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Transcript of 8 8 Chapter Time Value of Money Time Value of Money Slides Developed by: Terry Fegarty Seneca...
88Chapt
er
Chapt
er Time Value
of MoneyTime Value
of Money
Slides Developed by:
Terry FegartySeneca College
© 2006 by Nelson, a division of Thomson Canada Limited 2
Chapter 8 – Outline (1)
• The Time Value of Money Time Value Problems Amount Problems—Future Value Other Issues Financial Calculators Spreadsheet Solutions The Present Value of an Amount Finding the Interest Rate Finding the Number of Periods
© 2006 by Nelson, a division of Thomson Canada Limited 3
Chapter 8 – Outline (2)
• Annuities The Future Value of an Annuity The Future Value of an Annuity—Developing a
Formula The Future Value of an Annuity—Solving Problems The Sinking Fund Problem Compound Interest and Non-Annual Compounding The Effective Annual Rate The Present Value of an Annuity—Developing a
Formula The Present Value of an Annuity—Solving Problems
© 2006 by Nelson, a division of Thomson Canada Limited 4
Chapter 8 – Outline (3)
Amortized Loans Loan Amortization Schedules Mortgage Loans The Annuity Due Perpetuities Continuous Compounding
• Multipart Problems Uneven Streams Imbedded Annuities
© 2006 by Nelson, a division of Thomson Canada Limited 5
The Time Value of Money
• $100 in your hand today is worth more than $100 in one year Money earns interest
• The higher the interest, the faster your money grows
Q: How much would $1,000 promised in one year be worth today if the bank paid 5% interest?
A: $952.38. If we deposited $952.38 after one year we would have earned $47.62 ($952.38 × .05) in interest. Thus, our future value would be $952.38 + $47.62 = $1,000.
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© 2006 by Nelson, a division of Thomson Canada Limited 6
The Time Value of Money
• Present Value The amount that must be deposited today to
have a future sum at a certain interest rate The discounted value of a sum is its
present value In our example, what is the present value?
$952.38
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© 2006 by Nelson, a division of Thomson Canada Limited 7
The Time Value of Money
• Future Value The amount a present sum will grow into at a
certain interest rate over a specified period of time
In our example, • The present sum (value) is $952.38• The interest rate is 5%• The time is 1 year• The future value is $1,000
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© 2006 by Nelson, a division of Thomson Canada Limited 8
Time Value Problems
• Time value deals with four different types of problems Amount— a single amount that grows at
interest over time• Future value • Present value
Annuity— a stream of equal payments that grow at interest over time• Future value• Present value
© 2006 by Nelson, a division of Thomson Canada Limited 9
Time Value Problems
• 4 methods to solve time value problems Use formulas Use financial tables Use financial calculator Use financial functions in spreadsheet
© 2006 by Nelson, a division of Thomson Canada Limited 10
Amount Problems—Future Value
• The future value (FV) of an amount How much a sum of money placed at interest
(k) will grow into in some period of time• If the time period is one year
• FV1 = PV + kPV or FV1 = PV(1+k)
• If the time period is two years• FV2 = FV1 + kFV1 or FV2 = PV(1+k)2
• If the time period is generalized to n years• FVn = PV(1+k)n
© 2006 by Nelson, a division of Thomson Canada Limited 11
Amount Problems—Future Value
• The (1 + k)n depends on Size of k and n
• Can develop a table depicting different values of n and k and the proper value of (1 + k)n
• Can then use a more convenient formula
• FVn = PV [FVFk,n]
These values can be looked up in an interest
factor table.Q: If we deposited $438 at 6%
interest for five years, how much would we have?
