The Time Value of Money

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The Time Value of Money

• Economics 71a
• Spring 2007
• Mayo, Chapter 7
• Lecture notes 3.1

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Goals

• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More applications

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Compounding

• PV present or starting value
• FV future value
• R interest rate
• n number of periods

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First example

• PV 1000
• R 10
• n 1
• FV ?

FV 1000(1.10) 1,100
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Example 2Compound Interest

• PV 1000
• R 10
• n 3
• FV ?

FV 1000(1.1)(1.1)(1.1) 1,331
FV PV(1R)n
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Example 3The magic of compounding

• PV 1
• R 6
• n 50
• FV ?
• FV PV(1R)n 18
• n 100, FV 339
• n 200, FV 115,000

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Example 4Doubling times

• Doubling time time for funds to double

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Example 5Retirement Saving

• PV 1000, age 20, n 45
• R 0.05
• FV PV(10.05)45 8985
• Doubling 14
• R 0.07
• FVPV(10.07)45 21,002
• Doubling 10
• Small change in R, big impact

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Retirement Savings at 5 interest
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Goals

• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More applications

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Present Value

• Go in the other direction
• Know FV
• Get PV
• Answer basic questions like what is 100 tomorrow
  worth today

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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

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Goals

• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More applications

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Annuity

• Equal payments over several years
• Usually annual
• Types Ordinary/Annuity due
• Beginning versus end of period

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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

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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

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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!

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Application Lotteries

• Choices
• 16 million today
• 33 million over 33 years (1 per year)
• R 7
• PV12.75 million, take the 16 million today

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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

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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)

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Perpetuity

• This is an annuity with an infinite life

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Discounting to infinity

• Math review

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Present Value of a Constant Stream
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Perpetuity Examples and Interest Rate Sensitivity

• Interest rate sensitivity
• y100
• R 0.05, PV 2000
• R 0.03, PV 3333

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Goals

• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More applications

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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

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Mixed StreamInvestment Project

• Pays -1000 today
• Then 100 per year for 15 years
• R 0.05
• Implement project since PVgt0
• Technique Net present value (NPV)

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Goals

• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More applications

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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

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Term Structure
Discounting payments 1, 2, 3 years from now
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Goals

• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More examples

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Frequency and compounding

• APRAnnual 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

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General Formulas

• Effective annual rate (EFF) formula
• Limit as m goes to infinity
• For APR 0.06
• limit EFF 0.0618

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Goals

• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More examples

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More Examples

• Home mortgage
• Car loans
• College
• Calculating present values

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Home MortgageAmortization

• Specifications
• 100,000 mortgage
• 9 interest
• 3 years (equal payments) pmt
• Find pmt
• PV(pmt) 100,000

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Mortgage PV

• Find PMT so that
• Solve for PMT
• PMT 39,504

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Car Loan

• Amount 1,000
• 1 Year
• Payments in months 1-12
• 12 APR (monthly compounding)
• 12/121 per month
• PMT?

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Car Loan

• Again solve, for PMT
• PMT 88.85

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Total Payment

• 1288.85 1,066.20
• Looks like 6.6 interest
• Why?
• Paying loan off over time

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Payments and Principal

• How much principal remains after 1 month?
• You owe (10.01)1000 1010
• Payment 88.85
• Remaining 1010 88.85 921.15
• How much principal remains after 2 months?
• (10.01)921.15 930.36
• Remaining 930.36 88.85 841.51

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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

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Calculating Present Values

• Sometimes difficult
• Methods
• Tables (see textbook)
• Financial calculator (see book again)
• Excel spreadsheets (see book web page)
• Java tools (well use these sometimes)
• Other software (matlab)

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Discounting and Time Summary

• Powerful tool
• Useful for day to day problems
• Loans/mortgages
• Retirement
• We will use it for
• Stock pricing
• Bond pricing

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Goals

• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More examples