Download Chapter 4

Chapter 4
The Time Value
of Money
Copyright © 2011 Pearson Prentice Hall. All rights reserved.
Chapter Outline
4.1 The Timeline
4.2 The Three Rules of Time Travel
4.3 Valuing a Stream of Cash Flows
4.4 Calculating the Net Present Value
4.5 Perpetuities, Annuities, and Other
Special Cases
2-2
Example 4.1
You have just taken out a five-year loan from a bank
to buy an engagement ring. The ring costs $5000.
You plan to put down $1000 and borrow $4000.
You will need to make annual payments of $1000
at the end of each year. Show the timeline of the
loan from your perspective. How would the
timeline differ if you created it from the bank’s
perspective.
2-3
4.2 The Three Rules of Time Travel
•
•
•
•
Comparing and Combining Values
Moving Cash Flows Forward in Time
Moving Cash Flows Back in Time
Applying the Rules of Time Travel
2-4
Table 4.1 The Three Rules of Time Travel
2-5
Chapter 4, problem 3a, 4a
3a. Calculate the future value of $200 in 5 years at
an interest rate of 5% per year.
4a. What is the present value of $10,000 received
12 years from today when the interest rate is 4%
per year?
2-6
The Power of Compounding
This graph
illustrates the future
value of $1000
invested at a 10%
interest rate.
Because interest is
paid on past
interest, the future
value grows
exponentially—after
50 years, the money
grows 117-fold and
in 75 years (only 25
years later), it is
1272 times larger
than the value
today.
2-7
Present Value of a Stream of Cash Flows
Chapter 4, problem 13
You have a loan outstanding. It requires making three
annual payments at the end of the next three years of
$1000 each. Your bank has offered to allow you to
skip making the next two payments in lieu of making
one large payment at the end of the loan’s term in
three years. If the interest rate on the loan is 5%, what
final payment will the bank require you to make so that
it is indifferent between the two forms of payment?
2-8
Example 4.5 Net Present Value of an
Investment Opportunity
Chapter 4, problem 15
Marian Plunket owns her own business and is
considering an investment. If she undertakes the
investment, it will pay $4000 at the end of each of
the next three years. The opportunity requires an
initial investment of $1000 plus an additional
investment at the end of the second year of
$5000. What is the NPV of this opportunity if the
interest rate is 2% per year? Should Marian take
it?
2-9
4.5 Perpetuities, Annuities, and Other
Special Cases
• Perpetuities
• Historical Examples of Perpetuities
• Common Mistake: Discounting One Too Many
Times
• Annuities
• Growing Cash Flows
– Growing Perpetuity
– Growing Annuity
2-10
Chapter 4, problem 18
The British government has a consol bond outstanding
paying £100 per year forever. Assume the current
interest rate is 4% per year.
a. What is the value of the bond immediately after a
payment is made?
b. What is the value of the bond immediately before a
payment is made?
2-11
Present Value of an Annuity
Suppose that we now have a stream of equal cash
flows of $100 that last for 10 years (and not
forever). What is the PV of such a stream if interest
rates are 4%?
2-12
Chapter 4, problem 20
You are head of the Schwartz Family Endowment for
the Arts. You have decided to fund an arts school in
the San Francisco Bay area in perpetuity. Every five
years, you will give the school $1 million. The first
payment will occur fiver years from today. If the
interest rate is 8% per year, what is the preset value
of your gift?
2-13
Growing Perpetuity and Annuity
2-14
Chapter 4, problem 26
You work for a pharmaceutical company that has
developed a new drug. The patent on the drug will last
17 years, You expect that the drug’s profits will be $2
million in its first year and that this amount will grow at
a rate of 5% per year for the next 17 years. Once the
patent expires, other pharmaceutical companies will
be able to produce the same drug and competition will
likely drive profits to zero. What is the present value of
the new drug if the interest rate is 10% per year?
2-15
4.8 Solving for Variables Other Than
Present Value or Future Value
Chapter 4, problem 48
You are shopping for a car and read the following
advertisement in the newspaper: “Own a new
Spitfire! No money down. Four annual payment of
just $10,000.” You have shopped around and
know that you can buy a Spitfire for cash for
$32,500. What is the interest rate the dealer is
advertising (what is the IRR of the loan in the
advertisement?). Assume that you must make the
annual payments at the end of each year.
2-16
4.8 Solving for Variables Other Than
Present Value or Future Value
Your grandmother bought an annuity from Rock
Solid Life Insurance Company for $200,000 when
she retired. In exchange for the $200,000, Rock
Solid will pay her $25,000 per year until she dies.
The interest rate is 5%. How long must she live
after the day she retired to come out ahead (that
is, to get more in value than what she paid in)?
2-17
Chapter 5
Interest Rates
Copyright © 2011 Pearson Prentice Hall. All rights reserved.
5.1 Interest Rate Quotes and Adjustments
• The Effective Annual Rate
• Adjusting the Discount Rate to Different Time
Periods
• Annual Percentage Rates
• Application: Discount Rates and Loans
– Computing Loan Payments
– Computing the Outstanding Loan Balance
2-19
Table 5.1 Effective Annual Rates for a 6%
APR with Different Compounding Periods
2-20
Interest Rate Quotes
Chapter 5, problem 1
Your bank is offering you an account that will pay
20% interest in total for a two-year deposit.
