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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
    value of your prize?
     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
  value of your prize?
=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
    your monthly
    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
   Your monthly payments of
    $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.
   Payments are made monthly
    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
    your lifetime if you deposit $2,500 in a
    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
    receive a $100 perpetuity
    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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