MBA 643 Managerial Finance Lecture 4: Time Value of Money

1
MBA 643Managerial FinanceLecture 4 Time Value
of Money

• Spring 2006
• Jim Hsieh

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Goals

• Calculate the amount of investment needed today
  to generate some (positive) value in the future
• Example savings account
• Calculate the current value of cash flows
  expected (and known) in the future
• Example bond prices
• Convention
• Today will always be donated time 0.
  Successive time points will be denoted time 1,
  time 2, , etc.

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Future Values

• Definition The future value of a known current
  cash flow is obtained by compounding at a
  riskless rate or spot interest rate appropriate
  for the future period
• FVt C0(1r)t
• Example 1 How much wold 70,000 be worth in 14
  years _at_7.5?

0
1
2
FV2FV1(1r)C0(1r)2
FV1C0(1r)
C0
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Present Values

• Definition The present value of a known future
  cash flow is obtained by discounting at a
  riskless rate or spot interest rate appropriate
  for the future period
• PV Ct/(1r)t
• Example 2 What is the maximum price you would
  pay today for a 2-year pure discount bond (also
  called a zero-coupon bond or Treasury strip) with
  interest rate r0.08 and face value 100?

0
1
2
C2
PV1C2/(1r)
PVPV1/(1r)C2/(1r)2
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Additional Examples

• Example 3 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?
• Example 4 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.

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The Power of Compounding-- Longer compounding
period with higher rates
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The Power of Compounding-- Compounding more often

• (Suppose you have 1,000 now, how much will you
  have after 1 year? r0.1)
• Annual Compounding The interest is added to your
  investment once a year.
• Semiannual Compounding The interest is added to
  your investment twice a year.

0
1
1
0
2
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The Power of Compounding (contd)-- Compounding
more often

• Monthly Compounding The interest is added to
  your investment 12 times a year.
• Compounding n times The interest is added to
  your investment n times a year.
• A Generalized Formula If you invest PV in one
  account and the interest rate (r) is compounded n
  times per year, how much will you have after t
  years?

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The Power of Compounding (contd)-- Example 5

• Your bank representative gives you a quote of the
  interest rate in the savings account as 4
  compounded semiannually. If you deposit 100 at
  the beginning of the year, how much will you have
  in your account after 2 years? After 10 years?
  What if the interest rate is compounded
  quarterly? Monthly?

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Quoted Interest Rates vs. Effective Annual Rates
(EAR)

• Example 6 Suppose you are trying to open a
  savings account and 3 banks quote you the
  following rates
• Bank Annie 15 compounded daily (365 days a
  year)
• Bank Booboo 15.5 compounded quarterly
• Bank Charming 16 compounded annually
• Which of these is the best?
• Two different rates
• The quoted (stated) interest rate
• The EAR The interest rate expressed as if it
  were compounded once per year. If n is the
  number of times the interest is compounded per
  year, then
• EAR 1 (Quoted rate/n)n - 1

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Quoted Interest Rates vs. Effective Annual Rates
(EAR)

• Example 6 (contd)
• Comments
• The highest quoted rate is not necessarily the
  best.
• Compounding during the year can lead to a
  significant difference between the quoted rate
  and the effective rate.
• The quoted rate is quoted by financial
  institutions, but the effective rate is what you
  really get or what you really pay.

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Annual Percentage Rate (APR)

• Lenders are required by the Truth-in-lending laws
  in the U.S. to disclose an APR on almost all
  consumer loans.
• Example 7 If your bank charges you 1.6 per
  month on your credit card, then the APR must be
  reported as 1.612 19.2. Thus, an APR is
  actually a quoted rate. To compare loans with
  different APRs, you still need to convert APRs to
  EARs. Remember the EAR is the rate you actually
  pay.
• Question What is the EAR on your credit card?

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Multiple Cash Flows

• We have a cash flow stream, C1, C2, , CT, for T
  years
• Rule Discount (or compound) all cash flows to
  the present (or the future) and then add them up.
• PV0 C1/(1r) C2/(1r)2 CT/(1r)T
• FVT C1(1r)T-1 C2(1r)T-2 CT

0
1
2
T


C1
C2
CT
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Investing for More than One PeriodPresent
Values and Multiple Cash Flows

• Example 8 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?

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Multiple Cash Flows and Future Value

• Example 9 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 (year6)?

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Present Values and Multiple Cash Flows

• The price of an asset is the present value of the
  CFs produced by the asset.
• Ex. Stock ? dividends Bond ? interests and
  principal
• Two cash flows with the same PV are economically
  equivalent.
• We always prefer the CF stream with the highest
  PV.
• Example 10 Suppose r0.1. Which investment
  would you take?

