chapter 5 the time value of money. time value the process of expressing –the present in the future...
TRANSCRIPT
Chapter 5
The Time Value of Money
Time Value
• The process of expressing
–the present in the future (compounding)
–the future in the present (discounting)
Time Value
• Payments are either
–a single payment
–a series of equal payments (an annuity)
Time Value
• Time value of money problems may be solved by using
–Interest tables
–Financial calculators
–Software
Variables for Time Value of Money Problems
• PV = present value
• FV = future value
• PMT = annual payment
• N = number of time periods
• I = interest rate per period
Financial Calculators
• Express the cash inputs (PV, FV, and PMT) as cash inflows and cash outflows
• At least one of the cash variables must be
–an inflow (+)
–an outflow (-)
Future Value
• The future value of $1 takes a single payment in the present into the future
• The general equation for the future value of $1:
P0(1 + i)n = Pn
Future Value Illustrated
• PV = -100
• I = 8
• N = 20
• PMT = 0
• FV = ?
• = 466.10
Greater Terminal Values
• Higher interest rates
• Longer time periods
• Result in greater terminal values
Greater Terminal Values
Present Value
• The present value of $1 brings a single payment in the future back to the present
• The general equation for the present value of $1:P0 = Pn
(1+i)n
Present Value Illustrated
• FV = 100• I = 6• N = 5• PMT = 0• PV = ?
• = -74.73
Lower Present Values
• Higher interest rates
• Longer time periods
• Result in lower present values
Lower Present Values
Annuities - Future Sum
• The future sum of annuity takes a series of payments into the future
• Payments may be made–at the end of each time period
(ordinary annuity)–at the beginning of each time
period (annuity due)
FV Time Lines
•Ordinary annuity
Year Payment - $100 100 100
0 1 2 3
FV Time Lines
•Annuity due
Year Payment $100 100 100 -
0 1 2 3
Future Value of an Ordinary Annuity Illustrated
• PV = 0• PMT = -100• I = 5• N = 3• FV = ?
• = 315.25
Greater Terminal Values
• Higher interest rates
• Longer time periods
• Result in greater terminal values
Greater Terminal Values
Annuities - Present Value
• The present value of an annuity brings a series of payments in the future back to the present
Present Value of an Ordinary Annuity Illustrated
• FV = 0• PMT = 100• I = 6• N = 3• PV = ?
• = -267.30
Annuities - Present Value
• Higher interest rates result in lower present values
• But longer time periods increases the present value (because more payments are received)
Annuities - Present Value
Additional Time Value Illustrations
• The following is a series of problems or questions that use the time value of money.
Illustration 1
• You deposit $1,000 in an account at the end of each year for twenty years. What is the total amount in the account if you earn 6 percent annually?
Future Value of an Ordinary Annuity
• The unknown: FV
• The givens:
–PV = 0
–PMT = -1,000
–N = 20
–I = 6
The answer:$36,786
Interpretation
• For an annual cash payment of $1,000, you will have $36,786 after twenty years
• Of the $36,786
–$20,000 is the total cash outflow
–$16,786 is the earned interest
Illustration 2
• What is the present value of (or required cash outflow to purchase) an ordinary annuity of $1,000 for twenty years, if the rate of interest is 6 percent?
Present Value of an Annuity
• The unknown: PV
• The givens:
–FV = 0
–PMT = 1,000
–N = 20
–I = 6
The answer:$11,470
Interpretation
• For a present payment of $11,470, the individual will annually receive $1,000 for the next twenty years
• The $11,470 is an immediate cash outflow
• The $1,000 annual payment to be received is a cash inflow
Illustration 3
• What is the future value after ten years of a stock that cost $10 and appreciates at 9 percent annually?
Future Value of $1
• The unknown: FV
• The givens:
–PV = 10
–PMT = 0
–N = 10
–I = 9
The answer: $23.67
Interpretation
• A $10 stock will be worth $23.67 after 10 year if its price grows 9% annually.
Illustration 4
• What is the cost of a stock that was sold for $23.67, held for 10 years and whose value appreciated 9 percent annually?
Present Value of $1
• The unknown: PV
• The givens:
–FV = 23.67
–PMT = 0
–N = 10
–I = 9
The answer:$10
Interpretation
• $23.67 received after ten years is worth $10 today if the return rate is 9 percent.
Interpretation of Future and Present Values
These two problems are the same:
• In the first case the $10 is compounded into its future value ($23.67)
• In the second case the future value ($23.67) is discounted back to its present value ($10)
Illustration 5
• A stock was purchased for $10 and sold for $23.67 after 10 years. What was the return?
Future Value of I
• The unknown: I
• The givens:
–PV = 10
–PMT = 0
–N = 0
–FV = 23.67
The answer: 9%
Interpretation
• The yield on a $10 investment that was sold after 10 years for $23.67 is 9%.
Illustration 6
• If an investment pay $50 a year for 10 years and repays $1,000 after 10 years, what is this investment worth today if you can earn 6 percent?
Determination of Present Value
• The unknown: PV
• The givens:
–FV = 1,000
–PMT = 50
–I = 6
–N = 10
The answer: $926
Interpretation
• If you collect $50 a year for 10 years and receive $1,000 after 10 years, those cash inflows are currently worth $926 at 6 percent.
Illustration 7
• Time value is used to determine a loan repayment schedule such as a mortgage.
Loan Repayment Schedule
• Amount borrowed (PV) = $80,000
• Interest rate (I) = 8%
• Term of the loan (N) = 25 years
• No future value since loan is repaid
• Amount of the annual payment = $7,494.30
Loan Repayment Schedule
Principal Balance
Pmnt Interest Repayment Owed
1 $6,400.00 $1,094.15 $78,905.85
2 6,312.47 1,181.68 77,724.17
. . . .
. . . .
. . . .
25 555.13 6,939.17 .00
Illustration 8
• You have $115,000 and spend $24,000 a year. If you earn 8% annually, how long will your funds last?
Determination of Number of Years
• The unknown: N
• The givens:
–PV = 115,000
–I = 8
–FV = 0
–PMT = -24,000
The answer: 6.3 years
Interpretation
• If you have $115,000 and earn 8 percent annually, you can spend $24,000 per year for approximately 6 years and 4 months.
Non-annual Compounding
• More than one interest payment a year
• More frequent compounding
Non-annual Compounding
• Multiply number of years by frequency of compounding
• Divide interest rate by frequency of compounding
Periods less than One Year
• Same variables as in all time value problems except N < 1.
Illustration 9
• What is the return on an ivestment that costs $98,543 and pays $100,000 after 45 days?
Determination of Return
• The unknown: I
• The givens:
–PV = 98,543
–N = 0.1233
–FV = 100,000
–PMT = 0
The answer: 12.64%
Interpretation
• $98,543 invested for 45 days grows to $100,000 at 12.64 percent.