A: FV5 = $438(1.06)5 = $438(1.3382) = $586.13
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© 2006 by Nelson, a division of Thomson Canada Limited 12
Table 8-1: The Future Value Factor for k and n FVFk,n = (1+k)n
1.3382
6%
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1.33825
© 2006 by Nelson, a division of Thomson Canada Limited 13
Other Issues
• Problem-Solving Techniques Three of four variables are given
• We solve for the fourth
• The Opportunity Cost Rate The opportunity cost of a resource is the
benefit that would have been available from its next best use• Lost investment income is an opportunity cost
© 2006 by Nelson, a division of Thomson Canada Limited 14
Financial Calculators
• How to use a typical financial calculator in time value Five time value keys
• Use either four or five keys
Some calculators distinguish between inflows and outflows
• If a PV is entered as positive the computed FV is negative
© 2006 by Nelson, a division of Thomson Canada Limited 15
Financial Calculators
N
I/Y
PMT
FV
PV
Number of time periods
Interest rate (%)
Future Value
Present Value
Payment
Basic Calculator Keys
© 2006 by Nelson, a division of Thomson Canada Limited 16
Financial Calculators
N
I/Y
PMT
PV
FV
1
6
0
5000
4,716.98 Answer
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Q: What is the present value of $5,000 received in one year if interest rates are 6%?A: Input the following values on the calculator and compute the PV:
© 2006 by Nelson, a division of Thomson Canada Limited 17
Spreadsheet Solutions
• Time value problems can be solved on a spreadsheet such as Microsoft® Excel®
• Click on the fx (function) button• Select for the category Financial• Select the function for the unknown variable• For example, to solve for present value:
Use PV(k, n, PMT, FV) Interest rate (k) is entered as a decimal, not a
percentage
© 2006 by Nelson, a division of Thomson Canada Limited 18
Spreadsheet Solutions
• To solve for: FV use =FV(k, n, PMT, PV) PV use =PV(k, n, PMT, FV) k use =RATE(n, PMT, PV, FV) N use =NPER(k, PMT, PV, FV) PMT use =PMT(k, n, PV, FV) Of the three cash variables (FV, PMT or PV)
• One is always zero• The other two must be of the opposite sign• Reflects inflows (+) versus outflows (-)
Select the function for the unknown
variable, place the known variables in
the proper order within the
parentheses and input 0, for the
unknown variable.
© 2006 by Nelson, a division of Thomson Canada Limited 19
The Present Value of an Amount
• Either equation can be used to solve any amount problems
n
n
n n
Interest Factor
FV PV 1+k
Solve for PV
1PV = FV
1 k
k,nk,n
1FVF
PVF Solving for k or n involves
searching a table.
© 2006 by Nelson, a division of Thomson Canada Limited 20
Spreadsheet Solution—PV of Amount
Click on the fx (function) button Select for the category Financial In the insert function box, select PV You will see PV(rate,nper,pmt,fv,type) Press OK Fill in rate = B4, nper = C1, pmt = 0, fv = C2 Enter OK
Select the function for the unknown
variable, place the known variables in
the proper order within the
parentheses and input 0, for the
unknown variable.
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© 2006 by Nelson, a division of Thomson Canada Limited 21
Spreadsheet Solution—PV of Amount E
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© 2006 by Nelson, a division of Thomson Canada Limited 22
Example 8.3: Finding the Interest Rate
N
PV
I/Y
PMT
FV
3
-850
0
983.96
5.0 Answer
Calculator
Q: What interest rate will grow $850 into $983.96 in three years?A:
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Interest factor is 850.00 / 983.96 = 0.8639
Look up in Table A-2 with n = 3
Interest = 5%
Financial Table
© 2006 by Nelson, a division of Thomson Canada Limited 23
Example 8.3: Spreadsheet SolutionE
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© 2006 by Nelson, a division of Thomson Canada Limited 24
Example 8.3: Spreadsheet Solution
Click on the fx (function) button Select for the category Financial In the insert function box, select RATE You will see RATE(n, PMT, PV, FV) Press OK Fill in n = E1, PMT = 0, PV = -B2, FV = E2 Enter OK
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© 2006 by Nelson, a division of Thomson Canada Limited 25
Example 8.4: Finding the Number of Periods
PV
FV
N
I/Y
PMT
-1
2
14
0
5.29 Answer
Calculator
Interest factor is 2 / 1 = 2
Look up in Table A-2 with k = 14%
n = 5 – 6 years
Financial Table
Q: How long does it take money invested at 14% to double?
