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**The Time Value of Money (Part Two)**

Chapter 4 The Time Value of Money (Part Two)

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**1. Explain and illustrate an annuity.**

LEARNING OBJECTIVES 1. Explain and illustrate an annuity. 2. Determine the future value of an annuity. 3. Determine the present value of an annuity. 4. Adjust the annuity equation for present value and future value for an annuity due. 5. Distinguish between the different types of loan repayments. 6. Build and analyze amortization schedules. 7. Calculate waiting time and interest rates for an annuity.

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**4.1 Future Value of Multiple Payment Streams**

With unequal periodic cash flows, treat each of the cash flows as a lump sum and calculate its future value over the relevant number of periods. Sum up the individual future values to get the future value of the multiple payment streams.

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**Fig. 4.1 The time-line of a nest egg**

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**4.2 Future Value of an Annuity Stream**

Annuities are equal, periodic outflows/inflows at regular intervals, e.g. rent, lease, mortgage, car loan, and retirement annuity payments. An annuity stream can begin at the start of each period (annuity due) as is true of rent and insurance payments or at the end of each period, (ordinary annuity) as in the case of mortgage and loan payments. The formula for calculating the future value of an ordinary annuity stream is as follows: FV = PMT x (1+r)n -1 r where PMT is the term used for the equal periodic cash flow, r is the rate of interest, and n is the number of payments, one at the end of each period (ordinary annuity).

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**4.2 Future Value of an Annuity Stream**

Example: Future Value of an Ordinary Annuity Stream Jill has been faithfully depositing $2,000 at the end of each year over the past 10 years into an account that pays a guaranteed 8% per year. How much money has she have accumulated in the account?

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**4.2 Future Value of an Annuity Stream**

Example Answer (via the long way) Future Value of Payment One = $2,000 x = $3,998.01 Future Value of Payment Two = $2,000 x = $3,701.86 Future Value of Payment Three = $2,000 x = $3,427.65 Future Value of Payment Four = $2,000 x = $3,173.75 Future Value of Payment Five = $2,000 x = $2,938.66 Future Value of Payment Six = $2,000 x = $2,720.98 Future Value of Payment Seven = $2,000 x = $2,519.42 Future Value of Payment Eight = $2,000 x = $2,332.80 Future Value of Payment Nine = $2,000 x = $2,160.00 Future Value of Payment Ten = $2,000 x = $2,000.00 Total Value of Account at the end of 10 years $28,973.13

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**4.2 Future Value of an Annuity Stream**

Example Answer (short way) FORMULA METHOD FV = PMT x (1+r)n -1 r where, PMT = $2,000; r = 8%; and n=10. FVIFA =[((1.08)10 - 1)/.08] = , FV = $2000 x = $28, USING A FINANCIAL CALCULATOR N= 10; PMT = -2,000; I/Y = 8; PV=0; CPT FV = $28,973.13 4.2 Future Value of an Annuity Stream

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**4.2 Future Value of an Annuity Stream**

USING AN EXCEL SPREADSHEET Enter =FV(8%, 10, -2000, 0, 0); Output = $28, Rate = 0.08, Nper = 10, Pmt = -2,000, PV =0, and Type is 0, for ordinary annuities USING FVIFA TABLE (A-3), page 575 Find the FVIFA in the 8% column and the 10 period row; FVIFA = FV = 2000 x = $ (off by 7 cents)

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**FIGURE 4.3 Interest and principal growth with different interest rates for $100 annual payments.**

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**4.3 Present Value of an Annuity**

To calculate the value of a series of equal periodic cash flows at the current point in time, we can use the following simplified formula: The last portion of the equation is the Present Value Interest Factor of an Annuity (PVIFA). Practical applications include figuring out the nest egg needed prior to retirement or lump sum needed for college expenses.

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**FIGURE 4.4 Time line of present value of annuity stream.**

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**4.3 Present Value of an Annuity**

Example problem for the four solution methods You are now holding the winning lottery ticket that will pay the holder of the ticket $10,000 per year for the next 20 years. A friend has offered to buy the winning ticket from you. What should you sell the ticket for assuming you have a discount rate of 6% on future dollars (this is your opportunity rate for the future cash flow)? Four ways to solve Formula Calculator Spreadsheet Table

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**4.3 Present Value of an Annuity**

Formula Inputs? N = 20, r = 0.06, PMT = $10,000 and Compute PV, PV = $10,000 x [1 – 1/(1.06)20] / 0.06 = $114,699.21 Calculator Inputs? N = 20, I/Y = 6.0, PMT = 10,000, FV = 0 Compute PV PV = -$114,699.21

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**4.3 Present Value of an Annuity**

Spreadsheet, use PV function Inputs? Rate = 0.06, Nper = 20, Pmt = 10,000, Fv = 0 PV = -$114,699.21 Table First find the PVIFA with n = 20 and r = 6.0% on page 576, PVIFA = Calculate PV = $10,000 x = $114, (off by 21 cents)

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**4.4 Annuity Due and Perpetuity**

A cash flow stream such as rent, lease, and insurance payments, which involves equal periodic cash flows that begin right away or at the beginning of each time interval is known as an annuity due.

