Time Value of Money

1
Time Value of Money

• Goals
• Calculate the value today of cash flows expected
  in the future.
• Calculate the amount of money needed to today to
  generate some future value of money.

2
Time Value of Money

• Present values versus future values
• Interest rate conversions
• Annuities
• Perpetuities
• Growing cash flows
• Amortization (loan) payments
• Pricing bonds and stock

3
Future Values
0
1
2
C0
FV1 C0(1r)
FV2 FV1(1r) C0(1 r)2
FVt C0 (1 r)t How much would 70,000 be
worth in 14 years _at_ 7½? FV14 70,000(1.0725)14
186,492
4
Present Values
0
1
2
PV1 C2/(1 r)
C2
PV PV1/(1 r) C2/(1 r)2
PV Ct / (1 r)t What is the maximum price you
would pay today for a machine that generates a
single cash flow of 2,000,000 in 20 years?
Interest rate is 8 PV 2,000,000/(1.08)20
429,096
Note that this is the same formula as for FV
What if you sell this machine?
5
Multiple Cash Flows
C2
CT
C1
0
1
2
T

• Ct in year t, cash flows last for T years
• PV C1/(1r) C2/(1r)2 CT/(1r)T
• PV ? Ct / (1r)t

T
t 1
6
Multiple and Infinite Identical Cash Flows
C
C
C
C
?
0
1
2
T

• Annuity Finite stream of identical cash flows
• Perpetuity Infinite stream of identical cash
  flows
• Identical separated by an identical growth rate
  (g0 in this example)

7
Perpetuities and Annuities

• Perpetuity
• Annuity

How about future values?
Understand formula using timelines
8
Growing Cash Flows
Growing Perpetuity
C
C(1g)
C(1g)2
C(1g)3
0
1
2
3
4
Growing Annuity

• Growing Perpetuity Growing Annuity

What about a (growing) perpetuity starting in
year 4?
2 types of Time Value Formulae I A single cash
flow moved multiple time periods II Multiple
cash flows moved a single time period
9
Quick Quiz

• In 1934, the first edition of a book described by
  many as the bible of financial statement
  analysis was published. Security Analysis by
  Grham and Dodd has proven so popular among
  financial analysts that it has never been out of
  print.
• According to an item in The Wall Street Journal,
  a copy of the first edition was sold by a rare
  book dealer in 1996 for 7,500. The original
  price of the first edition was 3.37. What is the
  annually compounded rate of increase in the value
  of the book?

10
Future Values and Multiple Cash Flows

• Example Suppose your rich uncle offers to help
  pay for your business school education by giving
  you 5,000 each year for the next three years
  beginning today (year 0). You plan to deposit
  this money into an interest-bearing account so
  that you can attend business school six years
  from today. Assume you earn 4.25 per year on
  your account. How much will you have saved in
  six years?

(1.0425)6 1.2837 6418.39
(1.0425)5 1.2313 6156.73
(1.0425)4 1.1811 5905.74
18,480.86
11
Present Value

• Want to be a millionaire? No problem! Suppose you
  are currently 21 years old, and can earn 10
  percent on your money (about what the typical
  common stock has averaged over the last six
  decades - but more on that later). How much must
  you invest today in order to accumulate
  1 million by the time you reach age 65?
• FV PV ( 1 r )t ? PV FV / ( 1 r )t
• FV 1 million, r 0.10, and t 44 PV
  15,092

12
Present Values Multiple Periods

• Suppose you need 10,000 in three years. If you
  earn 5 each year, how much money do you have to
  invest today to make sure that you have the
  10,000 when you need it?
• PV 10,000 / (1.05)3 PV 8,638.38
• What is the maximum price youd be willing to pay
  for a promise to receive a 25,000 payment in 30
  years? You can invest your money somewhere else
  with similar risk and make a 24 annual return.
• PV 25,000 / (1.24)30 PV 39.38

13
Investing for More than One PeriodPresent
Values and Multiple Cash Flows

• Suppose your firm is trying to evaluate whether
  to buy an asset. The asset pays off 2,000 at
  the end of years 1 and 2, 4,000 at the end of
  year 3 and 5,000 at the end of year 4. Similar
  assets earn 6 per year. How much should your
  firm pay for this investment?
• Rule Discount cash flows to the present, one set
  of cash flows at a time and then add them up.