A: The future value is twice the present value. If the present value is $1, the future value is $2
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© 2006 by Nelson, a division of Thomson Canada Limited 26
Example 8.4: Finding the Number of Periods
Click on the fx (function) button Select for the category Financial In the insert function box, select NPER You will see NPER(k, PMT, PV, FV) Enter OK Fill in the cell references for
• k (= .14)• PMT = 0• PV (= -1)• FV = 2
Enter OK NPER = 5.29
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© 2006 by Nelson, a division of Thomson Canada Limited 27
Example 8.4: Spreadsheet SolutionE
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© 2006 by Nelson, a division of Thomson Canada Limited 28
Annuities
• Annuity A finite series of equal payments separated
by equal time intervals•Ordinary annuity
• Payments occur at the end of the time periods• Monthly lease, pension, and car payments are annuities
•Annuity due• Payments occur at the beginning of the time periods
© 2006 by Nelson, a division of Thomson Canada Limited 29
Figures 8.1 and 8.2: Ordinary Annuity and Annuity Due
Ordinary Annuity
Annuity Due
© 2006 by Nelson, a division of Thomson Canada Limited 30
Figure 8.3: Timeline Portrayal of an Ordinary Annuity
© 2006 by Nelson, a division of Thomson Canada Limited 31
Future Value of an Annuity
• Future value of an annuity The sum, at its end, of all payments and all
interest if each payment is deposited when received
© 2006 by Nelson, a division of Thomson Canada Limited 32
Figure 8.4: FV of a Three-Year Ordinary Annuity
© 2006 by Nelson, a division of Thomson Canada Limited 33
The Future Value of an Annuity—Developing a Formula• Thus, for a 3-year annuity, the formula is
FVFAk,n
0 1 2
0 1 2 n -1
n
nn i
ni=1
FVA = PMT 1+k PMT 1+k PMT 1+k
Generalizing the Expression:
FVA = PMT 1+k PMT 1+k PMT 1+k PMT 1+k
which can be written more conveniently as:
FVA PMT 1+k
Factoring PMT outside the summation, we
n
n i
ni=1
obtain:
FVA PMT 1+k
© 2006 by Nelson, a division of Thomson Canada Limited 34
The Future Value of an Annuity—Solving Problems
• There are four variables in the future value of an annuity equation The future value of the annuity itself The payment The interest rate The number of periods
© 2006 by Nelson, a division of Thomson Canada Limited 35
Example 8.5: The Future Value of an Annuity
Q: The Brock Corporation owns the patent to an industrial process and receives license fees of $100,000 a year on a 10-year contract for its use. Management plans to invest each payment until the end of the contract to provide funds for development of a new process at that time.
If the invested money is expected to earn 7%, how much will Brock have after the last payment is received?
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© 2006 by Nelson, a division of Thomson Canada Limited 36
Example 8.5: The Future Value of an Annuity
A: Use the future value of an annuity equation:
FVAn = PMT[FVFAk,n]
In Table A-3, look up the interest factor at an n of 10 and
a k of 7
Interest factor = 13.8164
Future value = $100,000[13.8164] = $1,381,640
FV
N
PMTI/Y
1,381,645 Answer
10
1000007
0PV
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© 2006 by Nelson, a division of Thomson Canada Limited 37
Example 8.5: Spreadsheet Solution
Click on the fx (function) button Select for the category Financial In the insert function box, select FV You will see FV(k, n, PMT, PV) Enter OK Fill in the cell references for
• k (=.07)• n (= 10)• PMT (=100000)• PV (=0)
Enter OK FV = 1,381,644.80
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© 2006 by Nelson, a division of Thomson Canada Limited 38
Example 8.5: Spreadsheet SolutionE
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© 2006 by Nelson, a division of Thomson Canada Limited 39
The Sinking Fund Problem
• Companies borrow money by issuing bonds for lengthy time periods No repayment of principal is made during the
bonds’ lives• Principal is repaid at maturity in a lump sum
• A sinking fund provides cash to pay off a bond’s principal at maturity
• Problem is to determine the periodic deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem
© 2006 by Nelson, a division of Thomson Canada Limited 40
Example 8.6: The Sinking Fund Problem
Q: The Greenville Company issued bonds totaling $15 million for 30 years. A sinking fund must be maintained after 10 years, which will retire the bonds at maturity. The estimated yield on deposited funds will be 6%.
How much should Greenville deposit each year to be able to retire the bonds?
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© 2006 by Nelson, a division of Thomson Canada Limited 41
Example 8.6: The Sinking Fund Problem
A: The time period of the annuity is the last 20 years of the bond issue’s life. On your calculator:
PMT
N
FVI/Y
407,768.35 Answer
20
150000006
0PV
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© 2006 by Nelson, a division of Thomson Canada Limited 42
Compound Interest and Non-Annual Compounding
• Compounding Earning interest on interest
• Compounding periods Interest is usually compounded annually,
semiannually, quarterly or monthly• Interest rates are quoted by stating the nominal
rate followed by the compounding period
© 2006 by Nelson, a division of Thomson Canada Limited 43
The Effective Annual Rate
• Effective annual rate (EAR) The annually compounded rate that pays the
same interest as a lower rate compounded more frequently
© 2006 by Nelson, a division of Thomson Canada Limited 44
The Effective Annual Rate—Example
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Q: If 12% is compounded monthly, what annually compounded interest rate will get a depositor the same interest?
