Chapter 04 - The Time Value of Money (Part 2) PDF

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Chapter(4( The(Time(Value(of(Money((Part(2)( LEARNING(OBJECTIVES(

(Slide(4E2)(

1. Compute the future value of multiple cash flows. 2. Determine the future value of an annuity. 3. Determine the present value of an annuity. 4. Adjust the annuity formula for present value and future value for an annuity due and understand the concept of a perpetuity. 5. Distinguish between the different types of loan repayments: discount loans, interestonly loans, and amortized loans. 6. Build and analyze amortization schedules. 7. Calculate waiting time and interest rates for an annuity. 8. Apply the time value of money concepts to evaluate the lottery cash flow choice. 9. Summarize the ten essential points about the time value of money.

IN(A(NUTSHELL…( In Part two of this 2-part unit on the time value of money topic, the author discusses and illustrates how the time value of money equation can be modified and used for calculations involving the compounding and discounting of interest in cash flow streams that are more complex that mere lump sums. Real life situations seldom involve single outflow/inflow types of cash flow streams. More often, we are faced with periodic outflows such as loan, rent, or lease payments and/or periodic inflows such as retirement annuities. In this chapter, we learn how to calculate the present and future values of more complex cash flow streams such as those involving unequal cash flows, ordinary annuities and annuities due. In addition, the different methods by which loans can be paid off; and the method of setting up and analyzing amortization schedules associated with mortgages and other installment loans are also covered.

LECTURE(OUTLINE( 4.1(Future(Value(of(Multiple(Payment(Streams( (Slides(4E3(to(4E7)( In the case of investments involving unequal periodic cash flows, we can calculate the future value of the cash flows by treating each of the cash flows as a lump sum and calculate its future value over the relevant number of periods. The individual future values are then summed up to get the future value of the multiple payment streams.

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It is best to use a time line, as shown in Figure 4-1 in the text, which clearly shows each cash flow, the respective number of periods over which interest is to be compounded, and the interest rate that will apply. There is a shorter alternative way to solve the future value of a stream of unequal periodic cash flows, which involves using the Net Present Value (NPV) function of a financial calculator or a spreadsheet. We can first compute the net present value (at t = 0) of the stream of uneven cash flows at the given rate of interest, and then using the NPV as the present value, we find the future value of a lump sum at the end point of the cash flow stream. This method is shown in Example 1 below. Example#1:#Future#Value#of#an#Uneven#Cash#Flow#Stream# Jim deposits $3,000 today into an account that pays 10% per year, and follows it up with 3 more deposits at the end of each of the next three years. Each subsequent deposit is $2,000 higher than the previous one. How much money will Jim have accumulated in his account by the end of three years? T0

T1

$3,000

T2

$5,000

$3,000 × (1.10)

3

T3

$7,000 $5,000 × (1.10)

2

$9,000

$7,000 × (1.10)1 $7,700.00 $6,050.00 $3,993.00

We use the Future Value of a Single Sum formula and compound each cash flow for the relevant number of years over which interest will be earned. Then we sum up the compounded values to get the accumulated value of Jim’s deposits at the end of three years as shown below: FV = PV × (1 + r)n FV of

= $3,000 × (1.10)3 = $3,000 × 1.331 = $3,993.00

FV of

= $5,000 × (1.10)2 = $5,000 × 1.210 = $6,050.00

FV of

= $7,000 × (1.10)1 = $7,000 × 1.100 = $7,700.00

FV of

= $9,000 × (1.10)0 = $9,000 × 1.000 = $9,000.00

Total = $26,743.00 Note: Students should be reminded that cash flows can only be added up if they occur at the same point in time as at the end of Year 3.

