Transcript Time Value of Money - Seattle University

Time Value of Money
Money Value of Time???
Interest Rates
 Why interest rates are positive?
– People have ‘positive time preference’
 Behavior of human beings
– Current resources have productive uses
 Technology and natural process
Simple vs. Compound Interest
 Simple Interest
– No interest is earned on interest money paid in
the previous periods
– Money grows at a slower rate
 Compound Interest
– Interest is earned on interest money paid in the
previous periods
– Money grows at a faster rate
Simple Interest Example
 $100 at 8% simple annual interest for 2
years
– First year interest
 100 x (.08) = $8
Total = 100 + 8 = $___
– Second year interest
 100 x (.08) = $8
Total = 100 + 8 + 8 = $___
– Total Interest after 2 years:
8 + 8 = $__
Another example
 You deposit $5000 into a savings account
that earns 13% simple annual interest.
What is the amount in the account after 6
years?
Answer:_________
 What is the total amount of interest earned?
Answer:_________
Compound Interest Example
 Invest $100 at 8% compounded annually for
2 years
– Total after first year:
 100 x (1 + .08) = $108
– Total after second year
 108 x (1 + .08) = $_____
– Total Interest = 116.64 - 100 = $______
Compound Interest Example
Year
1
2
3
4
5
Begin. Amount
$100.00
Interest Earned
$10.00
Ending Amount
$110.00
110.00
11.00
121.00
121.00
12.10
133.10
133.10
13.31
146.41
146.41
14.64
161.05
Total interest
$61.05
[What would be the total interest earned in
simple interest case? Ans: $_______ ]
Future Value for a Lump Sum
 Notice that
– 1. $110
= $100 (1 + .10)
– 2. $121
= $110 (1 + .10) = $100 * 1.1 * 1.1 = $100 * 1.12
– 3. $133.10 = $121 (1 + .10) = $100 * 1.1 * 1.1 * 1.1
= $100 ________
 In general, the future value, FVt, of $1 invested today at
r% for t periods is
FVt = $1 * (1 + r)t
 The expression (1 + r)t is called the future value factor.
FV on Calculator
 What is the FV of $5000 invested at 12%
per year for 4 years compounded annually?
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Clear all memory:
CLEAR ALL
Ensure # compounding periods is 1:
1
P/YR
Enter amount invested today: -5000 PV
Enter # of years: 4 N
Enter interest rate: 12 I/YR
Find Future Value: FV
 Answer: $___________
Notice..
 You entered $5000 as a negative amount
 You got FV answer as a positive amount
 Why the negative sign?
 It turns out that the calculator follows ‘cash
flow convention’
– Cash outflow is negative (i.e. money going out)
– Cash inflow is positive (i.e. money coming in)
Another example
 Calculate the future value of $500 invested today
at 9% per year for 35 years
 Answer: ________
Present Values
 Here you simply reverse the question
 You are given
– Future Value
– Number of Periods
– Interest Rate
 and need to find the sum (PRESENT
VALUE) needed today to achieve that FV
Present Value for a Lump Sum
 Q. Suppose you need $20,000 in three years to
pay tuition at SU. If you can earn 8% on your
money, how much do you need today?
 A. Here we know the future value is $20,000, the rate
(8%), and the number of periods (3). What is the
unknown present amount (called the present value)?
 From before:
FVt = PV x (1 + r)t
$20,000 = PV __________
Rearranging:
PV
= $20,000/(1.08)3
= $_____________
In general, the present value, PV, of a $1 to
be received in t periods when the rate is r is
PV =
FVt
(1+r)t
Present Value Factor = 1
(1+r)t
‘r’ is also called the discount rate
PV on Calculator
 Your friend promises to pay you $5,000
after 3 years. How much are you willing to
pay her today? You can earn 8%
compounded annually elsewhere.
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Clear all memory:
CLEAR ALL
P/YR
Ensure # compounding periods is 1:
1
Enter amount future value: 5000 FV
Enter # of years: 3 N
Enter interest rate: 8 I/YR
Find Present Value: PV
 Answer: $___________
Another PV example
 Vincent van Gogh painted Portrait of Dr.
Gachet in 1889. It sold in 1987 for $82.5
million. How much should he have sold it
in 1889 if annual interest rate over the
period was 9%?
 Answer: _____________
Vincent
Gogh
The Portrait of
Dr Gachet
Van
Present Value of $1 for Different Periods and Rates
1.00
r = 0%
.90
Present
value
of $1 ($)
.80
.70
.60
r = 5%
.50
.40
r = 10%
.30
.20
r = 15%
.10
r = 20%
1
2
3
4
5
6
7
8
9
10
Time
(years)
Notice...
 As time increases, present value declines
 As interest rate increases, present value declines
 The rate of decline is not a straight line!
Notice Four Components
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Present Value (PV)
Future Value at time t (FVt)
Interest rate per period (r)
Number of periods (t)
 Given any three, the fourth can be found
Finding ‘r’
 You need $8,000 after four years. You have
$7,000 today. What annual interest rate must
you earn to have that sum in the future?
Answer: __________
Finding ‘t’
 How many years does it take to double your
$100,000 inheritance if you can invest the
money earning 11% compounded annually?
Answer: __________
Note:
 When calculating future value what you are
doing is compounding a sum
 When calculating present value, what you are
doing is discounting a sum
FV - Multiple Cash Flows
 You deposit
$100 in one year
$200 in two years
$300 in three years
How much will you have in three years?
r = 7% per year.
 Answer: ____________
 Draw a time line!!!
PV - Multiple Cash Flows
 An investment pays:
$200 in year 1
$600 in year 3
$400 in year 2
$800 in year 4
You can earn 12% per year on similar
investments. What is the most you are
willing to pay now for this investment?
 Answer: __________
 Draw time line!!!
Important…
 You can add cash flows ONLY if they are
brought back (or taken forward) to the
SAME point in time
 Adding cash flows occurring at different
points in time is like adding apples and
oranges!
Level Multiple Cash Flows
 Examples of constant level cash flows for
more than one period
– Annuities
– Perpetuities
 Most of the time we assume that the cash
flow occurs at the END of the period
Examples of Annuities
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Car loan payments
Mortgage on a house
Most other consumer loans
Contributions to a retirement plan
Retirement payments from a pension plan
Saving a Fixed Sum
 You save $450 in a retirement fund every
month for the next 30 years. The interest
rate earned is 10%. What is the
accumulated balance at the end of 30 years?
 This is Future Value of an Annuity
Future Value Calculated
Save $2,000 every year for 5 years into an account
that pays 10%. What is the accumulated balance
after 5 years?
Future value
0
2
1
3
4
5
calculated by
Time
(years)
compounding each
cash flow separately
$2,000
$2,000
$2,000
$2,000
x 1.1
x 1.12
$2,000.0
2,200.0
2,420.0
x 1.13
2,662.0
2.928.2
x 1.14
$12,210.20
Total future value
FV of Annuity
 1  r   1
FVofAnnuity  C  

