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# Chapter 04 - PowerPoint by xiuliliaofz

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```									   Chapter 4 – Time Value of Money
Topics Covered
   Future Values
   Present Values
   Multiple Cash Flows
   Perpetuities and Annuities
   Inflation & Time Value
   Effective Annual Interest Rate
Future Values
Future Value - Amount to which an investment
will grow after earning interest.

Compound Interest - Interest earned on interest.

Simple Interest - Interest earned only on the
original investment.
Future Values
Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous
year’s balance.

Today          Future Years
1      2       3    4      5
Interest Earned           6.00 6.36 6.74 7.15 7.57
Value          100      106.00 112.36 119.10 126.25 133.82

Value at the end of Year 5 = \$133.82
Future Value
Future Value = Present Value of the investment
times (1 plus the interest rate) raised to the
number of periods

FV = PV(1+r)t
Future Value with Compounding
7000       Interest Rates

6000               0%
5%
5000               10%
FV of \$100

4000               15%

3000

2000

1000

0
20
22
24
26
28
30
10
12
14
16
18
0
2
4
6
8

Number of Years
Future Value

Example - FV
What is the future value of \$100 if interest is
compounded annually at a rate of 6% for five years?
Future Value

Example - FV
What is the future value of \$100 if interest is
compounded annually at a rate of 6% for five years?

FV =     PV(1+r) t        = \$100      (1.06) 5

= \$100 x 1.3382 = \$133.82
Future Values – Using Excel
Example - FV
What is the future value of
\$100 if interest is
compounded annually at a
rate of 6% for five years?
=FV(rate, nper, pmt, type)

=FV(6%, 5, 0, -100)
= \$133.82
Present Value

Present Value (PV) = Future Value/(1+r)t
Present Value
Example
You just bought a new computer for \$3,000. The
payment terms are 2 years same as cash. If you can earn
8% on your money, how much money should you set aside
today in order to make the payment when due in two
years?
Present Value

Present Value (PV) = Future Value/(1+r)t
=\$3,000/(1.08)2 =\$3,000/1.1664=\$2,572.02
Present Value

or by Excel=PV(rate,
nper, pmt, FV, type)
=PV(8%,2,0,3000)

PV = -\$2,572.02
(negative because you
give it up)
How to be a Millionaire
How much does a 21
year old have to save
each year and invest at
11% (historical return
on stocks) to have \$1
million at age 40?
How to be a Millionaire
How much does a 21 year
old have to save each year
and invest at 11% (historical
return on stocks) to have \$1
million at age 40?
=PMT(rate, nper, pv,
fv, type)
=PMT
(11%,19,0,\$1,000,000)
-\$17,562.50
Inflation
Inflation - Rate at which prices as a whole are
increasing.

Nominal Interest Rate - Rate at which money
invested grows.

Real Interest Rate - Rate at which the purchasing
power of an investment increases.
Inflation
Example
If the (nominal) interest rate on one year govt. bonds is 5.0% and the
inflation rate is 2.2%, what is the real interest rate?

1 + real interest rate = (1+ nominal rate)/(1 + inflation)
Savings
Bond
Inflation
Example
If the interest rate on one year govt. bonds is 5.0% and the inflation
rate is 2.2%, what is the real interest rate?
1 + real interest rate = (1+ nominal rate)/(1 + inflation)

1  real interestrate = 1+.022
1+.050               Savings
Bond
1  real interestrate = 1.027

real interestrate = .027 or 2.7%
Effective Interest Rates

Effective Annual Interest Rate - Interest rate
that is annualized using compound interest.

Annual Percentage Rate - Interest rate that is
annualized using simple interest.
Effective Interest Rates
example
Given a monthly rate of 1%, what is the Effective Annual
Rate(EAR)? What is the Annual Percentage Rate (APR)?
Effective Interest Rates
Example
Given a monthly rate of 1%, what is the
Effective Annual Rate(EAR)? What is the
Annual Percentage Rate (APR)?
EAR = (1 + .01)12 - 1 = r
EAR = (1 + .01)12 - 1 = .1268 or 12.68%

APR = .01 x 12 = .12 or 12.00%
A zero-coupon bond that will pay \$1,000 in 10 years is
selling for \$422.41 today. What interest rate does the
bond offer?

