Transcript Slide 1
Chapter 4 Time Value of Money
(cont.)
• Present value of multiple cash flows
• Nominal interest rate and real interest rate
• Effective interest rate
Multiple Cash Flows
• Usually an investment involve multiple/a stream of
(negative/positive) cash flows instead of just one
payment and one initial investment.
One term deposit
Several term deposits that end at the same time
Several withdrawals out of one deposit
FV of Multiple Cash Flows
• The future value of several cash flows paid (or
several cash flows received) at a certain point of time
can be calculated by adding up the future values of
each of the cash flows.
FV C0 (1 r ) N 0 C1 (1 r ) N 1 C2 (1 r ) N 2 ....
• N specifies how many periods away from now is the
FV that we want to calculated.
• Ct denotes the actual cash flow that is paid/received at
the end of the tth period.
FV of Multiple Cash Flows
FV C0 (1 r ) N 0 C1 (1 r ) N 1 C2 (1 r ) N 2 ....
Example: If you make one term deposit of $300 now and
another 2 of $200 at the end of each of the following two years,
and all the deposit expires at the end of the 4th year from now.
Interest rate is 8%. How much will your bank account balance
be? (draw a time line and assign values to variables in the formula)
FV of Multiple Cash Flows
FV C0 (1 r ) N 0 C1 (1 r ) N 1 C2 (1 r ) N 2 ....
Example: (cont.)
FV 300 (1 8%)40 200 (1 8%) 41 200 (1 8%) 42
408 252 233 893
PV of Multiple Cash Flows
• The present value of several cash flows paid
(or several cash flows received) in future can
be calculated by adding up the present values
of each of the cash flows.
Ct
C1
C2
PV C0
1
2
t
(1 r ) (1 r )
(1 r )
• Ct denotes the actual cash flow that is
paid/received at the end of the tth period.
PV of Multiple Cash Flows
PV C0
Ct
C1
C2
1
2
t
(1 r )
(1 r )
(1 r )
Example: If you need to make 3 payments at different point of
time: one of $250 now, a second payment of $300 at the end of
next year (the first year) and a third one of $500 at the end of
the year after next (the second year) . Interest rate is 8%. How
much money should you have in your bank account now so
that you would be able to make all the three payments at the
specified time? (draw a time line and assign values to variables in the
formula)
PV of Multiple Cash Flows
PV C0
Ct
C1
C2
1
2
t
(1 r )
(1 r )
(1 r )
Example: (cont.)
300
500
PV 250
1
(1 8%) (1 8%) 2
250 278 429 911
Multiple Cash Flows
• Using financial calculators:
– Calculate the FV/PV of each cash flows independently
then sum the results together
– Make sure the correct t (i.e. N) is used for each cash flow
• When there are several cash flows paid and
also several cash flows received, the formula
to be used are the same:
– Make sure the correct sign is given to each cash flow
Perpetuities & Annuities
Perpetuity:
A stream of level cash payments that never ends.
Annuity:
Equally spaced level stream of cash flows for a
limited period of time.
Perpetuities
Assume:
• Deposit $100
• Annual interest rate is 8% and it never changes
• Interests are withdrawn at the end of every year but
never the principal
Cash flows:
$100 8% $8
• Pay $100 now
• Receive $8 at the end of every year forever
Perpetuities
PV of Perpetuity: the value of all future cash flows from a
perpetuity in terms of a one time payment now
Formula: for a perpetuity whose cash flows occur at the end
of every period starting from now.
C = cash payment
r = interest rate / discount rate
PV
C
r
Perpetuities
Example - Perpetuity
In order to create an endowment, which pays
$100,000 per year, forever, how much money must be
set aside today if the rate of interest is 10%?
100, 000
PV
1, 000, 000
10%
Perpetuities
Example - continued
If the first perpetuity payment will not be received
until three years from today, how much money needs
to be set aside today?
100, 000
PV '
1, 000, 000
10%
PV '
PV
826, 446
2
(1 10%)
Annuities
• Annuity can be viewed as the difference
between two perpetuities
C
PV
r
PV
C
1
PV '
2
(1 r )
r (1 r ) 2
1
1
PV '' PV PV ' C
2
r
r
(1
r
)
Annuities
PV of Annuity: the value of all future cash flows from an annuity
in terms of a one time payment now
Formula: for an annuity whose cash flows occur at the end of
every period starting from now and lasting for t periods.
PV C
1
r
1
r ( 1 r ) t
C = cash payment every period
r = interest rate
t = number of periods cash payment is received
Annuities
PV Annuity Factor (PVAF) - The present value
of $1 a year for each of t years.
PVAF
1
r
1
r ( 1 r ) t
[Table A.3 on page 704 ]
• Find the appropriate PVAF according to the
right t and r
Annuities
Example - Annuity
To purchase a car, you are scheduled to make 3 annual
installments of $4,000 per year starting one year from now.
Given a rate of annual interest of 10%, what is the price you
are paying for the car (i.e. what is the PV)?
1
PV 4, 000 .10
.10(11.10)3
$4, 000 2.4869 $9,948
Annuities
• Example – Annuity (cont.)