14
1 / (1.06) 1886.79
1 / (1.06)2 1779.99
1 / (1.06)3 3358.48
1 / (1.06)4 3960.73
10,985.73
15
Finding the Number of Periods

• Sometimes we will be interested in knowing how
  long it will take our investment to earn some
  future value. Given the relationship between
  present values and futures value, we can also
  find the number of periods. We can solve for the
  number of periods by rearranging the following
  equation
• FV PV (1 r)t ? FV / PV (1 r)t
• ? ln(FV / PV) ln (1 r)t
• ? ln(FV) - ln(PV) t ln (1 r)
• ? t (ln(FV) - ln (PV)) /
  ln (1 r)

16
Finding the Number of Periods

• How long would it take to double your money at
  5?
• t (ln(FV) - ln (PV)) / ln (1 r)
• Approximately 14 years and 2 months
• Rule of thumb Rule of 72
• How long for your money to double at 9?
• How long for your money to triple at 11?

17
PV (Annuity) Calculation

• Assumes annuity payment occurs at the end of the
  period.
• Cash flows of an annuity are all the same
• Period covered by the interest rate r must
  correspond to the frequency of the annuity
  payment
• The present value of an annuity of C dollars per
  period for t periods when the rate interest rate
  is r is

18
Present Value of an Annuity Example

• Suppose you need 25,000 each year for business
  school. You need the first 25,000 at the end of
  12 months and the second 25,000 at the end of 24
  months. If you can earn 8 per year on your money
  how much do you need today to be able to afford
  business school?

19
Future Value of an Annuity
Suppose you plan to retire ten years from today.
You plan to invest 2,000 a year at the end of
each of the next ten years. You can earn 8 per
year on your money. How much will your
investment be worth at the end of the second
year? How much will it be after ten years?
20
Example Finding t

• Q. Suppose you owe 2000 on a VISA card, and the
  interest rate is 2 per month. If you make the
  minimum monthly payments of 50, how long will
  it take you to pay it off?
• A. A long time
• 2000 (50/0.02) x 1 - ( 1 / 1.02)t
• 2000 2500 1 - (1 / 1.02)t
• 2000/2500 -1 - (1 / 1.02t) - 0.2 - (1
  / 1.02t)
• 0.2 1.02t 1 1.02t 5 t ln(5) /
  ln(1.02)
• t ________ months, or about_______ years

81
6.5
21
Perpetuities

• A perpetuity is an annuity in which the stream of
  cash flows continues forever.
• Suppose we are examining a perpetuity that costs
  1,000 and offers a 12 rate of return. The cash
  flow each year is 1,0000.12 120. More
  generally, the present value of a perpetuity
  multiplied by the rate of interest must equal the
  cash flow

22

• The present value of a perpetual cash flow stream
  has a finite value (as long as the discount rate,
  r, is greater than 0). Heres a question
• How can an infinite number of cash payments have
  a finite value?
• Heres an example related to the question above.
  Suppose you are considering the purchase of a
  perpetual bond. The issuer of the bond promises
  to pay the holder 100 per year forever. If your
  opportunity rate is 10, what is the most you
  would pay for the bond today?
• One more question Assume you are offered a bond
  identical to the one described above (no
  principal repayment, just interest payments), but
  with a life of 50 years. What is the difference
  in value between the 50-year bond and the
  perpetual bond?