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**4.4 Annuity Due and Perpetuity**

Formula Adjustment PV annuity due = PV ordinary annuity x (1+r) FV annuity due = FV ordinary annuity x (1+r) PV annuity due > PV ordinary annuity FV annuity due > FV ordinary annuity Can you see why? Financial calculator Mode set to BGN for annuity due Mode set to END for an ordinary annuity Spreadsheet Type = 0 or omitted for an ordinary annuity Type = 1 for an annuity due.

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**4.4 Annuity Due and Perpetuity**

Example: Annuity Due versus Ordinary Annuity Let’s say that you are saving up for retirement and decide to deposit $3,000 each year for the next 20 years into an account which pays a rate of interest of 8% per year. By how much will your accumulated nest egg vary if you make each of the 20 deposits at the beginning of the year, starting right away, rather than at the end of each of the next twenty years?

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**4.4 Annuity Due and Perpetuity**

FV ordinary annuity = $3,000 x [((1.08)20 - 1)/.08] = $3,000 x = $137,285.89 FV of annuity due = FV of ordinary annuity x (1+r) FV of annuity due = $137, x (1.08) = $148,268.76 Difference is $10,982.87 Example Answer Given information: PMT = -$3,000; n=20; i= 8%; PV=0;

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**4.4 Annuity Due and Perpetuity**

Perpetuity A Perpetuity is an equal periodic cash flow stream that will never cease. The PV of a perpetuity is calculated by using the following equation:

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**4.4 Annuity Due and Perpetuity**

Example: PV of a perpetuity If you are considering the purchase of a consol that pays $60 per year forever, and the rate of interest you want to earn is 10% per year, how much money should you pay for the consol? Answer: r=10%, PMT = $60; and PV = ($60/0.10) = $600 $600 is the most you should pay for the consol. You can think of it this way, if you put $600 in the bank earning 10% you can withdraw the annual interest of $60 every year forever without touching the principal.

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**4.5 Three Payment Methods Discount loan Interest-only loan**

Loan payments can be structured in one of 3 ways: Discount loan Principal and interest is paid in lump sum at end Interest-only loan Periodic interest-only payments, principal due at end. Amortized loan Equal periodic payments of principal and interest

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4.5 Three Payment Methods Example: Discount versus Interest-only versus Amortized loans Roseanne wants to borrow $40,000 for a period of 5 years. The lenders offers her a choice of three payment structures: 1) Pay all of the interest (10% per year) and principal in one lump sum at the end of 5 years; 2) Pay interest at the rate of 10% per year for 4 years and then a final payment of interest and principal at the end of the 5th year; 3) Pay 5 equal payments at the end of each year inclusive of interest and part of the principal. Under which of the three options will Roseanne pay the least interest and why? Calculate the total amount of the payments and the amount of interest paid under each alternative.

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4.5 Three Payment Methods Method 1: Discount Loan. Since all the interest and the principal is paid at the end of 5 years we can use the FV of a lump sum equation to calculate the payment required, i.e. FV = PV x (1 + r)n FV5 = $40,000 x (1+0.10)5 = $40,000 x = $64, Interest paid = Total payment - Loan amount Interest paid = $64, $40,000 = $24,420.40

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**4.5 Three Payment Methods Annual Interest Payment (Years 1-4)**

Method 2: Interest-Only Loan. Annual Interest Payment (Years 1-4) = $40,000 x 0.10 = $4,000 each year ($16,000) Year 5 payment = Annual interest payment + Principal payment = $4,000 + $40,000 = $44,000 Total payment = $16,000 + $44,000 = $60,000 Interest paid = $60,000 - $40,000 = $20,000

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**4.5 Three Payment Methods Method 3: Amortized Loan.**

n = 5; I/Y = 10.0; PV=$40,000; FV = 0; CPT PMT= $10, Total payments = 5 x $10, = $52,759.50 Interest paid = Total Payments - Loan Amount Interest paid = $52, $40,000 Interest paid = $12,759.50

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**4.5 Three Payment Methods Loan Type Total Payment Interest Paid**

Discount Loan $64, $24,420.40 Interest-only Loan $60, $20,000.00 Amortized Loan $52, $12,759.31 Why does the equal annual payments of principal and interest each period have the lowest total interest?

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**4.6 Amortization Schedules**

Tabular listing of the allocation of each loan payment towards interest and principal reduction Helps borrowers and lenders figure out the payoff balance on an outstanding loan. Procedure: 1) Compute the amount of each equal periodic payment (PMT) using the ordinary annuity formula. 2) Calculate interest on unpaid balance at the end of each period, minus it from the PMT, reduce the loan balance by the remaining amount, 3) Continue the process for each payment period, until we get a zero loan balance.

