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. 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

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
• PV = present or starting value
• FV = future value
• R = interest rate
• n = number of periods
First example
• PV = 1000
• R = 10%
• n = 1
• FV = ?

FV = 1000*(1.10) = 1,100

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
• 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
• Doubling time = time for funds to double
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
Goals
• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More applications
Present Value
• Go in the other direction
• Know FV
• Get PV
• Answer basic questions like what is \$100 tomorrow worth today
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
Goals
• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More applications
Annuity
• Equal payments over several years
• Usually annual
• Types: Ordinary/Annuity due
• Beginning versus end of period
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
• 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?
• 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
• 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
• 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
• 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
• This is an annuity with an infinite life
Perpetuity Examples and Interest Rate Sensitivity
• Interest rate sensitivity
• y=100
• R = 0.05, PV = 2000
• R = 0.03, PV = 3333
Goals
• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More applications
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 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)
Goals
• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More applications
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 Structure

Discounting payments 1, 2, 3 years from now

Goals
• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More examples
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
• Effective annual rate (EFF) formula
• Limit as m goes to infinity
• For APR = 0.06
• limit EFF = 0.0618
Goals
• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More examples
More Examples
• Home mortgage
• Car loans
• College
• Calculating present values
Home MortgageAmortization
• Specifications:
• \$100,000 mortgage
• 9% interest
• 3 years (equal payments) pmt
• Find pmt
• PV(pmt) = \$100,000
Mortgage PV
• Find PMT so that
• Solve for PMT
• PMT = 39,504
Car Loan
• Amount = \$1,000
• 1 Year
• Payments in months 1-12
• 12% APR (monthly compounding)
• 12%/12=1% per month
• PMT?
Car Loan
• Again solve, for PMT
• PMT = 88.85
Total Payment
• 12*88.85 = 1,066.20
• Looks like 6.6% interest
• Why?
• Paying loan off over time
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
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
• 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
• Powerful tool
• Useful for day to day problems
• Loans/mortgages
• Retirement
• We will use it for
• Stock pricing
• Bond pricing
Goals
• Compounding and Future Values
• Present Value
• Valuing an income stream
• Annuities
• Perpetuities
• Mixed streams
• Term structure again
• Compounding
• More examples