   =RATE(nper, pmt,
pv, fv)
   = RATE(10,0,
-422.41, 1,000)
   9%
Present Value of Future Payments
   You have won the New York State Lottery
(can’t win if you don’t play). You get \$1 million
per year (at the end of each year) for 10 years. If
the cost of money is 9%, what is the present
Present Value of Future Payments
You have won the
New York State
Lottery. You get \$1
million per year (at
the end of each year)
for 10 years. If the
cost of money is 9%,
what is the present
=PV(rate, nper,pmt,FV)
=PV(9%,10,1000000,0)
=\$6,417,657.70
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.
(the lottery example is an annuity)
PV of a Perpetuity
= Payment/r
A share of preferred stock pays \$4 per
year forever. If the cost of funds is 8.5%,
what is a share worth?
PV of a Perpetuity
A share of
preferred stock
pays \$4 per year
forever. If the cost
of funds is 8.5%,
what is a share
worth?
PV = 4/.085

= \$47.06
Future Value of Annual Payments
You plan to save \$4,000 every year for 40 years and then retire.
Given a 10% rate of interest, what will be the FV of your
retirement account?
Future Value of Annual Payments
You plan to save \$4,000
every year for 40 years and
then retire. Given a 10%
rate of interest, what will
be the FV of your
retirement account?
=FV(rate,nper,pmt)
=FV(10%,40,-4000)
\$1,770,370.22
PV of a Perpetuity
A share of
preferred stock
pays \$4 per year
forever. If the cost
of funds is 8.5%,
what is a share
worth?
PV = 4/.085

= \$47.06
PROBLEMS
Chapter 4: Question 4
You deposit \$1,000 in your bank account. If
the bank pays 4 percent simple interest,
how much will you accumulate in your
account after 10 years? What if the bank
pays compound interest? How much of
your earnings will be interest on interest?
You deposit \$1,000 in your bank account. If the bank
pays 4 percent simple interest, how much will you
accumulate in your account after 10 years? What if
the bank pays compound interest? How much of
your earnings will be interest on interest?

   With simple interest, you earn 4% of \$1,000
   \$1,000 X .04 = \$40 each year.
    There is no interest on interest (only with
compounding)
   After 10 years, you earn total interest of \$400,
and your account accumulates to \$1,400.
You deposit \$1,000 in your bank account. If the
bank pays 4 percent simple interest, how much
will you accumulate in your account after 10
years? What if the bank pays compound
interest? How much of your earnings will be
interest on interest?
   With compound interest
=FV(rate, nper,pmt,pv))
   =FV(4%,10,0,-1000)
   Over 10 years your account
grows to: \$148.24 x 10 =
\$1480.24
   Interest on interest = \$80.24
Chapter 4:
Question 5
   You will require
\$700 in 5 years. If
you earn 5 percent
interest on your
funds, how much
will you need to
invest today in
order to reach your
savings goal?
You will require \$700 in 5 years. If you earn 5 percent
interest on your funds, how much will you need to
invest today in order to reach your savings goal?

   =PV(rate, nper,pmt,FV)
   =PV(5%,5,0,700)
   = -\$548.47
Chapter 4: Question 10
   How long will it take for \$400 to grow to \$1000
at the interest rate specified?
a. 4 percent
b. 8 percent
c. 16 percent
How long will it take for \$400 to grow to \$1000 at the
interest rate specified?
a. 4 percent       b. 8 percent       c. 16 percent

   In these problems, you can
either solve the equation
provided directly, or you can
use Excel.
   =NPER(rate, pmt,pv,fv)
   =NPER(rate,0,-400,1000)
   =23.36 @4%
   =11.91@8%
   =6.17 @16%
How long will it take for \$400 to grow to \$1000 at the
interest rate specified?
b. 8 percent
   PV = ()400
   FV = 1000
   PMT = 0
   i as specified by the problem.
   Then compute n on the calculator.
   \$400  (1.08)t = \$1,000 
    t = 11.91 periods
How long will it take for \$400 to grow to \$1000 at the
interest rate specified?
c. 16 percent
   In these problems, you can either solve the equation
provided directly, or you can use your financial
calculator. Setting:
   PV = ()400
    FV = 1000
    PMT = 0
    i as specified by the problem.
   Then compute n on the calculator.
   \$400  (1.16)t = \$1,000 
   t = 6.17 periods
Chapter 4: Question 19
   A zero-coupon bond
that will pay \$1,000
in 10 years is selling
for \$422.41 today.
What interest rate
does the bond offer?
A zero-coupon bond that will pay \$1,000 in 10 years is
selling for \$422.41 today. What interest rate does the
bond offer?