A: If your initial deposit were $100, you would have $112.68 after one year of 12% interest compounded monthly. Thus, an annually compounded rate of 12.68% [($112.68 $100) – 1] would have to be earned.
© 2006 by Nelson, a division of Thomson Canada Limited 45
The Effective Annual Rate
• EAR can be calculated for any compounding period using the following formula:
m
nominalEAR - 1k
1 m
Effect of more frequent compounding is greater at higher interest rates
© 2006 by Nelson, a division of Thomson Canada Limited 46
Impact of Compounding Frequency
$1,097
$1,098
$1,099
$1,100
$1,101
$1,102
$1,103
$1,104
$1,105
$1,106
Annual Semi-Annual
Quarterly Monthly Daily
$1,000 Invested at 10% Nominal Rate for One Year
© 2006 by Nelson, a division of Thomson Canada Limited 47
The Effective Annual Rate
• The APR and EAR Annual percentage rate (APR)
• Is actually the nominal rate and is less than the EAR
• Compounding Periods and the Time Value Formulas Time periods must be compounding periods Interest rate must be the rate for a single
compounding period• For instance, with a quarterly compounding period the knominal must be divided by 4 and the n must be multiplied by 4
© 2006 by Nelson, a division of Thomson Canada Limited 48
Example 8.7: The Effective Annual Rate
PMT
N
FV
I/Y
431.22 Answer
30
15000
1
0PV
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Q: You want to buy a car costing $15,000 in 2½ years. You plan to save the money by making equal monthly deposits in your bank account, which pays 12% compounded monthly. How much must you deposit each month?A: This is a future value of an annuity problem with a 1% monthly interest rate and a 30-month time period. On your calculator:
© 2006 by Nelson, a division of Thomson Canada Limited 49
The Present Value of an Annuity—Developing a Formula• Present value of an annuity
Sum of all of the annuity’s payments
2 3
1 2 3
1 2 n
PMT PMT PMTPVA =
1+k 1+k 1+k
which can also be written as:
PVA = PMT 1+k PMT 1+k PMT 1+k
Generalized for any number of periods:
PVA = PMT 1+k PMT 1+k PMT 1+k
Factoring PMT and using summation, we o
n
i
i=1
btain:
PVA PMT 1+k
PVFAk,n
© 2006 by Nelson, a division of Thomson Canada Limited 50
Figure 8.6: PV of a Three-Payment Ordinary Annuity
© 2006 by Nelson, a division of Thomson Canada Limited 51
The Present Value of an Annuity—Solving Problems
• There are four variables in the present value of an annuity equation The present value of the annuity itself The payment The interest rate The number of periods
• Problem usually presents 3 of the 4 variables
© 2006 by Nelson, a division of Thomson Canada Limited 52
Example 8.9: The Present Value of an Annuity
Q: The Shipson Company has just sold a large machine on an installment contract. The contract calls for payments of $5,000 every six months (semiannually) for 10 years. Shipson would like its cash now and asks its bank to it the present (discounted) value. The bank is willing to discount the contract at 14% compounded semiannually.
How much should Shipson receive?
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© 2006 by Nelson, a division of Thomson Canada Limited 53
Example 8.9: The Present Value of an Annuity
A: The contract represents an annuity with payments of $5,000. Adjust the interest rate and number of periods for semiannual compounding and solve for the present value of the annuity.
PV
N
FV
I/Y
52,970.07 Answer
20
0
7
5000PMT
This can also be calculated using the PVA Table A4.
Look up n = 20 and k = 7%.
PVA = $5,000[10.594]
= $52,970
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© 2006 by Nelson, a division of Thomson Canada Limited 54
Amortized Loans
• An amortized loan’s principal is paid off regularly over its life Generally structured so that a constant
payment is made periodically• Represents the present value of an annuity
© 2006 by Nelson, a division of Thomson Canada Limited 55
Example 8.10: Amortized Loans
PMT
N
PV
I/Y
293.75
48
10000
1.5
0FV
Answer
This can also be calculated using the PVA formula of
PVA = PMT[PVFAk, n]
n = 48 and k = 1.5% $10,000 = PMT[34.0426]
= $293.75.
Q: Suppose you borrow $10,000 over four years at 18% compounded monthly repayable in monthly installments. How much is your loan payment?
