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Chapter 5 Time Value of Money

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Time is 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: $ If we deposited $ after one year we would have earned $47.62 ($ × .05) in interest. Thus, our future value would be $ $47.62 = $1,000. Example

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**Time is 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

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**Outline of Approach Deal with four different types of problems Amount**

Present value Future value Annuity

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**Outline of Approach Mathematics Time lines**

For each type of problem an equation will be presented Time lines Graphic portrayal of a time value problem 1 2 Helps with complicated problems

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**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

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**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 Example

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**Table 5-1: The Future Value Factor for k and n FVFk,n = (1+k)n**

6% 5 1.3382

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**Other Issues Problem-Solving Techniques The Opportunity Cost Rate**

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

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**Financial Calculators**

Work directly with equations 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

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**Financial Calculators**

I/Y PMT FV PV Number of time periods Interest rate (%) Future Value Present Value Payment Basic Calculator Keys

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**Financial Calculators**

I/Y PMT PV FV 1 6 5000 4,716.98 Answer 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: Example

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**The Expression for the Present Value of an Amount**

The future and present values factors are reciprocals Either equation can be used to solve any amount problems Solving for k or n involves searching a table.

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**The Expression for the Present Value of an Amount—Example**

PV PMT FV 3 -850 983.96 5.0 I/Y Answer Or you could look up the interest factor of [850 ] in the proper table with n=3. Example Q: What interest rate will grow $850 into $ in three years? A: Using the FV equation of FV = PV (1 + i)n, the answer is:

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**The Expression for the Present Value of an Amount—Example**

Q: How long does it take money invested at 14% to double? A: The future value must be twice the present value, so make up some values for present and future values that meet that criterion, such as $1 doubling to $2. N PV PMT FV 5.29 -1 2 14 I/Y Answer Or you could look up the interest factor of 2.00 [2 1] in the proper table with k=14%. Example

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**Annuity Problems Annuity**

A finite series of equal payments separated by equal time intervals Ordinary annuities Payments occur at the end of the time periods Annuity due Payments occur at the beginning of the time periods

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**Figure 5.1: Ordinary Annuity**

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Figure 5.2: Annuity Due

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**The Future Value of an Annuity—Developing a Formula**

The sum, at its end, of all payments and all interest if each payment is deposited when received

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**Figure 5.4: FV of a Three-Year Ordinary Annuity**

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**The Future Value of an Annuity—Developing a Formula**

Thus, for a 3-year annuity, the formula is FVFAk,n

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**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 Helps to draw a time line

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**The Future Value of an Annuity—Solving Problems—Example**

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? A: Using the FVAn = PMT[FVFAk,n] equation and looking up the interest factor of the annuity at an n of 10 and a k of 7, we obtain an interest factor of This gives up a FVA10 of $100,000[ ] = $1,381,640. FV N PMT I/Y 1,381,645 10 100000 7 PV Answer Example

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**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

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**The Sinking Fund Problem –Example**

Q: The Greenville Company issued bonds totaling $15 million for 30 years. The bond agreement specifies that a sinking fund must be maintained after 10 years, which will retire the bonds at maturity. Although no one can accurately predict interest rates, Greenville’s bank has estimated that a yield of 6% on deposited funds is realistic for long-term planning. How much should Greenville plan to deposit each year to be able to retire the bonds with the money put aside? A: The time period of the annuity is the last 20 years of the bond issue’s life. Input the following keystrokes into your calculator. PMT N FV I/Y 407,768.35 20 15,000,000 6 PV Answer Example

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**Compound Interest and Non-Annual 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

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**Figure 5.5: The Effect of Compound Interest**

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**The Effective Annual Rate**

Effective annual rate (EAR) The annually compounded rate that pays the same interest as a lower rate compounded more frequently

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**The Effective Annual Rate—Example**

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 $ after one year of 12% interest compounded monthly. Thus, an annually compounded rate of 12.68% [($ $100) – 1] would have to be earned.

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**The Effective Annual Rate**

EAR can be calculated for any compounding period using the following formula: Effect of more frequent compounding is greater at higher interest rates

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Table 5.3

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**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

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**The Effective Annual Rate – Example**

PMT N FV I/Y 431.22 30 15,000 1 PV Answer 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. Input the following keystrokes into your calculator.

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**The Present Value of an Annuity—Developing a Formula**

Sum of all of the annuity’s payments Easier to develop a formula than to do all the calculations individually PVFAk,n

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**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

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**The Present Value of an Annuity—Solving Problems–Example**

Q: The Shipson Company has just sold a large machine to Baltimore Inc. on an installment contract. The contract calls for Baltimore to make payments of $5,000 every six months (semiannually) for 10 years. Shipson would like its cash now and asks its bank to discount the contract and pay it the present (discounted) value. Baltimore is a good credit risk, so the bank is willing to discount the contract at 14% compounded semiannually. How much should Shipson receive? 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. Example PV N FV I/Y 52,970.07 20 7 5000 PMT Answer This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 20 and a k of 7%, resulting in PVA = $5,000[10.594] = $52,970.