N
Input
Output
I/YR
3
PV
10%
PMT
4000
9,947.4
FV
0
(Ordinary) Annuity and Annuity Due
Annuity Due Calculation
• Adjust your financial calculator
• Switch from “End” to “Begin
• The inputs are the same as an ordinary annuity
• Example: start paying the installments right now
N
Input
Output
I/YR
3
PV
10%
PMT
4,000
10,942.1
FV
0
Switch From “End” to “Begin”
• HP
Press {shift} (i.e. the yellow button) and then press
{BEG/END}
• TI
• Press {2nd}, then {BGN}
• Press {2nd}, then {SET}
• Press {2nd}, then {QUIT}
• To switch back from “Begin” to “End”, just
repeat the procedure
Annuity Due Calculation (cont.)
• PV of and annuity due equals the multiple of the
PV of the ordinary annuity and (1+r)
– Both annuities have the same annual payment and
number of periods
• Example: start paying the installments right now
– Calculate the PV of corresponding ordinary annuity
N
Input
I/YR
3
Output
PV
10%
PMT
4,000
9,9947.4
– Multiply by (1+r)
9,947.4 (1 10%) 10,942.1
FV
0
Annuities Applications
PV C
1
r
1
r ( 1 r ) t
• Present Value of payments
• Implied interest rate for an annuity
• Calculation of periodic payments
– Mortgage payment
– Annual income from an investment payout
– Future Value of annuity
FV C PVAF (1 r ) t
Present Value of payments
• Example: In 1992, a nurse in a Reno casino
won the biggest jack pot - $9.3 million. That
sum was paid in 20 annual installments of
$465,000. What is the PV? r=10% (draw a time
line and assign values to variables in the annuity formula)
Present Value of payments
1
1
PV 465, 000
20
10%
10%
(1
10%)
10
465, 000 10
3,958,807.1
6.7275
N
Input
Output
I/YR
20
PV
10%
PMT
465,000
3,958,807
FV
0
Home Mortgages
• Example: Suppose you are buying a house that costs
$125,000, and you want to put down 20% ($25,000)
in cash. Assume that the mortgage loan lasts 30
years, i.e. 360 months. What will be your monthly
payment for each option, if the monthly interest rate
is 1%? (draw a time line and assign values to variables in the
annuity formula)
Home Mortgages
PMT C
100, 000
100, 000
1, 029
100
1
1
1% 1% (1 1%)360 100 35.9496
N
Input
Output
I/YR
360
10%
PV
PMT
100,000
FV
0
1029
Future Value of Annuity
Example - Future Value of annual payments
You plan to save $4,000 every year for 20 years
starting from the end of this year, and then retire.
Given a 10% rate of interest, what will be the balance
of your retirement account in 20 years?
Future Value of Annuity
N
Input
Output
I/YR
20
10%
PV
PMT
0
FV
4,000
229,100
Inflation
Inflation: Rate at which prices as a whole are
increasing.
• Consumer price index, CPI
Real Interest Rate: Rate at which the purchasing
power of the return of an investment increases.
• Real value of money
Nominal Interest Rate: Rate at which money
invested grows.
• Nominal value of money
• The quoted interest rate
Inflation
• Exact formula
• Approximation formula
Inflation
• Let r= real interest rate, i=inflation rate, and
R= nominal interest rate.
1 R
1 r
1 i
1 r 1 i 1 R
1 r i r i 1 R
r R i r i R i
Inflation
Example
If the interest rate on one year government bonds is
5.0% and the inflation rate is 2.2%, what is the real
interest rate?
Effective Interest Rates
• Effective Annual Interest Rate - Interest rate
that is annualized using compound interest.
Give the actual annual interests
• Annual Percentage Rate - Interest rate that is
annualized using simple interest.
Only a way to quote interest rates
Imposed by legal requirements
Effective Interest Rates
Example
Given APR of 12% and monthly compounding, what
is the Effective Annual Rate(EAR)?
• First, calculate month interest rate
APR 12%
monthly interest rate
1%
m
12
m : number of compounding periods per year
• Then, calculate the annual rate after compounding
EAR (1 compounding period interest rate)m 1
(1 1%)12 1 1.1268 1 12.68%
Amortizing Loan
• Mortgage Amortization (page 88)
Periodic Payment = Amortization + Periodic Interest
Periodic Interest = interest rate * prior period loan balance
Example: pay off 100,000 mortgage loan in 360 months at
interest rate of 1% per month
End of
Period
Loan
Balance
Periodic Interest
Amortization
Payment
0
100,000
0
0
1029
1
99,971
100,000*1%=1000
1029 -1000=29
1029
2
99,948 99,971 *1%= 999.7
1029 - 999.7=29.3
1029
…
1029
Amortizing Loan
Summary:
• Each periodic payment include amortization and
interests due.
• As the loan approaches maturity, the amortizations
paid increase every period.
• As the loan approaches maturity, the loan balances
and interests due decrease every period.
• The last amortization is just enough to payoff the last
part of principal.
Problem 25 on page 108 (4/e 24 on page 105)
Annuity Values
You want to buy a new car, but you can make an initial
payment of only $2,000 and can afford monthly payments of
at most $400.
a. If the APR on auto loans is 12% and you finance the
purchase over 48 months, what is the max price you can pay
for the car?
b. How much can you afford if you finance the purchase over
60 months?
Problem 28 on Page 109 (Problem 27 on Page 105)
• Rate on a Loan
If you take out an $8,000 car loan that calls for 48 monthly
payments of $240 each, what is the APR of the loan? What is
the EAR?
Problem 37 on Page 109 (Problem 36 on Page 106)
Amortizing Loan
You take out a 30-year $100,000 mortgage loan with an APR of 6% and
monthly payments. In 12 years you decide to sell your house and pay off the
mortgage. What is principal balance on the loan