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Perpetuity Multiple and Infinite Identical Cash
Flows

• PV(Perpetuity) C/(1r) C/(1r)2
• C/r
• Example 11 Suppose Martin Co. wants to sell
  preferred stock at 60 per share and offers a
  dividend of 3 every quarter. What rate of
  return will be for Martins preferred stock?

0
1
2
T
?


C
C
C
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Annuity Finite Stream of Identical Cash Flows

• PV(Annuity) C/(1r) C/(1r)2 C/(1r)T
• FV(Annuity) PV(Annuity)(1r)T

0
1
2
T

C
C
C
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Annuity Example 12 (Car Loan Payments)

• You want to buy a new sports coupe for 48,250,
  and the finance office at the dealership has
  quoted you a 9.8 APR loan for 60 months. What
  will your monthly payments be? What is the
  effective annual rate on this loan?

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Growing Perpetuity Multiple and Infinite Cash
Flows That Grows at a Fixed Rate (g)

• PV(Grow. Perp.) C/(1r) C(1g)/(1r)2
• C/(r-g)

0
1
2
T
?


C(1g)T-1
C
C(1g)
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Growing Annuity Multiple and Finite Cash Flows
That Grows at a Fixed Rate (g)

• PV(Grow. Annuity)
• C/(1r) C(1g)/(1r)2 C(1g)T-1/(1r)T

0
1
2
T

C(1g)T-1
C
C(1g)
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Multiple Growing Cash Flows Example 13

• A wealthy GMU-grad entrepreneur wishes to endow a
  chair in finance at the SOM. The first payment
  is 50,000, occurring at the end of the first
  year. The amount grows at 3 afterward. The
  rate of interest is 10. If she wants to provide
  an annual payment perpetually, what is the amount
  that must be set aside today?

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Example 13 (contd)

• If she wants to provide an annual payment for the
  next 20 years only, what is the amount that must
  be set aside today?

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Additional Examples Example 14 (Financing or
Rebate?)

Option 1 Rebate Option 2 5
Financing
SALE! SALE!
5 FINANCING OR 500 REBATEFULLY LOADED MUSTANG
only 10,999 5 APR on 36 month loan. If
United Bank is offering 10 car loans, should you
choose the 5 financing or 500 rebate?
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Additional Examples Example 15 (Mortgage)

• You have just signed closing documents associated
  with the purchase of a house for 350,000, and
  have arranged a 30-year, fixed rate mortgage bank
  loan at a 7 stated (quoted) annual rate.
  Because you have made a 30 down payment, the
  loan amount is 70 of the purchase price.
  Mortgage payments will be made at the end of each
  month, and your first payment will be due exactly
  one month from today. What is your monthly
  mortgage payment?

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Additional Examples Example 16 (Savings for
retirement)

• Your current salary is 60,000 per year, and is
  expected to grow by 5 per year until retirement.
  In 30 years (t30) you plan to retire and hope
  that at t30 to have amassed a 3 million
  retirement balance. If you invest part of your
  income each year in an account earning 9 per
  year, compounded annually, how much of your
  income must be invested to attain your retirement
  goal? Note that your first deposit will occur one
  year from today (t1) and your last deposit
  occurs at t30.

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Additional Examples Example 17 (College
planning)

• Your child will start school 18 years from today
  (t18). You have decided to start a college
  savings plan. You want to follow an aggressive
  investment strategy and plan to invest a lump sum
  at the end of each year for the next seventeen
  years (t117) into the Trust Me Mutual Fund with
  a discount rate of 10. At the end of the
  seventeenth year, you will deposit the money into
  a savings account that earns 4 annually. You
  will make tuition payments from this account. You
  estimate that tuition and room and board will
  cost 40,000 per year for four years. Assume the
  following
• 1. The expected return on the mutual fund will
  be the same each year for the next seventeen
  years.
• 2. The first deposit to the mutual fund will be
  made one year from today (t1).
• The first tuition payment is made 18 years from
  today (t18).
• Given your investment strategy, how much will you
  need to deposit in the Trust Me Mutual Fund each
  year?

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Example 17 (Contd)
0
1
17
18
19
20
21
2


C
C
P
P
P
P
C
r0.1
r0.04
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Exercise (It pays to start early)

• Mei Xiang and Tian Tian have different investment
  strategies for their retirement. Mei deposits
  4,000 in her IRA each year from t21 to 41 (20
  deposits) and keeps her savings in the account
  until t60. Tian starts his investment (4,000)
  at t31 until t60 (30 deposits). Assume both
  accounts earn 10 rate of return. How much will
  they have when they reach the age of 60?