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					Ch. 4 - The Time Value of
          Money
         Topics Covered
Future Values
Present Values
Multiple Cash Flows
Perpetuities and Annuities
Effective Annual Interest Rate
Inflation & Time Value
The Time Value of Money
Compounding and
Discounting Single
      Sums
             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 - Simple Interest
  Interest earned at a rate of 6% for five years on
  a principal balance of $100.

  Interest Earned Per Year = 100 x .06 = $ 6
            Future Values
Example - Simple Interest
 Interest earned at a rate of 6% for three years
 on a principal balance of $100.

                Today             Future Years
                           1         2    3
Interest Earned            6         6    6
Value           100      106       112 118

Value at the end of Year 3 = $118
             Future Values
Example - Compound Interest
  Interest earned at a rate of 6% for three years on
  the previous year’s balance.

Interest Earned Per Year =Prior Year Balance x
  .06
              Future Values
Example - Compound Interest
  Interest earned at a rate of 6% for three years
  on the previous year’s balance.

                  Today      Future Years
                             1        2    3
Interest Earned              6.00     6.36 6.74
Value             100      106.00 112.36 119.10

  Future Value of $100 compounded at 6% for three
  years = $119.10
Future Value of Single Cash Flow




    FV  PV  (1  r )   t
             Future Values
 FV  $100  (1  r )   t




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


FV  $100  (1  .06 )  $119 .10
                               3
      Future Values with Compounding
             7000       Interest Rates

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




             4000               15%

             3000

             2000

             1000

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




                                           Number of Years
Example: Mutual Fund Fees and
     Retirement Savings
Prof. Finance moves to a new university and has
$100,000 in retirement savings to invest (rollover)
into a new retirement account.
Prof. Finance wants to invest this money for 25
years into an indexed stock fund, which is
expected to return 9% annually.
Prof. has two choices: Vanguard Total Equity
Fund with a 0.4% annual expense fee and
Onguard Total Fencing Fund with an 1.2% annual
expense fee.
What is the difference in Prof. Finance’s expected
future retirement savings between the two funds?
            Present Values
 Present Value                     Discount Factor
Value today of a                   Present value of
  future cash                        a $1 future
     flow.                            payment.

                   Discount Rate
                 Interest rate used
                     to compute
                  present values of
                 future cash flows.
    Present Values

Present Value = PV

       Future Value after t periods
PV =              (1+r) t
 Example: Paying for Baby’s
           MBA
Just had a baby. You think the baby will
take after you and earn academic
scholarships to attend college to earn a
Bachelor’s degree. However, you want
send your baby to a top-notch 2-year MBA
program when baby is 25. You have
estimated the future cost of the MBA at
$85,000 for year 1 and $89,000 for year 2.
 Example: Paying for Baby’s
           MBA
Today, you want to finance both years of
baby’s MBA program with one payment
(deposit) into an account paying 8%
interest compounded annually.
How large must this deposit be?
    Time Value of Money
         (applications)

Value of Free Credit
Implied Interest Rates
Internal Rate of Return
Time necessary to accumulate funds
  Example : Finding Rate of
   Return or Interest Rate
A broker offers you an investment (a zero
coupon bond) that pays you $5,000 five
years from now for the cost of $3,700
today.
What is your annual rate of return?
    Important Time Value
       Relationships
Increasing interest rate and time
increases future value. POSITIVE
RELATIONSHIP.
Increasing interest rate and time
decreases present value. INVERSE
RELATIONSHIP.
    The Time Value of Money

      Compounding and
         Discounting
      Cash Flow Streams


0      1      2      3    4
            Perpetuities
Suppose you will receive a fixed payment
every period (month, year, etc.) forever.
This is an example of a perpetuity.
             Perpetuities
PV of Perpetuity Formula


           PV       C
                     r


C = cash payment
r = interest rate
      Perpetuities & Annuities
Example - Perpetuity
  You want to create an endowment to fund a
  football scholarship, which pays $15,000 per
  year, forever, how much money must be set
  aside today in the rate of interest is 5%?

         PV    15,PV 
                    000
                  .05
                      15, 000
                        .05
                                 $300 ,000
                                 $300 ,000
      Perpetuities & Annuities
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?


        PV     300, 000
                (1.05)3
                            $259,151
             Annuities
Annuity: a sequence of equal cash flows,
occurring at the end of each period. This
is known as an ordinary annuity.




0       1         2         3        4
PV                                   FV
Examples of Ordinary Annuities:
If you buy a bond, you will receive equal
semi-annual coupon interest payments
over the life of the bond.
If you borrow money to buy a house or a
car, you will pay a stream of equal
payments.
            Annuity-due
A sequence of periodic cash flows
occurring at the beginning of each period.




0       1          2          3        4
PV                                     FV
  Examples of Annuities-due
Monthly Rent payments: due at the
beginning of each month.
Car lease payments.
Cable TV and most internet service bills.
     Perpetuities & Annuities
PV of Ordinary Annuity Formula


     PV  C         1
                     r           1
                             r ( 1 r ) t   
C = cash payment
r = interest rate
t = Number of years cash payment is
  received
     Perpetuities & 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   
       Perpetuities & Annuities
Applications
 Value of payments
 Implied interest rate for an annuity
 Calculation of periodic payments
     Mortgage payment
     Annual income from an investment payout
     Future Value of annual payments
        FV   C  PVAF   (1  r )   t
     Perpetuities & Annuities
PV (and FV) of Annuity-dues = PV (or FV) of
 ordinary annuity x (1 + r) or BGN mode on


                                    (1 r)
 financial calculator.