23
Preferred Stock as a Perpetuity

• Preferred stock is an example of a perpetuity.
• The holder of preferred stock is promised a
  fixed cash dividend every period (usually
  quarter). It is called preferred because the
  dividend is paid before common stock dividends
  but after interest payments.
• Suppose GM wants to sell preferred stock at 100
  per share. A very similar issue of preferred
  stock outstanding has a price of 40 per share
  and offers a dividend of 1 every quarter.
• What dividend will GM have to offer if the
  preferred stock is to sell for 100?
• P2C2/r ? 401/r ? r0.025 ? P1C1/r ?
  100C1/0.025
• C1 2.50

24
Relation between annuities and perpetuities

• How to remember formulae for annuities?
• Difference between 2 perpetuities!!!
• PV(annuity) C/r minus discounted C/r
• FV(annuity) future value C/r minus C/r
• Draw timelines!

25
Growing Annuities and Perpetuities

• Cash flows grow g per time period
• C cash flow in first time period (t 1)
• If r g then PV TC / 1r
• Example What is the PV of a 10 payment, growing
  at 3 per year, for 4 years, with r 10?
• For a perpetual stream, growing at 3, we get C
  / (r - g) 10 / (0.07) 142.86

26
Comparing Interest Rates The Effect of
Compounding

• Stated or quoted rate The annual rate before
  considering any compounding effects, such as 10
  compounded semiannually.
• Effective Annual Rate (EAR) The rate, on an
  annual basis, that reflects compounding effects,
  such as 10 compounded semi-annually gives an
  effective rate of 10.25.

27
Effective Annual Rates

• Why is it important to work with EARs? Suppose
  you are interested in buying a new car. You have
  shopped around for loan rates and come up with
  the following three rates
• Bank A 12 compounded monthly
• Bank B 12 compounded quarterly
• Bank C 12.25 compounded annually
• Which is the best rate? We use effective annual
  rates to compare the above lending rates.

28
Calculating EARs

• What is the EAR for 12 compounded quarterly?
• Step 1 Divided the quoted rate by the number of
  times that interest is compounded during the
  year.
• Step 2 Add 1 to the result and raise it to the
  power of the number of times interest is
  compounded during the year.
• Step 3 Subtract 1 from your answer in Step 2.

29
Computing Present Values Using EARs

• What is the present value of 100 to be received
  at the end of two years at 10 compounded
  quarterly?
• Step 1 Calculate the effective annual rate
• Step 2 Calculate the present value of the cash
  flows.

EAR(1(0.10/4))4 - 110.38
PV 100 / (1.1038)2 82.07
30
Annual Percentage Rates (APRs)

• Annual Percentage Rate The rate per period times
  the of periods per year, making it a quoted or
  stated rate.
• What is the annual percentage rate if the
  interest rate is 1.25 per month?
• Example If you look at the credit agreement for
  your credit card, you will see that an annual
  percentage rate is charged. But what is the
  actual rate you pay on such a card if you do not
  make your payment?

31
APRs and EARs

• An APR of 18 with monthly payments is 0.015 or
  1.5 per month. What is the EAR?

EAR (1 (0.18/12))12 - 1 19.56
32
Compounding Periods, EARs, and APRs

• Compounding Number of times Effective
• period compounded annual rate
• Year 1 10.00000
• Quarter 4 10.38129
• Month 12 10.47131
• Week 52 10.50648
• Day 365 10.51558
• Hour 8,760 10.51703
• Minute 525,600 10.51709

33
Converting Interest Rates Summary

• Rule Convert Interest Rate to match the Cash
  Flow Periods
• Compounding APR or EAR?
• Periodic rate (i.e., monthly) APR / m
• m number of periods per year
• APR periodic rate m
• Effective Annual Rate
• EAR 1 (APR/m)m 1
• EAR 1 (periodic rate)m 1

34
Loan Payments

• You have decided to buy a new four-wheel drive
  sports vehicle and finance the purchase with a
  10-year loan. The loan is for 33,500. Interest
  starts accruing when the loan is taken. The first
  loan payment is one-month after the interest
  starts accruing. The interest rate on the loan
  is 8.5 (APR) per year for the ten-year period.
  
• What type of security is the series of loan
  payments?
• What is the present value of the loan?
• What discount rate should be used in the present
  value calculation?
• Calculate the monthly loan payment.
• How much have you paid off after 2 months?

35
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Option 1 Rebate Option 2 5-
Financing
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the 5 financing or 500 rebate?
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