Transcript Chapter 4 Future Value, Present Value and Interest Rates

Chapter 4
Future Value,
Present Value and
Interest Rates
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2008
Future Value, Present Value
and Interest Rates:
The Big Questions
1. How can we compare payments at different
dates?
2. What is an interest rate?
3. What is a bond?
4. What is the relationship between interest
rates and inflation?
4-2
Future Value, Present Value
and Interest Rates:
Roadmap
•
•
•
•
•
Future Value
Present Value
Internal Rate of Return
Bond Basics
Real vs. Nominal Interest Rates
4-3
A Brief History of Lending
• Lenders despised throughout history.
• Credit is so basic that we evidence of
loans going back 5 thousand years.
• Hard to imagine an economy without it.
• Yet, people still take a dim view of
lenders because they charge interest
4-4
Lending and Interest
• Why do lenders charge interest?
• The existence of alternatives means
that lenders face an opportunity cost.
• Borrowers rent resources from lenders.
Interest is the rent.
4-5
Valuing Monetary Payments
Now and in the Future
• Fundamental to studying financial
instruments is the ability to value
payments made at different times.
• Tools:
Future value and Present Value
4-6
Future Value:
Definition
The value on a future date of an
investment made today.
If you invest $100 today at 5 percent interest
per year, in one year you will have $105.
4-7
Future Value:
One Year
Future Value =
Present Value + Interest
FV = PV
$105 = $100
+
PVxi
+ $100x(0.05)
4-8
Future Value:
One Year
FV = PV + PVxi
= PVx(1+i)
Future Value in one year =
Present Value x (one plus interest rate)
4-9
Future Value:
Two Years
$100+$100(0.05)+$100(0.05) + $5(0.05) =$110.25
Present Value of the Initial Investment
+ Interest on the initial investment in the 1st Yr
+ Interest on the initial investment in the 2nd Yr
+ Interest on the Interest from the 1st Yr in the 2nd Yr
= Future Value in Two Years
4-10
Future Value:
General Formula
Future value of an investment of PV
in n years at interest rate i
FVn = PVx(1+i)n
(Remember: The interest rate is
measured is a decimal so if 5%, i = .05)
4-11
Future Value:
$100 Investment at 5% Annual Interest
After 10 years, $100 as grown to $162.89 – that’s the initial investment of
$100 plus interest of $62.89. Ignoring compounding, you would have just
multiplied 5 percent times 10 years and gotten $50.
The difference of $12.69 comes from compounding.
4-12
Future Value:
Caution
Time (n) & interest rate (i)
must be in same time units
If i is at annual rate, then n must be in years.
Future Value of $100 in 18 months at
5% annual interest rate is
FV = 100 x (1+.05)1.5
4-13
• Invest $100 at 5% annual interest
• How until you have $200?
• The Rule of 72:
– Divide the annual interest rate into 72
– So 72/5=14.4 years.
– 1.0514.4 = 2.02
4-14
Present Value:
Definition
Present Value (PV) is the value today
(in the present) of a payment that is
promised to be made in the future.
– At a 5 percent interest rate, the present value of
$105 one year from now is $100.
– Reverses the future value calculation
4-15
Present Value:
One Year
Solve the Future Value Formula for PV:
FV = PV x (1+i)
so
FV
PV 
(1  i )
Present Value = Future Value divided by
one plus interest rate
4-16
Present Value:
One Year Example
$100 received in one year, i=5%
FV
$100
PV 

 $95.24
(1  i) 1.05
Note: FV = PVx(1+i) = $95.24x(1.05) = $100
4-17
Present Value:
General Formula
Present Value of payment received
n years in the future:
FV
PV 
n
(1  i )
4-18
Present Value:
Example
Present Value of $100 received in
2½ yrs at interest rate of 8%.
FV
$100
PV 