r


t
Important to understand inputs
 ‘r’ is the interest rate per period
 ‘t’ is the # of periods.
 For example,
– if ‘t’ is # of years, ‘r’ is annual rate
– if ‘t’ is # of months, ‘r’ is the monthly rate
FV of Annuity Example
 You will contribute $5,000 per year for the
next 35 years into a retirement savings plan.
If your money earns 10% interest per year,
how much will you have accumulated at
retirement?
 Draw a time line!!!
Time Line
0
1
-5000
2
-5000
34
-5000
35
-5000
 Notice: Payment begins at the end of first year
FV of Annuity on Calculator
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Clear all memory:
CLEAR ALL
P/YR
Ensure # compounding periods is 1:
1
Enter payments: -5000 PMT
Enter # of payments: 35 N
Enter interest rate: 10 I/YR
Find Future Value: FV
 Answer: $___________
FV Annuity - A Twist..
 You estimate you will need $1 million to
live comfortably in retirement in 30 years.
How much must you save monthly if your
money earns 12% interest per year?
 Note: Payments are monthly, interest
quoted is annual!!!
Two ways to adjust for
compounding periods
 Divide annual interest rate by 12 and enter
interest rate per month into calculator as the
interest rate and leave “P/YR” as 1
OR
 Set “P/YR” on calculator as 12: 12
and enter the annual interest rate
P/YR
‘N’ on calculator
You can either:
 Enter # of periods directly (360 in the
example)
OR
 If you have set 12 as the P/YR then you can
N
also enter it as 30
– (notice it appears as 360)
FV Annuity on Calculator (2)
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Clear all memory:
CLEAR ALL
Monthly-> # compounding periods is 12: 12 P/YR
Enter Future Value: 1,000,000 FV
N
Enter # of payments: 30
Enter interest rate: 12 I/YR
Find payments: PMT
 Answer: $___________
Note the
difference!
Present Value of Annuities
 Here we bring multiple, level cash flows
back to the present (year 0)
 Typical examples are consumer loans where
the loan amount is the PV and the fixed
payments are the cash flows
PV of Annuity Example
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Cash flow per period (CFt) = $500
Number of periods (t) = 4 years
Interest Rate (r) = 9% per year
What is the present value (PV) = ?
 ALWAYS DRAW A TIME LINE!!!
PV of Annuity on Calculator
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Clear all memory:
CLEAR ALL
P/YR
Ensure # compounding periods is 1:
1
Enter payments: 500 PMT
Enter # of payments: 4 N
Enter interest rate: 9 I/YR
Find Present Value: PV
 Answer: $___________
PV of Annuity
1  1 
 (1  r ) t 
PVofAnnuity  C  

