NI/YR PV PMTFV 133.10 Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set:P/YR = 1, END. INPUTS OUTPUT Heres the setup to find FV

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11 10% Whats the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and its the reverse of compounding. 100 0123 PV = ?

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12 3 10 0100 N I/YR PV PMTFV -75.13 Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years. INPUTS OUTPUT Financial Calculator Solution

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13 20% 2 012? FV= PV(1 + I) N Continued on next slide Finding the Time to Double

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14 20 -1 0 2 NI/YR PV PMTFV 3.8 INPUTS OUTPUT Financial Calculator Solution

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15 ?% 2 0123 Finding the Interest Rate

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16 3 -1 0 2 NI/YR PV PMTFV 25.99 INPUTS OUTPUT Financial Calculator

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17 Ordinary Annuity PMT 0123 I% PMT 0123 I% PMT Annuity Due Ordinary Annuity vs. Annuity Due

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18 Whats the FV of a 3-year ordinary annuity of $100 at 10%? 100 0123 10% 110 121 FV= 331

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19 3 10 0 -100 331.00 NI/YRPVPMTFV Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT Financial Calculator Solution

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20 Whats the PV of this ordinary annuity? 100 0123 10% 90.91 82.64 75.13 248.69 = PV

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21 Have payments but no lump sum FV, so enter 0 for future value. 3 10 100 0 NI/YRPVPMTFV -248.69 INPUTS OUTPUT Financial Calculator Solution

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22 Find the FV and PV if the annuity were an annuity due. 100 0123 10% 100

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23 310 100 0 -273.55 NI/YRPVPMTFV INPUTS OUTPUT PV of Annuity Due: Switch from End to Begin

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24 310 0 100 -364.1 NI/YRPVPMTFV INPUTS OUTPUT FV of Annuity Due: Switch from End to Begin

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25 What is the PV of this uneven cash flow stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

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26 Input in CFLO register: CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.)

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27 Nominal rate (I NOM ) 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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28 Periodic rate (I PER ) I PER = I NOM /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: I PER = 8%/4 = 2%. 8% daily (365): I PER = 8%/365 = 0.021918%.

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29 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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30 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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31 FV Formula with Different Compounding Periods FV = PV1.+ I M N NOM M N

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32 $100 at a 12% nominal rate with semiannual compounding for 5 years = $100(1.06) 10 = $179.08 FV = PV1+ I M N NOM M N FV = $1001+ 0.12 2 5S 2x5

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33 FV of $100 at a 12% nominal rate for 5 years with different compounding FV(Annual)= $100(1.12) 5 = $176.23. FV(Semiannual)= $100(1.06) 10 =$179.08. FV(Quarterly)= $100(1.03) 20 = $180.61. FV(Monthly)= $100(1.01) 60 = $181.67. FV(Daily)= $100(1+(0.12/365)) (5x365) = $182.19.

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34 Effective Annual Rate (EAR = EFF%) The EAR is the annual rate which causes PV to grow to the same FV as under multi-period compounding.

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35 Effective Annual Rate Example Example: Invest $1 for one year at 12%, semiannual: FV = PV(1 + I NOM /M) M FV = $1 (1.06)2 = 1.1236. 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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36 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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37 = - 1.0 ( 1 + ) 0.12 2 2 = (1.06) 2 - 1.0 = 0.1236 = 12.36%. EFF% for a nominal rate of 12%, compounded semiannually EFF% = - 1 ( 1 + ) I NOM M M

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38 EAR (or EFF%) for a Nominal Rate of of 12% EAR Annual = 12%. EAR Q =(1 + 0.12/4) 4 - 1= 12.55%. EAR M =(1 + 0.12/12) 12 - 1= 12.68%. EAR D(365) =(1 + 0.12/365) 365 - 1= 12.75%.

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39 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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40 When is each rate used? I NOM : Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

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41 I PER : Used in calculations, shown on time lines. If I NOM has annual compounding, then I PER = I NOM /1 = I NOM. When is each rate used? (Continued)

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42 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 dont match interest compounding periods.

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43 Fractional Time Periods On January 1 you deposit $100 in an account that pays a nominal interest rate of 11.33463%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.)

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44 I PER = 11.33463%/365 = 0.031054% per day. FV=? 012273 0.031054% -100 Convert interest to daily rate

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45 273-100 0 108.85 INPUTS OUTPUT N I/YRPVFV PMT I PER =i NOM /M =11.33463/365 =0.031054% per day. Calculator Solution

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46 Non-matching rates and periods Whats 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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47 Time line for non-matching rates and periods 01 100 23 5% 45 6 6-mos. periods 100

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48 Non-matching rates and periods Payments occur annually, but compounding occurs each 6 months. So we cant use normal annuity valuation techniques.

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49 1st Method: Compound Each CF 01 100 23 5% 456 100100.00 110.25 121.55 331.80 FVA 3 = $100(1.05) 4 + $100(1.05) 2 + $100 = $331.80.

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50 2nd Method: Treat as an annuity, use financial calculator Find the EAR for the quoted rate: EAR = ( 1 + ) - 1 = 10.25%. 0.10 2 2

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51 3 10.25 0 -100 INPUTS OUTPUT N I/YR PVFV PMT 331.80 Use EAR = 10.25% as the annual rate in calculator.

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52 Whats the PV of this stream? 0 100 1 5% 23 100 90.70 82.27 74.62 247.59

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