the time value of money (part two). 4-2 1. explain and illustrate an annuity. 2. determine the...

The Time Value of Money (Part Two)

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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.

LEARNING OBJECTIVES

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

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Example: Future Value of an Ordinary Annuity StreamJill 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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Example Answer (via the long way)

Future Value of Payment One = $2,000 x 1.089 = $3,998.01Future Value of Payment Two = $2,000 x 1.088 = $3,701.86Future Value of Payment Three = $2,000 x 1.087 = $3,427.65Future Value of Payment Four = $2,000 x 1.086 = $3,173.75 Future Value of Payment Five = $2,000 x 1.085 = $2,938.66

Future Value of Payment Six = $2,000 x 1.084 = $2,720.98 Future Value of Payment Seven = $2,000 x 1.083 = $2,519.42 Future Value of Payment Eight = $2,000 x 1.082 = $2,332.80 Future Value of Payment Nine = $2,000 x 1.081 = $2,160.00 Future Value of Payment Ten = $2,000 x 1.080 = $2,000.00

Total Value of Account at the end of 10 years $28,973.1$28,973.133

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4.2 Future Value of an Annuity Stream USING AN EXCEL SPREADSHEET

Enter =FV(8%, 10, -2000, 0, 0); Output = Enter =FV(8%, 10, -2000, 0, 0); Output = $28,973.13$28,973.13

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 = 14.4866FVIFA = 14.4866FV = 2000 x 14.4866 = FV = 2000 x 14.4866 = $28.973.20 (off by 7 cents)$28.973.20 (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

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r

rPMTPV

n

1

11

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.

FIGURE 4.4 Time line of present value of annuity stream.

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4.3 Present Value of an AnnuityExample 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 AnnuityFormula

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$114,699.21Calculator

Inputs? N = 20, I/Y = 6.0, PMT = 10,000, FV = 0

Compute PVPV = -$114,699.21-$114,699.21

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4.3 Present Value of an AnnuitySpreadsheet, use PV function

Inputs? Rate = 0.06, Nper = 20, Pmt = 10,000, Fv = 0

PV = -$114,699.21-$114,699.21

TableFirst find the PVIFA with n = 20 and r = 6.0%

on page 576, PVIFA = 11.4699Calculate PV = $10,000 x 11.4699 =

$114,699.00 ($114,699.00 (off by 21 cents)

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4.4 Annuity Due and Perpetuity

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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.

4.4 Annuity Due and PerpetuityFormula AdjustmentPV annuity due = PV ordinary annuity x (1+r)FV annuity due = FV ordinary annuity x (1+r)PV annuity due > PV ordinary annuityFV annuity due > FV ordinary annuityCan you see why?

Financial calculatorMode set to BGN for annuity due Mode set to END for an ordinary annuity

SpreadsheetType = 0 or omitted for an ordinary annuity Type = 1 for an annuity due.

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4.4 Annuity Due and PerpetuityExample: Annuity Due versus Ordinary

AnnuityLet’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 PerpetuityExample Answer

Given information: PMT = -$3,000; n=20; i= 8%; PV=0;

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FV ordinary annuity = $3,000 x [((1.08)20 - 1)/.08]

= $3,000 x 45.76196$3,000 x 45.76196 = $137,285.89$137,285.89

FV of annuity due = FV of ordinary annuity x (1+r)FV of annuity due = $137,285.89 x (1.08) = $137,285.89 x (1.08) = $148,268.76$148,268.76

Difference is $10,982.87Difference is $10,982.87

4.4 Annuity Due and Perpetuity

PerpetuityA 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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r

PMTPV

4.4 Annuity Due and PerpetuityExample: 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) = r=10%, PMT = $60; and PV = ($60/0.10) = $600 $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 MethodsLoan payments can be structured in one of 3 ways: 1) Discount loan

• Principal and interest is paid in lump sum at end

2) Interest-only loan• Periodic interest-only payments, principal

due at end.

3) Amortized loan• Equal periodic payments of principal and

interest

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4.5 Three Payment MethodsExample: 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

amountof interest paid under each alternative.

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4.5 Three Payment MethodsMethod 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 FV = PV x (1 + r)= PV x (1 + r)nn

FVFV55 = $40,000 x (1+0.10) = $40,000 x (1+0.10)55 = $40,000 x 1.61051 = $40,000 x 1.61051 = = $64, 420.40$64, 420.40

Interest paid = Total payment - Loan amount Interest paid = $64,420.40 - $40,000 = Interest paid = $64,420.40 - $40,000 = $24,420.40$24,420.40

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4.5 Three Payment Methods

Method 2: Interest-Only Loan.

