7 - 1 Copyright © 2002 by Harcourt, Inc.All rights reserved. Future value Present value Rates of return Amortization CHAPTER 7 Time Value of Money

7 - 2 Copyright © 2002 by Harcourt, Inc.All rights reserved. Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF 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.

7 - 3 Copyright © 2002 by Harcourt, Inc.All rights reserved. Time line for a $100 lump sum due at the end of Year Year i%

7 - 4 Copyright © 2002 by Harcourt, Inc.All rights reserved. Time line for an ordinary annuity of $100 for 3 years i%

7 - 5 Copyright © 2002 by Harcourt, Inc.All rights reserved. Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through i% -50

7 - 6 Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? % 100 Finding FVs is compounding.

7 - 7 Copyright © 2002 by Harcourt, Inc.All rights reserved. FV 1 = PV + INT 1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $ After 2 years: FV 2 = PV(1 + i) 2 = $100(1.10) 2 = $ After 1 year:

7 - 8 Copyright © 2002 by Harcourt, Inc.All rights reserved. After 3 years: FV 3 = PV(1 + i) 3 = $100(1.10) 3 = $ In general, FV n = PV(1 + i) n.

7 - 9 Copyright © 2002 by Harcourt, Inc.All rights reserved. Four Ways to Find FVs Solve the equation with a regular calculator. Use tables. Use a financial calculator. Use a spreadsheet.

Copyright © 2002 by Harcourt, Inc.All rights reserved. FV n = PV(1 + i) n. Algebraic Solution FV 3 = 100(1 +.10) 3 FV 3 = 100(1.331) =

Copyright © 2002 by Harcourt, Inc.All rights reserved. FV n = PV(FVIF i,n ). Solution Using Tables FV 3 = 100(FVIF 10%, 3 ) Use FVIF table from pages A-6 & 7, Table A3 FV 3 = 100(1.331) =

Copyright © 2002 by Harcourt, Inc.All rights reserved. Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4th. FV n = PV(1 + i) n. Financial Calculator Solution

Copyright © 2002 by Harcourt, Inc.All rights reserved. Here’s the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set:P/YR = 1, END INPUTS OUTPUT NI/YR PV PMT FV

Copyright © 2002 by Harcourt, Inc.All rights reserved. 10% What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding PV = ?

Copyright © 2002 by Harcourt, Inc.All rights reserved. Solve FV n = PV(1 + i ) n for PV:  PV = = FV n. FV n (1 + i) n  i PV= $100 = $100(PVIF i,n ) Table A1 = $100(0.7513) = $ So the (PVIF 10%,3 ) =.7513  n

Copyright © 2002 by Harcourt, Inc.All rights reserved. Financial Calculator Solution N I/YR PV PMTFV Either PV or FV must be negative. Here PV = Put in $75.13 today, take out $100 after 3 years. INPUTS OUTPUT

Copyright © 2002 by Harcourt, Inc.All rights reserved. If sales grow at 20% per year, how long before sales double? Solve for n: FV n = $1(1 + i) n ; $2 = $1(1.20) n ln 2 = ln 1.2n =.18232n = n Use calculator to solve, see next slide.

Copyright © 2002 by Harcourt, Inc.All rights reserved. If sales grow at 20% per year, how long before sales double? Solve for n: FV n = PV(FVIF i,n ); $2 = $1(FVIF 20,n ) 2.00 = (FVIF 20,n ) n between 3 and 4 years and Use calculator to solve, see next slide.

Copyright © 2002 by Harcourt, Inc.All rights reserved N I/YR PV PMTFV 3.8 Graphical Illustration: FV 3.8 Year INPUTS OUTPUT

Copyright © 2002 by Harcourt, Inc.All rights reserved. Compound Growth How do you find the compound growth rate for your company to analyze sales growth ? (1 + i ) 9 = (1 + i ) 9 = / (1 + i) 9 = (1 + i ) = (1 + i) = i =.195 or 19.5% Can use either PV or FV formula, use FV

Copyright © 2002 by Harcourt, Inc.All rights reserved. Tabular Solution (PVIF i,9 ) = PVIF i,9 = / PVIF i,9 = Use table A-3, for 9 Periods, find i is between 18% and 20% Use PV formula and table A-1

Copyright © 2002 by Harcourt, Inc.All rights reserved. Calculator Solution, Compound Growth INPUTS OUTPUT NI/YRPVPMTFV 19.51

Copyright © 2002 by Harcourt, Inc.All rights reserved. Ordinary Annuity PMT 0123 i% PMT 0123 i% PMT Annuity Due What’s the difference between an ordinary annuity and an annuity due?

Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the FV of a 3-year ordinary annuity of $100 at 10%? % FV= 331

Copyright © 2002 by Harcourt, Inc.All rights reserved. FVA =( PMT)* ( 1 +i) n – 1 I FVA =( 100)* ( 1 +.1) 3 – 1.1 FVA =( 100)* – 1 = 100 * 3.31 =.1 FVA = Algebraic Solution

Copyright © 2002 by Harcourt, Inc.All rights reserved. FVA i,n =( PMT) * (FVIFA i,n ) Use Table A-4 on pages A-8 & 9 FVA 10%,3 =( 100) * (FVIFA 10%,3 ) FVA 10%,3 =( 100)* 3.31 = FVA 10%,3 = Tabular Solution

Copyright © 2002 by Harcourt, Inc.All rights reserved Financial Calculator Solution Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT I/YRNPMTFVPV

Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the PV of this ordinary annuity? % = PV

Copyright © 2002 by Harcourt, Inc.All rights reserved. 1. PVA =( PMT)* 1 – (1 + i) n i 1. PVA =( 100) * 1 – (1 +.1) 3.1 PVA =( 100)* = 100 * =.1 PVA = Algebraic Solution

Copyright © 2002 by Harcourt, Inc.All rights reserved. PVA i,n =( PMT) * (PVIFA i,n ) Use Table A-2 on pages A-4 & 5 PVA 10%,3 =( 100) * (PVIFA 10%,3 ) PVA 10%,3 =( 100)* = PVA 10%,3 = Tabular Solution

Copyright © 2002 by Harcourt, Inc.All rights reserved. Have payments but no lump sum FV, so enter 0 for future value INPUTS OUTPUT NI/YRPVPMTFV

Copyright © 2002 by Harcourt, Inc.All rights reserved. Find the FV and PV if the annuity were an annuity due % 100 Easiest way, multiply results by (1 + i).

Copyright © 2002 by Harcourt, Inc.All rights reserved. 1. PVAD i,n =( PMT)* 1 – (1 + i) n * (1 + i) i 1. PVAD 10%,3 =( 100) * 1 – (1 +.1) 3 * (1 +.1).1 PVAD 10%,3 =( 100)* * (1.1) = * * (1.1)= PVAD 10%,3 = Algebraic Solution

Copyright © 2002 by Harcourt, Inc.All rights reserved. PVAD i,n =( PMT) * (PVIFA i,n )* (1 + i) Use Table A-2 on pages A-4 & 5 PVAD 10%,3 =( 100) * (PVIFA 10%,3 ) * (1 + i) PVAD 10%,3 =( 100)* * 1.1 = PVAD 10%,3 = Tabular Solution

Copyright © 2002 by Harcourt, Inc.All rights reserved Switch from “End” to “Begin.” Then enter variables to find PVA 3 = $ Then enter PV = 0 and press FV to find FV = $ INPUTS OUTPUT NI/YRPVPMTFV

Copyright © 2002 by Harcourt, Inc.All rights reserved. What is the PV of this uneven cash flow stream? % = PV

Copyright © 2002 by Harcourt, Inc.All rights reserved. Input in “CFLO” register: CF 0 = 0 CF 1 = 100 CF 2 = 300 CF 3 = 300 CF 4 = -50 Enter I = 10, then press NPV button to get NPV = $ (Here NPV = PV.)

Copyright © 2002 by Harcourt, Inc.All rights reserved. A 20-year old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095) in an online stock account. The stock account has an expected annual return of 12%. The Power of Compound Interest

Copyright © 2002 by Harcourt, Inc.All rights reserved. How much money by the age of 65? ,487, INPUTS OUTPUT NI/YRPVPMTFV If she begins saving today, and sticks to her plan, she will have $1,487, by the age of 65.

Copyright © 2002 by Harcourt, Inc.All rights reserved. How much would a 40-year old investor accumulate by this method? , INPUTS OUTPUT NI/YRPVPMTFV Waiting until 40, the investor will only have $146,000.59, which is over $1.3 million less than if saving began at 20. So it pays to get started early.

Copyright © 2002 by Harcourt, Inc.All rights reserved. How much would the 40-year old investor need to save to accumulate as much as the 20-year old? , INPUTS OUTPUT NI/YRPVPMTFV The 40-year old investor would have to save $11, every year, or $30.56 per day to have as much as the investor beginning at the age of 20.

