Time Value of Money
Chapter 4
TVM on YouTube
Barton College
2
Don’t forget to look at the Notes Pages
(See Below how to do this..)
Barton College
3
Chapter Topics
• TVM Introduction
• Single Period Investments
• Multiple Period Investments
• Compound Interest
• Future Value (FV) and Compounding
• Present Value (PV) & Discounting
• Determining the Discount Rate given the PV or FV
• Determining the Number of Periods given the PV or FV
• A few Problems for practice…..
Barton College
4
Introduction to the Time
Value of Money (TVM)
• Someone wins the PowerBall Lottery and the payout is $110M. Does
the recipient really receive this amount or something less?
• One person invests $1M in savings bonds and another person invest
$1M in a franchise restaurant chain. Are these investments similar,
different, or contain different amounts of risk…..?
– Understanding how to interpret the TVM will help the investor/manager choose a
decision that best fits the particular purpose.
– It also helps the investor understand why a dollar today is more than a dollar
tomorrow (if held in one’s pocket and not invested)
Barton College
5
Basic Definitions
• Future Value – later money on a time line
• Present Value – earlier money on a time line
• Interest rate – “exchange rate” between earlier money
and later money
– Discount rate
– Cost of capital (wacc)
– Opportunity cost of capital
– Required return
– Historical return
– Expected return
Barton College
6
Future Value
Future Value is the amount an investment is worth after one or more
periods.
Board Example
• Single Period and Simple Interest
– Suppose you invest $1000 for one year at 5% per year. What is the
future value in one year?
•
•
•
Interest = 1000(.05) = $50
Value in one year = principal + interest = 1000 + 50 = $1050
Future Value (FV) = 1000(1 + .05) = $1050
• Multiple Periods: Suppose you leave the money in for another year.
How much will you have two years from now?
– FV = 1000(1.05)(1.05) = 1000(1.05)2 = $1102.50
Barton College
7
Compounding Effects
Simple Interest vs. Compound Interest
• Consider the earlier example
– FV with simple interest = 1000 + 50 + 50 = $1100
– FV with compound interest = 1102.50
•
FV = 1000(1.05)(1.05) = 1000(1.05)2 = $1102.50
– The extra $2.50 comes from the interest of .05(50) = 2.50
earned on the first interest payment
Barton College
8
Future Values: General Formula
• FV = PV(1 + r)t
– FV = future value
– PV = present value
– r = period interest rate, expressed as a decimal
– t = number of periods
• Future value interest factor = (1 + r)t
Barton College
Future and Present
Value Interest Factors
can be found at the
back of your book.
9
A few basic Definitions
• Compounding – The process of accumulation interest on an
investment over time to earn more interest.
• Interest on Interest – Interest earned on the reinvestment of previous
interest.
• Compound Interest – Interest earned on both the initial principal and
the interest reinvested from prior periods.
• Simple interest – just the interest earned on the original principle.
Barton College
10
Future Values – Example 2
• Suppose you invest the $1000 from the previous example for 5 years.
How much would you have?
– i = 5%, t = 5, PV = $1000
– FV = PV(1+r)t
– FV = 1000(1+.05) 5
– FV = 1000(1.05)5 = 1276.28
• The effect of compounding is small for a small number of periods, but increases
as the number of periods increases. (Simple interest would have a future value of
$1250, for a difference of $26.28.)
Example of (100 @ 10% rate) = $10.00
• Simple interest provides $10 x 5 = $150
• Compound Interest provides $161.05
–
–
–
–
–
Period 1:
Period 2:
Period 3:
Period 4:
Period 5:
Barton College
100(1.10) = $110
110(1.10) = $121
121(1.10) = $133.10
133.10(1.10) = $146.41
146.41(1.10) = $161.05
FV = 100(1.10)5 = $161.05
161.05 – 150 = $11.05
11
Future Values – Example 3
• Suppose you had a deposit $10 at 5.5% interest 200 years ago. How
much would the investment be worth today?
– FV = PV(1+r)n = 10(1.055)200 = 447,189.84
• What is the effect of compounding?
– Simple interest = 10 + 200(10)(5.5%) = 210.55
Compounding added $446,979.29 to the value of the investment
Calculator: N = 200; I/Y = 5.5; PV = 10; CPT FV = -447,198.84
Barton College
12
FV Calculator Keys
• Texas Instruments BA-II Plus
– FV = future value
– PV = present value
– I/Y = period interest rate
•
•
• Use Appendix D to
help set up your
calculator
P/Y must equal 1 for the I/Y to be the period rate
Interest is entered as a percent, not a decimal
– N = number of periods
– Remember to clear the registers (CLR TVM) after
each problem
– Other calculators are similar in format
Calculator Page
Barton College
13
Finding the Present Value
or
(Discounting a Future Value back to the Present)
Barton College
14
Present Values
• How much do I have to invest today to have some amount in the future?
– FV = PV(1 + r)t
– Rearrange to solve for PV = FV / (1 + r)t
• When we talk about discounting, we mean finding the present value of
some future amount.
• When we talk about the “value” of something, we are talking about the
present value unless we specifically indicate that we want the future value.
Barton College
15
Present Value – One Period
Example
• Suppose you need $10,000 in one year for the down payment on a new car. If you
can earn 7% annually, how much do you need to invest today?
