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

€

FVPV

= 2 = (1+ R)n

log(2) = n log(1+ R)

n =log(2)

log(1+ R)

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

Retirement Savings at 5% interest

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

€

PV =FV

(1+ R)n

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

€

PV =100

(1+ R)ii=1

10

∑ = 772

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

€

FV = 100(1.05)i

i=0

9

∑

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

€

PV =1

(1+ 0.07)ii=1

33

∑

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

Discounting to infinity

• Math review:

€

s = ai

i=1

∞

∑

as = ai+1

i=1

∞

∑ = ai

i=2

∞

∑

s −as = a

s =a

1−a

Present Value of a Constant Stream

€

a =1

1 + R

PV =y

(1 + R)ii=1

∞

∑ a =1

1 + R

PV = y ai

i=1

∞

∑

PV =a

1 − ay =

1(1+ R)

1 −1

(1 + R)

(y) =

1(1 + R)

1 + R1 + R

−1

(1 + R)

=yR

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

€

PV =5001.05

+1000

(1.05)2 +100000(1.05)3 = 87,767

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)€

PV = −1000 +100

(1.05)ii=1

15

∑ = 38

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

€

PV =y1

(1+ R1)+

y2

(1+ R2 )2 +y3

(1+ R3)3

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

1APREFFe=−€

EFF = (1+APRm

)m −1

€

EFF = eAPR −1

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€

PV =PMT

(1.09)ii=1

3

∑ =100000

PMT1

(1.09)ii=1

3

∑ =100000

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€

PV =PMT(1.01)ii=1

12

∑ =1000

PMT1

(1.01)ii=1

12

∑ =1000

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