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The Time Value of Money. Economics 71a Spring 2007 Mayo, Chapter 7 Lecture notes 3.1. Goals. Compounding and Future Values Present Value Valuing an income stream Annuities Perpetuities Mixed streams Term structure again Compounding More applications. Compounding.

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the time value of money

The Time Value of Money

Economics 71a

Spring 2007

Mayo, Chapter 7

Lecture notes 3.1

goals
Goals
  • Compounding and Future Values
  • Present Value
  • Valuing an income stream
    • Annuities
    • Perpetuities
  • Mixed streams
  • Term structure again
  • Compounding
  • More applications
compounding
Compounding
  • PV = present or starting value
  • FV = future value
  • R = interest rate
  • n = number of periods
first example
First example
  • PV = 1000
  • R = 10%
  • n = 1
  • FV = ?

FV = 1000*(1.10) = 1,100

example 2 compound interest
Example 2Compound Interest
  • PV = 1000
  • R = 10%
  • n = 3
  • FV = ?

FV = 1000*(1.1)*(1.1)*(1.1) = 1,331

  • FV = PV*(1+R)^n
example 3 the magic of compounding
Example 3:The magic of compounding
  • PV = 1
  • R = 6%
  • n = 50
  • FV = ?
    • FV = PV*(1+R)^n = 18
    • n = 100, FV = 339
    • n = 200, FV = 115,000
example 4 doubling times
Example 4:Doubling times
  • Doubling time = time for funds to double
example 5 retirement saving
Example 5Retirement Saving
  • PV = 1000, age = 20, n =45
  • R = 0.05
    • FV = PV*(1+0.05)^45 = 8985
    • Doubling 14
  • R = 0.07
    • FV=PV*(1+0.07)^45 = 21,002
    • Doubling = 10
  • Small change in R, big impact
goals10
Goals
  • Compounding and Future Values
  • Present Value
  • Valuing an income stream
    • Annuities
    • Perpetuities
  • Mixed streams
  • Term structure again
  • Compounding
  • More applications
present value
Present Value
  • Go in the other direction
  • Know FV
  • Get PV
  • Answer basic questions like what is $100 tomorrow worth today
example given a zero coupon bond paying 1000 in 5 years
ExampleGiven a zero coupon bond paying $1000 in 5 years
  • How much is it worth today?
  • R = 0.05
  • PV = 1000/(1.05)^5 = $784
  • This is the amount that could be stashed away to give 1000 in 5 years time
goals13
Goals
  • Compounding and Future Values
  • Present Value
  • Valuing an income stream
    • Annuities
    • Perpetuities
  • Mixed streams
  • Term structure again
  • Compounding
  • More applications
annuity
Annuity
  • Equal payments over several years
    • Usually annual
  • Types: Ordinary/Annuity due
    • Beginning versus end of period
present value of an annuity
Present Value of an Annuity
  • Annuity pays $100 a year for the next 10 years (starting in 1 year)
  • What is the present value of this?
  • R = 0.05
future value of an annuity
Future Value of An Annuity
  • Annuity pays $100 a year for the next 10 years (starting in 1 year)
  • What is the future value of this at year 10?
  • R = 0.05
why the funny summation
Why the Funny Summation?
  • Period 10 value for each
    • Period 10: 100
    • Period 9: 100(1.05)
    • Period 8: 100(1.05)(1.05)
    • Period 1: 100(1.05)^9
  • Be careful!
application lotteries
Application: Lotteries
  • Choices
    • $16 million today
    • $33 million over 33 years (1 per year)
  • R = 7%
  • PV=$12.75 million, take the $16 million today
another way to view an annuity
Another Way to View An Annuity
  • Annuity of $100
    • Paid 1 year, 2 year, 3 years from now
  • Interest = 5%
  • PV = 100/(1.05) + 100/(1.05)^2 + 100/(1.05)^3
  • = 272.32
cost to generate from today
Cost to Generate From Today
  • Think about putting money in the bank in 3 bundles
  • One way to generate each of the three $100 payments
  • How much should each amount be?
    • 100 = FV = PV*(1.05)^n (n = 1, 2, 3)
    • PV = 100/(1.05)^n (n = 1, 2, 3)
  • The sum of these values is how much money you would have to put into bank accounts today to generate the annuity
  • Since this is the same thing as the annuity it should have the same price (value)
perpetuity
Perpetuity
  • This is an annuity with an infinite life
perpetuity examples and interest rate sensitivity
Perpetuity Examples and Interest Rate Sensitivity
  • Interest rate sensitivity
    • y=100
    • R = 0.05, PV = 2000
    • R = 0.03, PV = 3333
goals25
Goals
  • Compounding and Future Values
  • Present Value
  • Valuing an income stream
    • Annuities
    • Perpetuities
  • Mixed streams
  • Term structure again
  • Compounding
  • More applications
mixed stream apartment building
Mixed StreamApartment Building
  • Pays $500 rent in 1 year
  • Pays $1000 rent 2 years from now
  • Then sell for 100,000 3 years from now
  • R = 0.05
mixed stream investment project
Mixed StreamInvestment Project
  • Pays -1000 today
  • Then 100 per year for 15 years
  • R = 0.05
  • Implement project since PV>0
  • Technique = Net present value (NPV)
goals28
Goals
  • Compounding and Future Values
  • Present Value
  • Valuing an income stream
    • Annuities
    • Perpetuities
  • Mixed streams
  • Term structure again
  • Compounding
  • More applications
term structure
Term Structure
  • We have assumed that R is constant over time
  • In real life it may be different over different horizons (maturities)
  • Remember: Term structure
  • Use correct R to discount different horizons
term structure30
Term Structure