Annual Interest Payment (Years 1-4) Annual Interest Payment (Years 1-4) = $40,000 x 0.10 = $4,000 each year ($16,000)

Year 5 payment Year 5 payment = Annual interest payment + Principal payment = $4,000 + $40,000 = $44,000

Total payment Total payment = $16,000 + $44,000 = $60,000

Interest paid = $60,000 - $40,000 = Interest paid = $60,000 - $40,000 = $20,000$20,0004-25


Method 3: Amortized Loan. n = 5; I/Y = 10.0; PV=$40,000; FV = 0; n = 5; I/Y = 10.0; PV=$40,000; FV = 0;

CPT PMT= CPT PMT= $10,551.89923$10,551.89923

Total payments = 5 x $10,551.89923 = $52,759.50

Interest paid = Total Payments - Loan AmountInterest paid = $52,759.50 - $40,000Interest paid = $12,759.50$12,759.50

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Loan Type Total Payment Interest PaidDiscount Loan $64,420.40 $24,420.40Interest-only Loan $60,000.00 $20,000.00Amortized Loan $52,759.31 $12,759.31$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 SchedulesTabular listing of the allocation of each loan payment towards interest and principal reductionHelps 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 SchedulesExample: 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; PV = $40,000; n=5; I/Y=10.0; FV=0;

CPT PMT = CPT PMT = $10,551.90 (rounded to nearest $10,551.90 (rounded to nearest whole cent)whole cent)

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The loan payoff amount at the end of 2 years is $26,241.01$26,241.01

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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 33,448.10 10,551.90 3,344.81 7,207.09 26,241.01

3 26,241.01 10,551.90 2,264.10 7,927.80 18,313.21

4 18,313.21 10,551.90 1,831.32 8,720.58 9,592.64

5 9,592.64 10,551.90 959.26 9,592.64 0.00

Amortization Table

4.7 Waiting Time and Interest Rates for Annuities

Problems involving annuities typically have 4 variables, i.e. PV or FV, PMT, r, nIf 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 involvedMartha 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 ? 8.0 0 -8000 100000TVM KEYS N I/Y PV PMT FVCompute Compute 9.0064679.006467

Method 3: Using an Excel spreadsheetUsing the “NPER” function we enter the following:

Rate = 8%; Pmt = -8000; PV = 0; FV = 100000; Type = 0 or omitted;

display in excel = NPER(8%,-8000,0,100000,0)The cell displays 9.006467.9.006467.

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4.7 Waiting Time for AnnuitiesMethod 1: Formula (uses natural logs)

N = ln ([FV x r]/PMT + 1) / ln (1+r)N = ln ([100,000 x 0.08]/8,000 + 1) / ln 1.08N = ln 2 / ln 1.08 = 0.693147181 / 0.07691041 = 9.0064679.006467

Method 4: TablesYou need to interpret from the tables…Take FV / PMT to find the FVIFA, 12.50Look for 12.50 under the 8% column, find its close to n = 9 (its between 9 and 10 but very close to 99)

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4.8 Finding the interest rateSolving a Lottery Problem

In the case of lottery winnings, 2 choices

1) Annual lottery payment for fixed number of years, OR2) 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 rateExample: Calculating an implied rate of return given an annuityLet’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 rateUsing the TVM keys of a financial calculator, enter:PV=26,000,000; FV=0; N=30; PMT = -1,500,000; CPT I/Y = CPT I/Y = 3.98%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 rateWe 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 17.3333)

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 Equation1. Amounts of money can be added or subtracted

only if they are at the same point in time.2. The timing and the amount of the cash flow

are what matters.3. It is very helpful to lay out the timing and

amount of the cash flow with a timeline.4. Present value calculations discount all future

cash flow back to current time.5. 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 Equation6. An annuity is a series of equal cash payments at

regular intervals across time.7. 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.

8. 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 Equation9. There are 3 basic ways to repay a loan:

(1) Discount loans, (2) Interest-only loans, and (3) Amortized loans.

10.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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the time value of money (part two). 4-2 1. explain and illustrate an annuity. 2. determine the...

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