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**Spreadsheet Solutions**

Time value problems can be solved on a spreadsheet such as Microsoft Excel™ or Lotus 1-2-3™ 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) 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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**Spreadsheet Solutions**

Complications Interest rates in entered as decimals, not percentages 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 (-)

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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

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**Amortized Loans—Example**

PMT N PV I/Y 293.75 48 10,000 1.5 FV Answer This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 48 and a k of 1.5%, resulting in $10,000 = PMT[ ] = $ 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 and input the following keystrokes into your calculator.

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**Amortized Loans—Example**

PV N FV I/Y 15,053.75 36 1 500 PMT Answer Example This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 36 and a k of 1%, resulting in PVA = $500[ ] = $15, 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 and input the following calculator keystrokes.

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**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)

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**Loan Amortization Schedules—Example**

Q: Develop an amortization schedule for the loan demonstrated in Example 5.12. Note that the Interest portion of the payment is decreasing while the Principal portion is increasing.

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**Mortgage Loans Mortgage loans (AKA: mortgages)**

Loans used to buy real estate Often the largest single financial transaction in an average 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

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**Mortgage Loans Implications of mortgage payment pattern**

Early mortgage payments provide a large tax savings which reduces the effective cost of a loan Halfway through a mortgage’s life half of the loan has not been paid off Long-term loans like mortgages result in large total interest amounts over the life of the loan

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**Mortgage Loans—Example**

PMT N FV I/Y 1,015.65 360 .5979 150,000 PV Answer $150,944 Net interest cost $64,690 Tax 30% $215,634 Total Interest $150,000 - Original Loan $365,634 Total payments X number of payments $1,015.65 Monthly 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.

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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

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**The Annuity Due—Example**

FV N PMT I/Y 3,020,099 x 1.02 = 3,080,501 40 50,000 2 PV Answer 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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**The Annuity Due The present value of an annuity due**

Formula 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

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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

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**Perpetuities—Example**

You may also work this by inputting a large n into your calculator (to simulate infinity), as shown below. PV N PMT I/Y 250 999 5 2 FV Answer Q: The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Money markets are such that investors can earn about 8% compounded quarterly on their money. How much can Longhorn sell this special security for? A: Convert the k to a quarterly k and plug the values into the equation.

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**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:

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**Continuous Compounding—Example**

A: To determine the EAR of 12% compounded continuously, find the future value of $100 compounded continuously in one year, then calculate the annual return. Example NOTE: Some advanced calculators have a function to solve for the EAR with continuous compounding. Q: The First National Bank of Charleston is offering continuously compounded interest on savings deposits. Such an offering is generally more of a promotional feature than anything else. If you deposit $5,000 at 6½% compounded continuously and leave it in the bank for 3½ years, how much will you have? Also, what is the equivalent annual rate (EAR) of 12% compounded continuously? A: To determine the future value of $5,000, plug the appropriate values into the equation.

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**Table 5.5: Time Value Formulas**

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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

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**Multipart Problems—Example**

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. Exeter’s cash flow is good now but may not be later, so 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 has offered an account that will pay 12% compounded monthly. How much should Exeter deposit with the bank each month to have enough reserved for the product launch?

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**Multipart Problems—Example**

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 two problems that must be handled sequentially. 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’ll solve a future value of an annuity problem for the payment

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**Multipart Problems—Example**

To find the future value of the $75,000… PV N PMT I/Y 102,643 8 4 75,000 FV Answer To find the savings annuity value 14,731 24 1 397,355 $500,000 - $102,645 = $397,355

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**Uneven Streams and Imbedded Annuities**

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

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**Uneven Streams and Imbedded Annuities**

$100 $200 $300 Example A: We’ll start with a guess of 12% and discount each amount separately at that rate. This value is too low, so we need to select a lower interest rate. Using 11% gives us $ The answer is between 8% and 9%. Q: Calculate the interest rate at which the present value of the stream of payments shown below is $500.

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**Calculator Solutions for Uneven Streams**

Financial calculators and spreadsheets have the ability to handle uneven streams with a limited number of payments Generally programmed to find the present value of the streams or the k that will equate a present value to the stream

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Imbedded Annuities Sometimes uneven streams of cash flows will have annuities embedded within them We can use the annuity formula to calculate the present or future value of that portion of the problem

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