     PV  C      1
                  r
                           1
                       r (1 r ) t


C = cash payment
r = interest rate
t = Number of years cash payment is
  received
Example: Invest Early in an IRA
How much would you have at age 65 if
you deposit $2,500 at the end of each
year in an account paying 9% annually
starting at:
    (A) age 41?
    (B) age 22?
            Why an IRA?
Imagine in the last example, you didn’t take
advantage of the tax-sheltered environment of
an IRA.
Your annual investment return would be taxed!
With a 28% tax rate, our annual after-tax return
would fall from 9% to 6.48% (=9%(1-.28)).
At age 65: I would have $135,519 vs. $191,975.
You would have $535,392 vs. $1,102,114: 52%
less!! The IRS killed Kenny,…!
    Example: Enjoying your
        Retirement
You go ahead and make the contributions
starting at age 22 in the last example, giving you
$1,102,114 at age 65.
You expect to live to age 85. So, you want to
make 20 annual withdrawals from your IRA
paying 9% at the beginning of each year starting
at age 65.
How large can this annual withdrawal be?
 More annuity fun, enjoying your
     release from baseball
Bob B. is released from the last year of his
guaranteed contract from a New York
baseball team. He is due $5.9 million from
the last year of this contract. Bob and the
team agree to defer the $5.9 million at 8%
interest for 15 years. At this time (15 years
from today), the team will begin the first of 15
equal annual payments at 8% interest.
The press is reporting the payments will total
$30 million. Are they correct?
          Non-Annual Interest
            Compounding
When interest is compounded more frequently than once
a year.
Important non-annual compounding terms and things to
know:
   Quoted Annual Rate, or Annual Percentage Rate (APR):
    Stated nominal annual rate before compounding.
   Effective Annual Rate (EAR): the actual (effective) annual
    interest rate earned or paid.
   Periodic Interest Rate: the interest rate paid or charged
    each interest compounding period = quoted rate/m, where
    m = number of compounding periods per year.
       Effective Interest Rates
example
  Given a monthly rate of 1%, what is the Effective
  Annual Rate(EAR)? What is the Annual
  Percentage Rate (APR)?
                     12
      EAR = (1 + .01) - 1 = r
                     12
      EAR = (1 + .01) - 1 = .1268 or 12.68%


      APR = .01 x 12 = .12 or 12.00%
   FV and PV with non-annual
     interest compounding
  n = number of years
  m = number of times interest is paid per
  year
  APR = nominal annual rate (APR)
  APR/m = periodic rate
Single CF
FVnm = PV(1+ARR/m)nm
PV = FVnm/(1+APR/m)nm
      Non-annual annuities
Ordinary:
 PV = C(PVAFAPR/m,nm)
 FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm

Annuity-Due:
 PV = C(PVAFAPR/m,nm)(1+APR/m)
 FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm+1
    Example: Low Rate or Rebate?

The Frontier family want to buy a sport ut (SUV). They decide
on a 4-wheel drive Jeep Grand Cherokee. The purchase price
with tax of the vehicle is $32,500. The Frontiers have $4,000
as a down payment.
Jeep offers the choice of two incentives on the 4-door Grand
Cherokee.
   0% APR Financing for 60 months, or
   $3,000 rebate which would be applied toward the purchase price.
    If the Frontiers elect to take the rebate, they can get 4.49% APR
    financing for 60 months.
   Question: Which incentive would give the Frontiers the lowest
    monthly payment?
  Example: The $200 national ISP
signup credit: good deal for whom?
 A national ISP all provide $200 for new customers to use
 at a particular electronics store chain if they sign-up for a
 2-year internet service contract at $21.95/month.
 What interest rate (APR) are you paying on this “free
 money” if you wanted internet service and could get it for
 free? (200 PV, -21.95 PMT, 24 N 0 FV, CPT I/Y =
 9.8%/month x 12= 117.8% APR!!)
 What interest rate (APR) are you paying on this “free
 money” if you wanted internet service and could get it for
 $9.95/month?(-12 PMT CPT I/Y = 3.15%/mo x 12 =
 37.8%! Thanks, but no thanks!
                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
                          1+ nominal interest rate
 1  real interest rate =     1+inflation rate


   approximation formula


Real int. rate  nominal int. rate - inflation rate
                  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+.050
1  real int erestrat e = 1+.022       Savings

1  real int erestrat e = 1.027          Bond

real int erestrat e = .027 or 2.7%

         ion
Approximat = .050 - .022 = .028or 2.8%
Example: Real retirement income
Going back to your retirement in 43 years,
you expect 3% inflation along with your
9% nominal investment rate annually and
want to withdraw $32,000 in real terms at
the beginning of each year for 20 years
once you retire.
How will this change your retirement
saving plans?

				
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posted:11/14/2010
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