r




 Again: ‘r’ and ‘t’ must match – i.e. if t is # of
months, r must be monthly rate
Car Loan Example
 Car costs $ 20,000
 Interest rate per month = 1%
 5-year loan ---> number of months = t = 60
 What is the monthly payment?
 Answer: ___________
Mortgage payments
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House cost $250,000
Mortgage Rate = 7.5% annually
Term of loan = 30 years
Payments made monthly
 What are your payments?
 Answer: _____________
To Reiterate...
 Be VERY careful about compounding
periods
 Problem can state annual interest rate, but
the cash flows can be monthly, quarterly…
 The convention is to state interest rate
annually (Annual Percentage Rate)
Perpetuity
 Annuity forever
 Examples: Preferred Stock, Consols
Perpetuity
C
PV 
r
 Note: C and r measured over same interval
Perpetuity Example
 Preferred stock pays $1.00 dividend per quarter.
The required return, r, is 2.5% per quarter.
 What is the stock value?
Perpetuity Example
 Steve Forbes’s flat-tax proposal was
expected to save him $500,000 a year
forever if passed. He spent $40,000,000 of
his own money for campaign
 Charge: He was running for presidency for
personal gain
 Did the charge make sense
Forbes continued...
 What should be ‘r’ in the example?
 At what ‘r’ would Forbes have gained from
being a president and steamrolling flat-tax
proposal?
Compounding Periods
 Interest can be compounded
– Annually
– Monthly
- Semiannually
- Daily
- Continuously
 Smaller the compounding period, faster is
the growth of money
 The same PV or FV formula can be used:
BUT UNDERSTAND THE INPUTS!!
Compounding example
 Invest $5,000 in a 5-year CD
 Quoted Annual Percentage Rate (APR) = 15%
 Calculate FV5 for annual, semi-annual,
monthly and daily compounding
 Key: Adjust “P/YR” on calculator
Answers:
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Annual:
Semi-annual
Monthly:
Daily:
$10,056.78
$10,305.16
$10,535.91
$10,583.37
 Continuous Compounding???
Continuous compouding
 Compounded every instant “microsecond”
FVt  PV  e
rt
 r = interest rate per period
 t = number of periods
 Previous example answer: $ 10,585.00
Continuous compounding
example
 Invest $4,500 in an account paying 9.5%
compounded continuously
 What is the balance after 4 years?
Answer: _________
Quoted vs. Effective Interest
Rates
 Quoted Rate: Usually stated annually along with
compounding period (APR)
– e.g. 10% compounded quarterly
 Effective Annual Rate (EAR): Interest rate
actually earned IF the compounding period were
one year
EAR
m
(QuotedRate) 

