© The McGraw-Hill Companies, Inc., 2008 McGraw-Hill/Irwin Chapter 4 Future Value, Present Value and...
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Transcript of © The McGraw-Hill Companies, Inc., 2008 McGraw-Hill/Irwin Chapter 4 Future Value, Present Value and...
© The McGraw-Hill Companies, Inc., 2008McGraw-Hill/Irwin
Chapter 4
Future Value, Present Value and
Interest Rates
4-2
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?
4-3
Future Value, Present Value and Interest Rates:
Roadmap
• Future Value
• Present Value
• Internal Rate of Return
• Bond Basics
• Real vs. Nominal Interest Rates
4-4
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
4-5
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.
4-6
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
4-7
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.
4-8
Future Value:One Year
Future Value =Present Value + Interest
FV = PV + PVxi
$105 = $100 + $100x(0.05)
4-9
Future Value:One Year
FV = PV + PVxi
= PVx(1+i)
Future Value in one year =Present Value x (one plus interest rate)
4-10
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
4-11
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)
4-12
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.
4-13
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
4-14
• 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
4-15
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
4-16
Present Value:One Year
Solve the Future Value Formula for PV:
FV = PV x (1+i)so
Present Value = Future Value divided by one plus interest rate
)1( i
FVPV
4-17
Present Value:One Year Example
$100 received in one year, i=5%
Note: FV = PVx(1+i) = $95.24x(1.05) = $100
24.95$05.1
100$
)1(
i
FVPV
4-18
Present Value:General Formula
Present Value of payment received n years in the future:
ni
FVPV
)1(
4-19
Present Value:Example
Present Value of $100 received in 2½ yrs at interest rate of 8%.
Note: FV = PVx(1+i)n=$82.50x (1.08)2.5 = $100
20.85$)08.1(
100$
)1( 5.2
ni
FVPV
4-20
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)
4-21
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.
4-22
Present Valueof $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
4-23
• 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
4-24
• 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)
4-25
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.
4-26
Internal Rate of Return:Definition
The interest rate that equates the present value of an investment
with its cost.
4-27
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?
4-28
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?
4-29
Internal Rate of Return:Example
Solve for i:
$1,000,000
10321 )1(
000,150$......
)1(
000,150$
)1(
000,150$
)1(
000,150$
iiii
Solving for i, i=.0814 or 8.14%
4-30
Can you retire when you’re 40?
• Assume– Live to 85– Interest rate = 4%– Want to have $100,000 per year
• You will need004,072,2$
)04.1(
000,100$
)04.1(
000,100$
)04.1(
000,100$
)04.1(
000,100$
)04.1(
000,100$4544321
4-31
Bond Basics
• Bond: A promise to make a series of payments on specific future date
• Bond Price = Present Value of payments
4-33
Coupon Bond
• A type of loan:– Interest paid during the life of the loan– Loan repaid at maturity
• Coupon Rate: the annual interest the borrower pays (ic)
• Maturity Date: when the payments stop and the loan is repaid (n)
• Principal: the final payment (F)
4-34
Coupon Bond:Valuing the Principal
nnBP ii
FP
)1(
100$
)1(
Present value of Bond Principal = Payment divided by one plus the interest rate raised to n
4-35
Coupon Bond:Valuing the Coupon Payments
nCP i
C
i
C
i
C
i
CP
)1(......
)1()1()1( 321
Value of Coupon Payments = Present value of the sequence
Note that C= ic x F
4-36
Price of Coupon Bond:Principal + Coupons
Price of Coupon Bond (PCB) =
Present value of Coupon Payments (PCP) + Present Value of the Principal (PBP)
nnBPCPCB i
F
i
C
i
C
i
C
i
CPPP
)1()1(......
)1()1()1( 321
4-37
Bond Pricing:Important Property
The price of a bond (PCB) and the interest rate (i) are inversely related:
i PCB
4-38
• Credit cards are useful.
• But lenders charge high interest rates.
• Pay off your balance as fast as you can.
4-39
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.
4-40
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.
4-41
Real and Nominal Interest Rates
Nominal interest rate = Real Interest Rate + Expected Inflation
i = r + e (This is called the “Fisher Equation”)
4-42
Nominal Interest Rate, Inflation Rate and Real Interest Rate
Nominal Interest Rate = Real Interest Rate + Expected Inflation
4-43
Real and Nominal Interest Rates
Countries with high nominal interest rates have high inflation:
i