Discounting payments 1, 2, 3 years from now

goals31
Goals
  • Compounding and Future Values
  • Present Value
  • Valuing an income stream
    • Annuities
    • Perpetuities
  • Mixed streams
  • Term structure again
  • Compounding
  • More examples
frequency and compounding
Frequency and compounding
  • APR=Annual percentage rate
  • Usual quote:
    • 6% APR with monthly compounding
  • What does this mean?
    • R = (1/12)6% every month
  • That comes out to be
    • (1+.06/12)^12-1
    • 6.17%
  • Effective annual rate
general formulas
General Formulas
  • Effective annual rate (EFF) formula
  • Limit as m goes to infinity
  • For APR = 0.06
  • limit EFF = 0.0618
goals34
Goals
  • Compounding and Future Values
  • Present Value
  • Valuing an income stream
    • Annuities
    • Perpetuities
  • Mixed streams
  • Term structure again
  • Compounding
  • More examples
more examples
More Examples
  • Home mortgage
  • Car loans
  • College
  • Calculating present values
home mortgage amortization
Home MortgageAmortization
  • Specifications:
    • $100,000 mortgage
    • 9% interest
    • 3 years (equal payments) pmt
  • Find pmt
    • PV(pmt) = $100,000
mortgage pv
Mortgage PV
  • Find PMT so that
  • Solve for PMT
    • PMT = 39,504
car loan
Car Loan
  • Amount = $1,000
  • 1 Year
    • Payments in months 1-12
  • 12% APR (monthly compounding)
  • 12%/12=1% per month
  • PMT?
car loan39
Car Loan
  • Again solve, for PMT
  • PMT = 88.85
total payment
Total Payment
  • 12*88.85 = 1,066.20
  • Looks like 6.6% interest
  • Why?
    • Paying loan off over time
payments and principal
Payments and Principal
  • How much principal remains after 1 month?
    • You owe (1+0.01)1000 = 1010
    • Payment = 88.85
    • Remaining = 1010 – 88.85 = 921.15
  • How much principal remains after 2 months?
    • (1+0.01)*921.15 = 930.36
    • Remaining = 930.36 – 88.85 = 841.51
college should you go
CollegeShould you go?
  • 1. Compare
      • PV(wage with college)-PV(tuition)
      • PV(wage without college)
  • 2. What about student loans?
  • 3. Replace PV(tuition) with PV(student loan payments)
  • Note: Some of these things are hard to estimate
  • Second note: Most studies show that the answer to this question is yes
calculating present values
Calculating Present Values
  • Sometimes difficult
  • Methods
    • Tables (see textbook)
    • Financial calculator (see book again)
    • Excel spreadsheets (see book web page)
    • Java tools (we’ll use these sometimes)
    • Other software (matlab…)
discounting and time summary
Discounting and Time: Summary
  • Powerful tool
  • Useful for day to day problems
    • Loans/mortgages
    • Retirement
  • We will use it for
    • Stock pricing
    • Bond pricing
goals45
Goals
  • Compounding and Future Values
  • Present Value
  • Valuing an income stream
    • Annuities
    • Perpetuities
  • Mixed streams
  • Term structure again
  • Compounding
  • More examples