Copyright © 2002 by Harcourt, Inc.All rights reserved. Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

Copyright © 2002 by Harcourt, Inc.All rights reserved. Rules for Non-annual Compounding 95% of the time, the method for adjusting for non-annual compounding is: Divide i by m, m being the # of compounding periods in a year. Multiply n by m

Copyright © 2002 by Harcourt, Inc.All rights reserved. FV of $100 after 3 years under 10% semiannual compounding? Quarterly? = $100(1.05) 6 = $ FV 3Q = $100(1.025) 12 = $ FV = PV1.+ i m n Nom m*n       FV = $ S 2*3      

Copyright © 2002 by Harcourt, Inc.All rights reserved % % Annually: FV 3 = $100(1.10) 3 = $ Semiannually: FV 6 = $100(1.05) 6 = $

Copyright © 2002 by Harcourt, Inc.All rights reserved. Exam Question (Example) Your uncle has given you a choice between receiving $20,000 today on your 18 th birthday, or waiting until your 25 th birthday and receiving $40,000. If you would invest in a junk bond fund if you took the $20,000, expecting to average 10% per year, compounded semiannually, which would you prefer?

Copyright © 2002 by Harcourt, Inc.All rights reserved. Exam Question (Example) See board for timeline. Algebraic solution: FV = PV(1 + i/m) n*m FV = 20,000 (1 +.1/2) 7 * 2 FV = 20,000 ( ) = 39, Prefer the $40,000 in 7 years.

Copyright © 2002 by Harcourt, Inc.All rights reserved. Exam Question (Example) Tabular solution: PV = FV(PVIF i/2,n*2 ) PV = 40,000 (PVIF 10/2,7*2 ) Table A-1 PV = 40,000 (.505) = 20,202 Prefer the $40,000 in 7 years (same conclusion.

Copyright © 2002 by Harcourt, Inc.All rights reserved. Financial Calculator Solution ? 0 40, , INPUTS OUTPUT NI/YRPVPMTFV Could have solved for FV inputting PV calcuation: P/Y set to 2

Copyright © 2002 by Harcourt, Inc.All rights reserved. We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year. i Per = periodic rate. EAR= EFF% =. effective annual rate

Copyright © 2002 by Harcourt, Inc.All rights reserved. i Nom is stated in contracts. Periods per year (m) must also be given. Examples: l 8%; Quarterly l 8%, Daily interest (365 days)

Copyright © 2002 by Harcourt, Inc.All rights reserved. Periodic rate = 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. Examples: 8% quarterly: i Per = 8%/4 = 2%. 8% daily (365): i Per = 8%/365 = %.

Copyright © 2002 by Harcourt, Inc.All rights reserved. Effective Annual Rate (EAR = EFF%): The annual rate that causes PV to grow to the same FV as under multi-period compounding. Example: EFF% for 10%, semiannual: FV = (1 + i Nom /m) m = (1.05) 2 = EFF% = 10.25% because (1.1025) 1 = Any PV would grow to same FV at 10.25% annually or 10% semiannually.

Copyright © 2002 by Harcourt, Inc.All rights reserved. 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.

Copyright © 2002 by Harcourt, Inc.All rights reserved. How do we find EFF% for a nominal rate of 10%, compounded semiannually? EFF= – 1 Or use a financial calculator. = – 1.0 = (1.05) 2 – 1.0 = = 10.25%.  1 +  i Nom m  1 +  m

Copyright © 2002 by Harcourt, Inc.All rights reserved. EAR = EFF% of 10% EAR Annual = 10%. EAR Q =( /4) 4 – 1= 10.38%. EAR M =( /12) 12 – 1= 10.47%. EAR D(365) =( /365) 365 – 1= 10.52%.

Copyright © 2002 by Harcourt, Inc.All rights reserved. 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.

Copyright © 2002 by Harcourt, Inc.All rights reserved. When is each rate used? i Nom :Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

Copyright © 2002 by Harcourt, Inc.All rights reserved. i Per :Used in calculations, shown on time lines. If i Nom has annual compounding, then i Per = i Nom /1 = i Nom.

Copyright © 2002 by Harcourt, Inc.All rights reserved. (Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.) EAR = EFF%: Used to compare returns on investments with different payments per year.

Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? % 45 6-mos. periods 100 6

Copyright © 2002 by Harcourt, Inc.All rights reserved. Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

Copyright © 2002 by Harcourt, Inc.All rights reserved. 1st Method: Compound Each CF % FVA 3 = $100(1.05) 4 + $100(1.05) 2 + $100 = $

Copyright © 2002 by Harcourt, Inc.All rights reserved. Could you find FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: 2nd Method: Treat as an Annuity EAR = ( 1 + ) – 1 = 10.25%

Copyright © 2002 by Harcourt, Inc.All rights reserved. Or, to find EAR with a calculator: NOM% = 10. P/YR = 2. EFF% =

Copyright © 2002 by Harcourt, Inc.All rights reserved. EFF% = P/YR = 1 NOM% = INPUTS OUTPUT NI/YRPVFVPMT b. The cash flow stream is an annual annuity. Find k Nom (annual) whose EFF% = 10.25%. In calculator, c.

Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the PV of this stream? %