– PV = FV / (1+r)n = 10,000 / (1+.07) 1
– PV = 10,000 / (1.07)1 = 9345.79
– Calculator
• 1= N
• 7 = I/Y
• 10,000 = FV
• CPT PV = -9345.79
Calculator Example of Present Value (PV)
• http://movies.atomiclearning.com/movie/k12/21142/play
Barton College
16
Present Values – Example 2
• You want to begin saving for you daughter’s college education and you
estimate that she will need $150,000 in 17 years. If you feel confident that
you can earn 8% per year, how much do you need to invest today?
• PV = FV / (1+r)n
• PV = 150,000 / (1.08)17 = $40,540.34
Key strokes: 1.08 yx 17 = 3.70002 and $150,000/3.70002 = $40, 540.34
Calculator: N = 17; I/Y = 8; FV = 150,000; CPT PV = -40,540.34
Barton College
17
Present Values – Example 3
• Your parents set up a trust fund for you 10 years ago that is now
worth $19,671.51. If the fund earned 7% per year, how much did your
parents invest?
• PV = FV / (1+r)n
– PV = 19,671.51 / (1.07)10 = 10,000
Calculator: N = 10; I/Y = 7; FV = 19,671.51; CPT PV = -10,000
Barton College
18
Present Value – Important
Relationship I
• For a given interest rate – the longer the time period, the lower the
present value
– What is the present value of $500 to be received in 5 years? 10 years?
The discount rate is 10%
– 5 years: PV = 500 / (1.1)5 = 310.46
– 10 years: PV = 500 / (1.1)10 = 192.77
A dollar decreases in value more and more over time…..
Or… the discount factor increases over time….
Barton College
19
The Basic PV Equation Refresher
• PV = FV / (1 + r)t
• There are four parts to this equation
– PV, FV, r and t (I will sometimes mix n and t for the same thing…)
– If we know any three, we can solve for the fourth
• If you are using a financial calculator, be sure and remember the
sign convention or you will receive an error when solving for r
or t
Barton College
20
Discount Rate
• Often we will want to know what the implied interest rate is in an
investment
• Rearrange the basic PV equation and solve for “r”
–
–
–
–
FV = PV(1 + r)t
FV/PV = (1+r)t
(FV/PV)1/t = ((1+r)t)1/t
(FV/PV)1/t = 1+r
– r = (FV / PV)1/t – 1
• If you are using formulas, you will want to make use of both the yx
and the 1/x keys
Barton College
21
Discount Rate – Example 1
• You are looking at an investment that will pay $1200 in 5 years if you invest $1000
today. What is the implied rate of interest?
– FV = 1200
– N=5
– PV = 1000
r = ______
r = (FV / PV)1/t – 1
– r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%
– Calculator – the sign convention matters!!!
• N= 5
• PV = -1000 (you pay 1000 today)
• FV = 1200 (you receive 1200 in 5 years)
• CPT I/Y = 3.714%
Barton College
22
Discount Rate – Example 2
• Suppose you are offered an investment that will allow you to
double your money in 6 years. You have $10,000 to invest. What is
the implied rate of interest?
– r = (FV / PV)1/t – 1
– r = (20,000 / 10,000)1/6 – 1 = .122462 = 12.25%
Calculator: N = 6; FV = 20,000; PV = 10,000; CPT I/Y = 12.25%
Barton College
23
Discount Rate – Example 3
• Suppose you have a 1-year old son and you want to provide $75,000
in 17 years towards his college education. You currently have $5000
to invest. What interest rate must you earn to have the $75,000
when you need it?
– r = (FV / PV)1/t – 1
– r = (75,000 / 5,000)1/17 – 1 = .172688 = 17.27%
Calculator: N = 17; FV = 75,000; PV = 5,000; CPT I/Y = 17.27%
Barton College
24
Solving for the Number of Periods
– Example 1
• You want to purchase a new car and you are willing to pay $20,000. If you can
invest at 10% per year and you currently have $15,000, how long will it be
before you have enough money to pay cash for the car?
Given: FV = PV (1+r)t
Solve for t
– FV/PV = (1+r)t
– Using the ln and “The Power Rule for natural logs”
– ln (FV/PV) = t ln(1+r)
– t = ln(FV/PV) / ln(1+r)
– t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years
– ln xt = t ln x
Barton College
(The Power Rule)
25
Number of Periods – Example 2 Continued
• To buy a house you have $15,000 but you need $21,750 that includes all
your closing costs. Rates are currently 7.5%. How long will it take to reach
the desired amount?
– Down payment = .1(150,000) = 15,000
– Closing costs = .05(150,000 – 15,000) = 6,750
– Total needed = 15,000 + 6,750 = 21,750
• Compute the number of periods
– PV = -15,000
– FV = 21,750
– I/Y = 7.5
– CPT N = 5.14 years
To buy a House???
t = ln (FV/PV) / ln (1+r)
• Using the formula
– t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years
Barton College
26
Spreadsheet Example
• Use the following formulas for TVM calculations
–
–
–
–
FV(rate,nper,pmt,pv)
PV(rate,nper,pmt,fv)
RATE(nper,pmt,pv,fv)
NPER(rate,pmt,pv,fv)
• The formula icon is very useful when you can’t remember the exact
formula
• Click on the Excel icon to open a spreadsheet containing four
different examples.
Barton College
27
Practice Problems
• Complete the following problems in your book.
Questions and Problems in your Book:
– 1-5, 12, 13, 14 & 15
– Try the problems several different ways, e.g., using the basic formulas, your
calculator and Excel.
– If you need help with using your calculator, please see the FNC 330
website. I have several user guide links to the HP 10-B and TI BA-II.
Barton College
28
Future Value and Present
Value Formulas
Barton College
29