Determine the equivalent discount rate for a
period length of
a. Six months
b. One year.
c. One month.
2-21
Chapter 5, problem 14
You have decided to refinance your mortgage. You
plan to borrow whatever is outstanding on your
current mortgage. The current monthly payment is
$2356 and you have made every payment on time.
The original term of the mortgage was 30 years,
and the mortgage is exactly four years and eight
months old. You have just made your monthly
payment. The mortgage interest rate is 6 3/8%
(APR). How much do you owe on the mortgage
today?
2-22
Chapter 5, problem 22
You need a new car and the dealer has offered you
a price of $20,000, with the following payment
options: (a) pay cash and receive a $2000 rebate,
or (b) pay a $5000 down payment and finance the
rest with a 0% APR loan over 30 months. But
having just quit your job and started and MBA
program, you are in debt and you expect to be in
debt for at least the next 2.5 years. You plan to use
credit cards to pay your expenses; luckily you have
one with a low (fixed) rate of 15% APR (monthly).
Which payment option is best for you?
2-23
More on interest rates
Interest rates are typically quoted as APR “r%”
per year, compounded n times per year. That has
an interpretation that you will earn an interest rate
of r/n per period, for n periods. In introductory
finance we learn that if you invest $1, in 1 year
you will have.
 r
1  
 n
n
3-24
Interest rates
We also say, that the EAR (Effective interest rate)
is
n
 r
1    1
 n
The bottom line is that, the more compounding
periods in a year, the larger the EAR.
3-25
Compounding Periods for 10% APR
Compounding
period
• Year
• Quarter
• Month
• Week
• Day
• Hour
• Minute
Number of times
compounded
1
4
12
52
365
8,760
525,600
Effective
annual rate
10.00000%
10.38129
10.47131
10.50648
10.51558
10.51703
10.51709
26
Continuous compounding
If compounding is continuous – that is , if interest
accrues every instant, then we can use the
exponential function to compute future values. For
example with a 10% continuous compound rate,
after 1 year we will have a future value of
Thus, our EAR=10.5171%
e 0.1  $1.105171
3-27
Continuous compounding
 r
rt
e  lim 1  
n 
 n
nt
By definition
so that is why we can use exp() to calculate future
value or EAR. Note also that if you know how
much you earned from a $1 investment (i.e., the
future value) – you can easily figure out the
corresponding continuously compounded rate of
return of your investment by taking the natural log.
rt
ln(
e
)
rt
ln( e )  rt or r 
t
3-28
Example: continuous rate
• Suppose you have a zero-coupon bond that
matures in 5 years. The price today is $62.092 for
a bond that pays $100.
(1) What is the annually compounded rate of return
(EAR)?
(2) What is the continuously compounded rate of
return?
3-29
Two nice properties of continuously
compounded returns.
1. Additive property of rate being in the power
e e e
x
y
x y
2. Symmetry for increases and decreases
Result: When trying to explain dynamics of asset
prices (especially of derivatives), it is much
simpler and more intuitive to use the quoted
continuous rate.
3-30
Examples
• If the continuous rate is 10%, what will be the
future value of $100 in 3 years and 6 months.
• What is the present value of a $500 payment that
will be paid 2 years from now, if the continuous
rate is 15%.
3-31
5.3 The Determinants of Interest Rates
•
•
•
•
Inflation and Real Versus Nominal Rates
Investment and Interest Rate Policy
The Yield Curve and Discount Rates
The Yield Curve and the Economy
2-32
The Term Structure of Interest Rates
• The term structure describes the relationship between
investment length and nominal interest rate on
government bonds (in annual terms). This is often
referred to as the relationship between interest rate and
zero-coupon default free bonds because zero-coupon
bonds pay all of their cash flow at maturity.
• In essence, the term-structure shows the interest rates
that exist in the economy as a function or time, when
there is no risk and other affects (e.g., liquidity of the
bond).
• In regular times, the term structure should be upward
sloping – because the only things that affect it should be
the real interest rate, expected inflation, and interest rate
risk.
8-33
8-34
8-35
Figure 5.1 U.S. Interest Rates
and Inflation Rates,1960–2009
2-36
Figure 5.2 Term Structure of Risk-Free U.S. Interest
Rates, November 2006, 2007, and 2008
2-37
Figure 5.3 Short-Term Versus Long-Term
U.S. Interest Rates and Recessions
2-38
Chapter 5, problem 32
Suppose the current one-year interest rate is 6%. One year from
now, you believe the economy will start to slow and the one-year
interest rate will fall to 5%. In two years, you expect the economy
to be in the midst of a recession, causing the Federal Reserve to
cut interest rates drastically and the one-year interest rate to fall
to 2%. The one year interest rate will then risk to 3% the following
year, and continue to rise by 1% per year until it return to 6%,
where it will remain from then on.
a. If you were certain regarding these future interest rate
changes, what two-year interest rate would be consistent
with these expectations.
b. What current term structure of interest rates, for terms of 1 to
10 years, would be consistent with these expectations?
c. Plot the yield curve in this case. Howe does the one-year
interest rate compare to the ten year interest rate.
2-39
5.3 Risk and Taxes
2-40