A: Adjust your interest rate and number of periods for monthly compounding. On your calculator:
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© 2006 by Nelson, a division of Thomson Canada Limited 56
Example 8.11: Amortized Loans
Q: Suppose you want to buy a car and can afford to make payments of $500 a month. The bank makes three-year car loans at 12% compounded monthly. How much can you borrow toward a new car?
A: Adjust your k and n for monthly compounding. On your calculator:
PV
N
FV
I/Y
15,053.75 Answer
36
0
1
500PMT
This can also be calculated using the PVA Table A4.
Look up n = 36 and k = 1%.
PVA = $500[30.1075]
= $15,053.75
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© 2006 by Nelson, a division of Thomson Canada Limited 57
Loan Amortization Schedules
• Detail the interest and principal in each loan payment
• Show the beginning and ending balances of unpaid principal for each period
• Need to know Loan amount (PVA) Payment (PMT) Periodic interest rate (k)
© 2006 by Nelson, a division of Thomson Canada Limited 58
Example 8.11: Loan Amortization Schedule
Q: Develop an amortization schedule for the loan demonstrated in Example 8.11
Note that the Interest portion of the payment is decreasing while the Principal portion is
increasing.
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© 2006 by Nelson, a division of Thomson Canada Limited 59
Mortgage Loans
• Mortgage loans (AKA: mortgages) Loans used to buy real estate
• Often the largest single financial transaction in a person’s life Typically an amortized loan over 30 years
• During the early years of the mortgage nearly all the payment goes toward paying interest
• This reverses toward the end of the mortgage
Halfway through a mortgage’s life half of the loan has not been paid off
© 2006 by Nelson, a division of Thomson Canada Limited 60
Mortgage Loans
• Implications of mortgage payment pattern Long-term loans like mortgages result in large total
interest amounts over the life of the loan• At 6% interest, compounded monthly, over 25 years,
borrower pays almost the amount of the loan just in interest!
• Canadian banks compound semi-annually, thus lowering interest charges
Early mortgage payments and more frequent mortgage payments provide a large interest saving
© 2006 by Nelson, a division of Thomson Canada Limited 61
Mortgage Loans—Example E
xam
ple
PMT
N
FV
I/Y
1,015.65
360
0
0.5979
150000PV
Answer143.76%Interest / Principal
$215,634Total Interest
$150,000- Original Loan
$365,634Total payments
360X # of payments
$1,015.65Monthly payment
Q: Calculate the monthly payment for a 30-year 7.175% mortgage of $150,000. Also calculate the total interest paid over the life of the loan.
A: Adjust the n and k for monthly compounding and input the following calculator keystrokes.
© 2006 by Nelson, a division of Thomson Canada Limited 62
The Annuity Due
• In an annuity due payments occur at the beginning of each period
• The future value of an annuity due Because each payment is received one period
earlier, it spends one period longer in the bank earning interest
n -1
n
n k,n
FVAd = PMT + PMT 1+k PMT 1+k 1 k
which written with the interest factor becomes:
FVAd PMT FVFA 1 k
© 2006 by Nelson, a division of Thomson Canada Limited 63
Figure 8.7: The Future Value of a Three-Period Annuity Due
© 2006 by Nelson, a division of Thomson Canada Limited 64
Example 8.12: The Annuity Due
FV
N
PMT
I/Y
3,020,099 x 1.02 = 3,080,501 Answer
40
50000
2
0PV
NOTE: Advanced calculators allow you to switch from END (ordinary annuity) to BEGIN
(annuity due) mode.
Q: The Baxter Corporation started making sinking fund deposits of $50,000 per quarter today. Baxter’s bank pays 8% compounded quarterly, and the payments will be made for 10 years. What will the fund be worth at the end of that time?
A: Adjust the k and n for quarterly compounding and input the following calculator keystrokes.
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© 2006 by Nelson, a division of Thomson Canada Limited 65
The Annuity Due
• The present value of an annuity due Formula
k,nPVAd PMT PVFA 1 k
Recognizing types of annuity problems Always represent a stream of equal payments Always involve some kind of a transaction at one
end of the stream of payments• End of stream—future value of an annuity• Beginning of stream—present value of an annuity
© 2006 by Nelson, a division of Thomson Canada Limited 66
Perpetuities
• A perpetuity is a stream of regular payments that goes on forever An infinite annuity
• Future value of a perpetuity Makes no sense because there is no end point
• Present value of a perpetuity A diminishing series of numbers
• Each payment’s present value is smaller than the one before
p
PMTPV
k
© 2006 by Nelson, a division of Thomson Canada Limited 67
Example 8.13: Perpetuities
p
PMT $5PV $250
k 0.02 You may also work this by inputting
a large n into your calculator (to simulate infinity), as shown below.