   =RATE(nper,pmt,pv
, fv)
   =RATE(10,0,-
421.41,1000)
   = 9%
Chapter 4: Question 22
   If you take out an
\$8,000 car loan that
calls for 48 monthly
payments at an APR of
10 percent, what is
payment? What is the
effective annual
interest rate on the
loan?
If you take out an \$8,000 car loan that calls for 48
monthly payments at an APR of 10 percent, what is
your monthly payment? What is the effective annual
interest rate on the loan?

   =PMT(rate,nper,pv,fv)
   =PMT(10%/12,48,8000)
   =-\$202.90
   Effective annual interest
= (1+.00833)12 – 1
   = .1047 = 10.47%
Chapter 4: Question 24
Professor’s Annuity Corp. offers a lifetime annuity to
retiring professors. For a payments of \$80,000 at age
65, the firm will pay the retiring professor \$600 per
month until death.

a.If the professor’s remaining life expectancy is 20 years,
what is the monthly rate on this annuity?

b.If the monthly interest rate is .5 percent, what monthly
annuity payment can the firm offer to the retiring
professor?
Professor’s Annuity Corp. offers a lifetime annuity to
retiring professors. For a payment of \$80,000 at age 65, the
firm will pay the retiring professor \$600/ month until death.
a. If the professor’s remaining life expectancy is 20 years,
what is the monthly rate on this annuity?

   =RATE(nper,pmt,pv,fv)
   =RATE(20*12,600,-80000)
   = 0.548%
Professor’s Annuity Corp. offers a lifetime annuity to
retiring professors. For a payment of \$80,000 at age
65, the firm will pay the retiring professor \$600/
month until death.
b. If the monthly interest rate is .5 percent, what
monthly annuity payment can the firm offer to the
retiring professor?

=PMT(rate,nper,pv,fv)
=PMT(.5%,20*12,-
80000)

   CPT PMT = \$573.14
Chapter 4: Question 25
   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 percent and
you finance the purchase over 48 months, what
is the maximum price you can pay for the car?
   B. How much can you afford if you finance the
purchase over 60 months?
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. If the APR on auto loans is 12 percent and
you finance the purchase over 48 months, what is the
maximum price you can pay for the car?
   =PV(rate,nper,pmt,pv,fv)
   =PV(12%/12,48,-400)
   =\$15,189.58
   Your monthly payments of \$400 can
support a loan of \$15,189.58
   With a down payment of \$2,000, you
can pay at most \$17,189.58 for the car.
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. How much can you afford if you finance the
purchase over 60 months?
   =PV(rate,nper,pmt,pv,fv)
   =PV(12%/12,60,-400)
   CPT PV = \$17,982.02
\$400 can support a loan of
\$17,982.02
   With a down payment of
\$2,000, you can pay at most
\$19,982.02 for the car.
Chapter 4: Question 32
   A store offers two payment plans. Under the
installment method, you pay 25 percent down
and 25 percent of the purchase price in each of
the next three years. If you pay the entire bill
immediately, you can take a 10 percent discount
from the purchase price. Which is a better deal
if you can borrow or lend funds at a 5 percent
interest rate?
A store offers two payment plans. Under the installation
method, you pay 25 percent down and 25 percent of the
purchase price in each of the next three years. If you pay
the entire bill immediately, you can take a 10 percent
discount from the purchase price. Which is a better deal if
you can borrow or lend funds at a 5 percent interest rate?
 Compare the present value of the payments. Assume the
product sells for \$100.
Installment plan:               Pay in full:
•=PV(rate,nper,pmt,pv,fv)          Payment net of discount = \$90
•=PV(5%,3,-25)
•= \$68.08
•+ \$25 downpayment = \$93.08