 $85.20
n
2.5
(1  i)
(1.08)
Note: FV = PVx(1+i)n=$82.50x (1.08)2.5 = $100
4-19
Present Value:
Important Properties
Present Value is higher:
1. The higher future value of the payment.
(FV bigger)
2. The shorter time period until payment.
(n smaller)
3. The lower the interest rate.
(i smaller)
4-20
Present Value:
$100 at 5% interest rate
Note rate of decline of Present Value. At a 5% interest rate, a $100
payment made in 14.4 years has a PV=$50.
4-21
Present Value
of $100 Payment
As the interest rate
rises from 1% to 5%,
a payment due
•1 year falls by $3.77
•10 years falls by $29.14
4-22
• Divine law of Islamic religion (Shari’a)
forbids paying interest
• Banks developed alternatives.
• Liabilities
– Deposit accounts: No interest
– Investment accounts: Share in bank’s profits or
losses
• Assets
– Profit share in exchange for loan
4-23
• Investment grows 0.5% per month
• What is the compound annual rate?
FVn=PV(1+i)n = 100x(1.005)12=106.17
Compound annual rate = 6.17%
(Note: 6.17 > 12x0.05=6.0)
4-24
To decide you need to compare
1. The value of the extra savings you will
accumulate from waiting that allows you
to purchase a more expensive care
2. The value of having the new care sooner.
4-25
Internal Rate of Return:
Definition
The interest rate that equates the
present value of an investment
with its cost.
4-26
Internal Rate of Return:
Example
You run a sports equipment factory.
Should you purchase new tennis racquet machine?
• Cost: $1 million
• Produces 3000 racquets per year
• Sell racquets for $50 apiece
• The machine lasts 10 years and collapses
with no resale value.
• Should you buy the machine?
4-27
Internal Rate of Return:
Example
• Balance the cost of the machine against the
revenue
• $1 million today vs.
$150,000 a year for ten years.
• Is the $150,000 revenue enough to make
payments on a $1 million loan?
4-28
Internal Rate of Return:
Example
Solve for i:
$1,000,000
$150,000 $150,000 $150,000
$150,000



 ......
1
2
3
(1  i)
(1  i)
(1  i)
(1  i)10
Solving for i, i=.0814 or 8.14%
4-29
Can you retire when you’re 40?
• Assume
– Live to 85
– Interest rate = 4%
– Want to have $100,000 per year
• You will need
$100,000 $100,000 $100,000
$100,000 $100,000






 $2,072,004
(1.04)1
(1.04) 2
(1.04) 3
(1.04) 44
(1.04) 45
4-30
Bond Basics
• Bond: A promise to make a series of
payments on specific future date
• Bond Price = Present Value of payments
4-31
Coupon Bond
$1000 Face Value
50-yr, 3½% coupon
bond issued on
May 1, 1945.
Coupons
4-32
Coupon Bond
• A type of loan:
– Interest paid during the life of the loan
– Loan repaid at maturity
• Coupon Rate: the annual interest the
borrower pays (ic)
• Maturity Date: when the payments stop
and the loan is repaid (n)
• Principal: the final payment (F)
4-33
Coupon Bond:
Valuing the Principal
PBP
F
$100


n
n
(1  i)
(1  i)
Present value of Bond Principal =
Payment divided by one plus the interest rate raised to n
4-34
Coupon Bond:
Valuing the Coupon Payments
PCP
C
C
C
C



 ......
1
2
3
(1  i)
(1  i)
(1  i)
(1  i) n
Value of Coupon Payments = Present value of the sequence
Note that C= ic x F
4-35
Price of Coupon Bond:
Principal + Coupons
 C
C
C
C 
F
PCB  PCP  PBP  


 ......

1
2
3
n
n
(
1

i
)
(
1

i
)
(
1

i
)
(
1

i
)
(
1

i
)


Price of Coupon Bond (PCB) =
Present value of Coupon Payments (PCP)
+ Present Value of the Principal (PBP)
4-36
Bond Pricing:
Important Property
The price of a bond (PCB) and the
interest rate (i) are inversely related:
i  PCB 
4-37
• Credit cards are useful.
• But lenders charge high
interest rates.
• Pay off your balance as
fast as you can.
4-38
Real and Nominal Interest Rates
• Borrowers care about the resources
required to repay.
• Lenders care about the purchasing
power of the payments they received.
• Neither cares solely about the number
of dollars, they care about what the
dollars buy.
4-39
Real and Nominal Interest Rates
Nominal Interest Rates (i)
Interest Rates expressed in current
dollar terms.
Real Interest Rates (r)
Nominal Interest Rate adjusted for
inflation.
4-40
Real and Nominal Interest Rates
Nominal interest rate =
Real Interest Rate + Expected Inflation
i = r + e
(This is called the “Fisher Equation”)
4-41
Nominal Interest Rate,
Inflation Rate and Real Interest Rate
Nominal Interest Rate = Real Interest Rate + Expected Inflation
4-42
Real and Nominal Interest Rates
Countries with high
nominal interest
rates have high
inflation:
  i
4-43
Chapter 4
End of Chapter
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2008