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**4.6 Amortization Schedules**

Example: Loan amortization schedule. Prepare a loan amortization schedule for the amortized loan option given in the previous Example with the five annual payments for the $40,000 at 10% annual interest rate. What is the loan payoff amount at the end of 2 years? Step One, determine the annual payment: PV = $40,000; n=5; I/Y=10.0; FV=0; CPT PMT = $10, (rounded to nearest whole cent)

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Amortization Table Year Beg. Bal Payment Interest Prin. Red End. Bal 1 40,000.00 10,551.90 4,000.00 6,551.90 33,448.10 2 3,344.81 7,207.09 26,241.01 3 2,264.10 7,927.80 18,313.21 4 1,831.32 8,720.58 9,592.64 5 959.26 0.00 The loan payoff amount at the end of 2 years is $26,241.01

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**4.7 Waiting Time and Interest Rates for Annuities**

Problems involving annuities typically have 4 variables, i.e. PV or FV, PMT, r, n If any 3 of the 4 variables are given, we can easily solve for the fourth one. This section deals with the procedure of solving problems where either n or r is not given. For example: Finding out how many deposits (n) it would take to reach a retirement or investment goal; Figuring out the rate of return (r) required to reach a retirement goal given fixed monthly deposits,

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**4.7 Waiting Time and Interest Rates for Annuities**

Example: Solving for the number of annuities involved Martha wants to save up $100,000 as soon as possible so that she can use it as a down payment on her dream house. She figures that she can easily set aside $8,000 per year and earn 8% annually on her deposits. How many years will Martha have to wait before she can buy that dream house?

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**4.7 Waiting Time for Annuities**

Method 2: Using a financial calculator INPUT ? TVM KEYS N I/Y PV PMT FV Compute Method 3: Using an Excel spreadsheet Using the “NPER” function we enter the following: Rate = 8%; Pmt = -8000; PV = 0; FV = ; Type = 0 or omitted; display in excel = NPER(8%,-8000,0,100000,0) The cell displays

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**4.7 Waiting Time for Annuities**

Method 1: Formula (uses natural logs) N = ln ([FV x r]/PMT + 1) / ln (1+r) N = ln ([100,000 x 0.08]/8, ) / ln 1.08 N = ln 2 / ln 1.08 = / = Method 4: Tables You need to interpret from the tables… Take FV / PMT to find the FVIFA, Look for under the 8% column, find its close to n = 9 (its between 9 and 10 but very close to 9)

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**4.8 Finding the interest rate**

Solving a Lottery Problem In the case of lottery winnings, 2 choices 1) Annual lottery payment for fixed number of years, OR 2) Lump sum payout. How do we make an informed judgment? Need to figure out the implied rate of return of both options using TVM functions.

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**4.8 Finding the interest rate**

Example: Calculating an implied rate of return given an annuity Let’s say that you have just won the state lottery. The authorities have given you a choice of either taking a lump sum of $26,000,000 or a 30-year annuity of $1,500,000. Both payments are assumed to be after-tax. What will you do? The missing variable is the implied interest rate on the two payment choices.

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**4.8 Finding the interest rate**

Using the TVM keys of a financial calculator, enter: PV=26,000,000; FV=0; N=30; PMT = -1,500,000; CPT I/Y = 3.98% 3.98% = rate of interest used to determine the 30-year annuity of $1,500,000 versus the $26,000,000 lump sum pay out. Choice: If you can earn an annual after-tax rate of return higher than 4.0% over the next 30 years, go with the lump sum. Otherwise, take the annuity option.

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**4.8 Finding the interest rate**

We could use the Spreadsheet functions (Rate function) to find the 3.98%. We could use the Tables to estimate the interest rate by looking at the PVIFA at 30 years with the PVIFA calculated as PV / PMT but again we will need to estimate between to interest rates although in this case it will be very close to 4.0% (PVIFA is ) We can not use the formula to solve for interest rate, it is an iterative process (trial and error)

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**4.9 Ten Important Points about the TVM Equation**

Amounts of money can be added or subtracted only if they are at the same point in time. The timing and the amount of the cash flow are what matters. It is very helpful to lay out the timing and amount of the cash flow with a timeline. Present value calculations discount all future cash flow back to current time. Future value calculations value cash flows at a single point in time in the future

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**4.9 Ten Important Points about the TVM Equation**

An annuity is a series of equal cash payments at regular intervals across time. The time value of money equation has four variables but only one basic equation, and so you must know three of the four variables before you can solve for the missing or unknown variable. There are three basic methods to solve for an unknown time value of money variable: (1) Using equations and calculating the answer; (2) Using the TVM keys on a calculator; (3) Using financial functions from a spreadsheet.

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**4.9 Ten Important Points about the TVM Equation**

There are 3 basic ways to repay a loan: Discount loans, Interest-only loans, and Amortized loans. Despite the seemingly accurate answers from the time value of money equation, in many situations not all the important data can be classified into the variables of present value, i.e., time, interest rate, payment, or future value.

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Chapter 3 Mathematics of Finance

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