EAR  1 

1

m

m = number of compounding periods in a year
EAR on Calculator
 What is the EAR for quoted rate of 15% per
year compounded quarterly?
 Set number of periods per year: 4 P/YR
 Enter quoted annual rate: 15 I/YR
EFF%
 Compute EAR:
 Answer: _______
EAR Example
 Compute EAR for 12% compounded
–
–
–
–
Annually
Quarterly
Monthly
Daily
 Answers: ____ , ____ , ____ , ____
EAR for Continuous
compounding
EAR  e  1
q
 Example: Quoted rate is 10% compounded
continuously
 EAR = _____%
Complicatons to TVM
 When payments begin beyond year 1
 PV and FV combined
 When payments begin in year 0 (Annuities Due)
Payments beyond year 1
 A car dealer offers ‘no payments for next 12
months’ deal on a $15,000 car. After that,
you will pay monthly payments for the next
4 years. r = 10% APR. What are your
monthly payments?
 Answer: ___________
PV and FV combined
 How much must you invest per year to have an
amount in 20 years that will provide an annual
income of $12,000 per year for 5 years? r = 8%
annually.
 Answer: ___________
PV and FV combined 2
 You have 2 options:
– Receive $100 for next 10 years only
– Receive $100 forever beginning in year 11
 If r = 10% which one would you prefer?
 At what interest rate are you indifferent
between the two options?
Annuities Due
 Payments begin in year 0
– Ex. Rent/Lease Payments
 Trick:
 Adjust BEG/END on calculator to BEG
OR
 Leave to END, but multiply (1+r) for both
PV and FV
Annuity Due Example
 Find PV of a 4-year (5 payment), $400
annuity due. r = 10%
 Find FV in year 5 of the above annuity due
0
1
2
3
4
5
Time
(years)
$400
$400
$400
 Answers:
– PV = $1,667.95
– FV5 = $2,686.24
$400
$400
FV
Another Example
 You start to contribute $500 every month to
your IRA account beginning immediately.
How much will you accumulate at the end
of first year? The return on your investment
is 20% per year.
 Note: ‘Return’ here is just another term for
the interest rate
 Answer: $_______
Tricky but Legal...
 Add-on Interest
Called ‘add-on’ interest because interest is
added on to the principal before the
payments are calculated
 Points on a Loan:
Percentage of loan amount reduced up front
– Used in home mortgages
Example: Add-on Interest
 You are offered the opportunity to borrow $1,000
for 3 years at 12% ‘add-on’ interest. The lender
calculates the payment as follows:
Amt. owed in 3 years: $1000 x (1+.12)3 = 1,405
Monthly Payment = $1,405 / 36 = $39
 What is the effective annual rate (EAR)?
 Steps:
– Calculate the APR interest (I/YR)
– Use answer to calculate the EAR
Add-on Example (2)
 Calcuate the EAR on a 6-year, $7,000 loan at
13% ‘add-on’ interest. The payments are
monthly.
 Answer: ________
Example: Points on a Loan
 1-year loan of $100. r = 10% + 2 points
[Note: 1 point = 1% of loan amount. Hence
you pay upfront $2 to lender. Hence you
are actually getting only $98, not $100]
 What is the EAR?
 $110 = $98 (1+r)
r = 12.24%
Points on a loan (2)
 Calculate the EAR on a 10-year, $110,000
mortgage when interest rate quoted is 7.75% + 1
point. The payments are monthly
 Answer: _________
Balloon Payments
 Amount on the loan outstanding after a
certain number of payments have been
made
– Sometimes called ‘residual’ on a loan
 e.g. when you want to pay off a loan early
Balloon Example
 You borrowed $90,000 on a house for 30
years 10 years ago. The annual interest rate
then was 17%. The payments are monthly.
Since interest rate has fallen, you want to
payoff the remaining amount on the loan
and refinance it. What is the outstanding
amount to be paid off?
(Note: Payments are $1,283.11)
 Answer: $__________
Two ways to calculate Balloons
 First calculate payments
 Take the present value of the remaining
(unpaid) payments
OR
 Use amortization function on calculator
 Enter the period : period INPUT
AMORT , and then =
=
=
 Enter
Another Example..
 What is the outstanding balance on a 5 year
$19,000 car loan at 11% interest after 2-1/2 years
have passed? The payments are monthly.
 Answer: $____________
TVM TIPS
 Draw time line!
 Check & set BEG/END on calculator
 Check & set P/YR on calculator
 Check & set # of decimal places to 4
TVM Tips Continued...
 Clear all previously stored #’s in memory
– Especially true when same problem requires
multiple TVM calculations
 Make sure that for FV and PV calculation,
you have correctly signed (+/-) the cash
flows