PV
N
PMTI/Y
250
999
52
0FVAnswer
Q: The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Investors can earn 8% compounded quarterly on their money. How much can Longhorn sell this security for?
A: Convert the k to a quarterly k and plug the values into the equation.
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© 2006 by Nelson, a division of Thomson Canada Limited 68
Continuous Compounding
• Compounding periods can be shorter than a day As the time periods become infinitesimally short,
interest is said to be compounded continuously
• To determine the future value of a continuously compounded value:
Where k = nominal rate, n = number of years, e = 2.71828
knnFV PV e
© 2006 by Nelson, a division of Thomson Canada Limited 69
Example 8.15: Continuous Compounding
Q: The First National Bank of Cardston is offering continuously compounded interest on savings deposits.
If you deposit $5,000 at 6½% compounded continuously and leave it in the bank for 3½ years, how much will you have?
What is the equivalent annual rate (EAR) of 12% compounded continuously?
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© 2006 by Nelson, a division of Thomson Canada Limited 70
Example 8.15: Continuous Compounding
A: To determine the future value of $5,000, plug the appropriate values into the equation
n = knFV PV e
To determine the EAR of 12% compounded continuously, find the future value of $100 compounded continuously in one year, then calculate the annual return
.065 3.53.5=FV $5,000(2.71828) = $6.277.29
11
.12=FV $100 e
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=
.12= $100 2.71828 = $112.75
$112.75-$100EAR= =12.75%
$100
NOTE: Some advanced calculators have a function to solve for the EAR with continuous compounding
© 2006 by Nelson, a division of Thomson Canada Limited 71
Table 8.5: Time Value Formulas
© 2006 by Nelson, a division of Thomson Canada Limited 72
Multipart Problems
• Time value problems are often combined due to complex nature of real situations A time line portrayal can be critical to
keeping things straight
© 2006 by Nelson, a division of Thomson Canada Limited 73
Example 8.16: Multipart Problems
Q: Exeter Inc. has $75,000 invested in securities that earn a return of 16% compounded quarterly. The company is developing a new product that it plans to launch in two years at a cost of $500,000. Management would like to bank money from now until the launch to be sure of having the $500,000 in hand at that time. The money currently invested in securities can be used to provide part of the launch fund. Exeter’s bank account will pay 12% compounded monthly.
How much should Exeter deposit with the bank each month ?
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© 2006 by Nelson, a division of Thomson Canada Limited 74
Example 8.16: Multipart Problems
A: Two things are happening in this problem Exeter is saving money every month (an annuity) and The money invested in securities (an amount) is growing
independently at interest
We have three steps First, we need to find the future value of $75,000 Then, we subtract that future value from $500,000 to
determine how much extra Exeter needs to save via the annuity
Then, we solve a future value of an annuity problem for the payment
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© 2006 by Nelson, a division of Thomson Canada Limited 75
Example 8.16: Multipart Problems E
xam
ple
To find the future value of the $75,000…
PV
N
PMT
I/Y
102,645
8
0
4
75000FV Answer
To find the savings annuity value
PMT
N
PV
I/Y
14,731
24
0
1
397355FVAnswer
$500,000 - $102,645= $397,355
© 2006 by Nelson, a division of Thomson Canada Limited 76
Uneven Streams
• Many real world problems have sequences of uneven cash flows These are NOT annuities
• For example, if you were asked to determine the present value of the following stream of cash flows
$100 $200 $300
Must discount each cash flow individually Not really a problem when attempting to determine either a
present or future value Becomes a problem when attempting to determine an interest rate
© 2006 by Nelson, a division of Thomson Canada Limited 77
Example 8.18: Uneven Streams
$100 $200 $300
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A: Start with a guess of 12% and discount each amount separately at that rate.
This value is too low; select a lower interest rate. Using 11% gives us $471.77. The answer is
between 8% and 9%.
1 k,1 2 k,2 3 k,3
12,1 12,2 12,3
PV = FV PVF FV PVF FV PVF
$100 PVF $200 PVF $300 PVF
$100 .8929 $2
$462.2
00 .7972 $300 .71 8
7
1
Q: Calculate the interest rate at which the present value of the stream of payments shown below is $500.
© 2006 by Nelson, a division of Thomson Canada Limited 78
Imbedded Annuities
• Sometimes uneven streams of cash flows will have annuities imbedded within them We can use the annuity formula to calculate
the present or future value of that portion of the problem