copyright ©2003 south-western/thomson learning chapter 4 the time value of money
TRANSCRIPT
![Page 1: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/1.jpg)
Copyright ©2003 South-Western/Thomson Learning
Chapter 4The Time Value Of Money
![Page 2: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/2.jpg)
Introduction
• This chapter introduces the concepts and skills necessary to understand the time value of money and its applications.
![Page 3: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/3.jpg)
Simple and Compound Interest
• Simple Interest – Interest paid on the principal sum only
• Compound Interest – Interest paid on the principal and on prior
interest that has not been paid or withdrawn
![Page 4: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/4.jpg)
t to denote time
PV0 = principal amount at time 0
FVn = future value n time periods from time 0
PMT to denote cash payment
PV to denote the present value dollar amount
T to denote the tax rate
I to denote simple interest
i to denote the interest rate per period
n to denote the number of periods
Notation
![Page 5: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/5.jpg)
Future Value of a Cash Flow
• At the end of year n for a sum compounded at interest rate i is FVn = PV0 (1 + i)n Formula
• In Table I in the text, (FVIFi,n) shows the future value of $1 invested for n years at interest rate i: FVIFi,n = (1 + i)n Table I
• When using the table, FVn = PV0 (FVIFi,n)
![Page 6: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/6.jpg)
Tables Have Three Variables
• Interest factors (IF)
• Time periods (n)
• Interest rates per period (i)
• If you know any two, you can solve algebraically for the third variable.
![Page 7: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/7.jpg)
Present Value of a Cash Flow
• PV0 = FVn [ ] Formula
• PVIFi, n = Table II
• PV0 = FVn(PVIFi, n) Table II
1 (1 + i)n
1 (1 + i)n
![Page 8: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/8.jpg)
Example Using Formula
• What is the PV of $100 one year from now with 12 percent interest compounded monthly?
PV0 = $100 1/(1 + .12/12)(12 1)
= $100 1/(1.126825)
= $100 (.88744923)
= $ 88.74
![Page 9: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/9.jpg)
Example Using Table II
PV0 = FVn(PVIFi, n)
= $100(.887) From Table II
= $ 88.70
![Page 10: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/10.jpg)
Annuity
• A series of equal dollar CFs for a specified number of periods
• Ordinary annuity is where the CFs occur at the end of each period.
• Annuity due is where the CFs occur at the beginning of each period.
![Page 11: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/11.jpg)
FVIFAi, n = Formula for IF
FVANn = PMT(FVIFAi, n) Table III
Future Value of an Ordinary Annuity
(1 + i)n – 1i
![Page 12: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/12.jpg)
Derivation of the FVAN formula(1)
nn-1 n-2 n-n
FVAN =PMT 1+i +PMT 1+i + +PMT 1+i
The FVAN formula is a geometric series because each term on the right side is equal to the previous term multiplied by a common factor: 1/(1+i).
Multiply both sides of the equation above by the common factor to create a second equation.
![Page 13: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/13.jpg)
Derivation of the FVAN formula(2)
1 n-1 1 n-2 1 n-n 1FVAN =PMT 1+i +PMT 1+i + +PMT 1+in1+i 1+i 1+i 1+i
1 n-2 n-3 -1FVAN =PMT 1+i +PMT 1+i + +PMT 1+in1+i
Subtract this new equation from the original equation on the previous slide. The result:
n-1 -1n n
1FVAN - FVAN =PMT 1+i -PMT 1+i
1+i
Solve for FVAN.
n
n
1+i -1FVAN =PMT
i
![Page 14: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/14.jpg)
Present Value of an Ordinary Annuity
PVIFAi, n = Formula
PVAN0 = PMT( PVIFAi, n) Table IV
1 (1 + i)n
1 –
i
![Page 15: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/15.jpg)
Annuity Due
• Future Value of an Annuity Due– FVANDn = PMT(FVIFAi, n)(1 + i) Table III
• Present Value of an Annuity Due – PVAND0 = PMT(PVIFAi, n)(1 + i) Table IV
![Page 16: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/16.jpg)
Other Important Formulas
• Sinking Fund– PMT = FVANn/(FVIFAi, n) Table III
• Payments on a Loan– PMT = PVAN0/(PVIFAi, n) Table IV
• Present Value of a Perpetuity– PVPER0 = PMT/i
![Page 17: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/17.jpg)
Interest Compounded More Frequently Than Once Per Year
Future Valuenm
nom0n
m
i1PVFV )( +=
Present Value
)nm
minom(1 +
FVnPV0 =
m = # of times interest is compoundedn = # of years
![Page 18: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/18.jpg)
Interest Compounded More Frequently than Once Per Year
• Texas Instruments BA II Plus Calculator – set the number of compounding periods to 12 per year:
• 2nd, P/Y, , 12, ENTER, CE/C, CE/C
• When finished: 2nd, CLR TVM
• And, reset compounding to once per year: 2nd, P/Y, , 1, ENTER, CE/C, CE/C
![Page 19: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/19.jpg)
Effective Annual Rates
A nominal rate of interest (or Annualized Percentage Rate) is found by multiplying the rate charged or paid per period by the number of periods during the year.
#periodsRatei =APR=nom yearperiod
This rate does not include the effect of compounding of interest at the end of each period of the year.
![Page 20: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/20.jpg)
Effective Annual Rates
For comparison purpose, we need an effective annual rate that includes the effect of compounding.
1
m
inom1+i 1+ meff
Solve for the rate that gives the same effect with once per year compounding as the APR gives with more frequent compounding than annual.
minomi =1+ -1meff
![Page 21: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/21.jpg)
Effective Annual Rates
If compounding is done continuously,
minomi = lim 1+ -1 mmeff
inom i = e - 1eff
![Page 22: Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649ca55503460f94966818/html5/thumbnails/22.jpg)
Compounding and Effective Rates
• Rate of interest per compounding period
im = (1 + ieff)1/m – 1
• Effective annual rate of interest ieff = (1 + inom/m)m – 1