Choose     the second plan for lower present value of payments.
Chapter 4: Question 39

   You’ve borrowed \$4,248.68 and agreed to pay back
the loan with monthly payment of \$200. If the interest
rate is 12 percent stated as an APR, how long will it
take you to pay back the loan? What is the effective
annual rate on the loan?
You’ve borrowed \$4,248.68 and agreed to pay back the loan
with monthly payment of \$200. If the interest rate is 12
percent stated as an APR, how long will it take you to pay
back the loan? What’s the effective annual rate on the loan?
   The loan repayment is an
annuity with PV equal to
\$4,248.68.
and the monthly interest rate is
1%.
   We need to solve for the
number of months, NPER
   =NPER(1%,-200,4248.68)
   = 24.
   Therefore, the solution is n =
24 months, or 2 years.
You’ve borrowed \$4,248.68 and agreed to pay back the loan
with monthly payment of \$200. If the interest rate is 12
percent stated as an APR, how long will it take you to pay
back the loan? What’s the effective annual rate on the loan?

   The effective annual rate on
the loan is:
   (1.01)12  1 =
    0.1268 =
   EAR = 12.68%
Chapter 4: Question 43
   You’ve borrowed \$100,000 to buy a condo. You
will repay the loan in equal monthly payments of
\$804.62 over the next 30 years. What monthly
interest rate are you paying on the loan? What is
the effective annual interest rate on the loan?
What rate is the lender more likely to quote on
the loan?
You’ve borrowed \$100,000 to buy a condo. You will repay
the loan in equal monthly payments of \$804.62 over the
next 30 years. What monthly interest rate are you paying on
the loan?
   =RATE(12*30,-804.62,100000)
   CPT %i = .75%
You’ve borrowed \$100,000 to buy a condo. You will repay
the loan in equal monthly payments of \$804.62 over the
next 30 years. What monthly interest rate are you paying on
the loan? What is the effective annual interest rate on the
loan?

   The effective annual rate is: (1.00750)12  1 =
   0.0938 =
   9.38%
You’ve borrowed \$100,000 to buy a condo. You will repay
the loan in equal monthly payments of \$804.62 over the
next 30 years. What monthly interest rate are you paying on
the loan? What is the effective annual interest rate on the
loan? What rate is the lender more likely to quote on the
loan?
   The effective annual rate is: (1.00750)12  1 =
   0.0938 =
   9.38%

   The lender is more likely to quote the APR
   0.750%  12 =
   9% which is lower than the effective annual rate of 9.38%
(and is required by the Truth-in-Lending Law
Chapter 4: Question 49
   A local bank will pay you \$100 a year for
bank today. If you plan to live forever,
what interest rate is the bank paying?
A local bank will pay you \$100 a year for your lifetime
if you deposit \$2,500 in a bank today. If you plan to
live forever, what interest rate is the bank paying?

   If you live forever, you will
that has present value
equal to: \$100/r
   Therefore: \$100/r =
\$2,500
   r = 4 percent
Chapter 4: Question 60
   You believe you will need to have saved
\$500,000 by the time you retire in 40 years in
order to live comfortably. If the interest rate is 6
percent per year, how much must you save each
year to meet your retirement goals?
You believe you will need to have saved \$500,000 by the
time you retire in 40 years in order to live comfortably. If
the interest rate is 6 percent per year, how much must you
save each year to meet your retirement goals?
   =PMT(6%,40,0,500000)
   =\$3,230.77
Chapter 4: Question 65
   An engineer in 1950 was earning \$6,000 per
year. Today, she earns \$60,000 per year.
However, on average, goods today cost 6 times
what they did in 1950. What is her real income
today in terms of constant 1950 dollars?
An engineer in 1950 was earning \$6,000 per year.
Today, she earns \$60,000 per year. However, on
average, goods today cost 6 times what they did in
1950. What is her real income today in terms of
constant 1950 dollars?

   \$60,000/6 =
   \$10,000.
   Her real income increased from \$6,000 to
\$10,000.
   Chapter 4 – Time Value of Money

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