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ALTERNATIVE(METHOD:( Using the Cash Flow (CF) key of the calculator, enter the respective cash flows. That is, CF0 = -$3000; CF1 = -$5000; CF2 = -$7000; CF3 = -$9000; Next calculate the NPV using I = 10%; ; Finally, using PV = -$20,092.41; n = 3; i = 10%; PMT = 0;

4.2(Future(Value(of(an(Annuity(Stream(

(Slides(4E8(to(4E13)(

Often, we are faced with financial situations which involve equal, periodic outflows/inflows. Such payment streams are known as annuities. Examples of an annuity stream include rent, lease, mortgage, car loan, and retirement annuity payments. An annuity stream can begin at the start of each period as is true of rent and insurance payments or at the end of each period, as in the case of mortgage and loan payments. The former type is called an annuity due while the latter is known as an ordinary annuity stream. This section covers ordinary annuities. Annuities due will be covered in a later section. Although the future value of an ordinary annuity stream can be calculated by using the same process that was explained in section 4.1 above, there is a simplified formula which makes the process much easier. The formula for calculating the future value of an annuity stream is as follows:

฀ (1+ r )n − 1฀ ฀ FV = PMT × ฀ 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 periods involved. The item that PMT is multiplied by is known as the Future Value Interest Factor of an Annuity (FVIFA). It can be either calculated using the equation or got from a table provided in Appendix A-3. Of course, table values are only available for discrete interest rates and time periods. Note: The length of the period can be a day, week, quarter, month, or any other equal unit of time, not just a year as is often misunderstood by students. The rate of interest, however, is often given on an annual basis and must be accordingly adjusted and used in the problem. Example#2:#Future#Value#of#an#Ordinary#Annuity#Stream# Jill has been faithfully depositing $2,000 at the end of each year since the past 10 years into an account that pays 8% per year. How much money will she have accumulated in the account? This problem could be solved by first calculating the FV of each of the year end deposits for the respective number of years involved, and then summing them all up at the end of year 10 as shown below: Future Value of Payment One = $2,000 × 1.089 = 8

Future Value of Payment Two = $2,000 × 1.08 =

$3,998.01 $3,701.86

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Future Value of Payment Three = $2,000 × 1.087 = $3,427.65 Future Value of Payment Four = $2,000 × 1.086 = 5

$3,173.75

Future Value of Payment Five = $2,000 × 1.08 =

$2,938.66

Future Value of Payment Six = $2,000 × 1.084 =

$2,720.98

3

Future Value of Payment Seven = $2,000 × 1.08 = $2,519.42 Future Value of Payment Eight = $2,000 × 1.082 = $2,332.80 Future Value of Payment Nine = $2,000 × 1.081 = $2,160.00 Future Value of Payment Ten = $2,000 × 1.080 =

$2,000.00

Total Value of Account at the end of 10 years $28,973.13 FORMULA(METHOD( It is much quicker to solve this problem using the following formula:

฀ (1+ r )n − 1฀ ฀ FV = PMT × ฀ r where, PMT = $2,000; r = 8%; and n=10 i.e. the number of deposits involved. The FVIFA would equal [((1.08)10 - 1)/.08] = 14.486562, and the FV = $2000 × 14.486562 = $28,973.13 USING(A(FINANCIAL(CALCULATOR( N= 10; PMT = -2,000; I = 8; PV=0; CPT FV = 28,973.13 USING(AN(EXCEL(SPREADSHEET( Enter =FV(8%, 10, -2000, 0, 0); Output = $28,973.125 Rate, Nper, Pmt, PV,Type Type is 0 for ordinary annuities and 1 for annuities due USING(FVIFA(TABLE(AE3( Find the FVIFA in the 8% column and the 10 period row; FVIFA = 14.486 FV = 2000 × 14.4865 = $28.973.12

4.3(Present(Value(of(an(Annuity(

(Slides(4E14(to(4E19)(

If we are interested in finding out the value of a series of equal periodic cash flows at the current point in time, we can either sum up the discounted values of each periodic cash flow (PMT) for the related number of periods or use the following simplified formula:

PV = PMT ×

฀ ฀ 1 ฀฀ ฀ 1−฀ n ฀฀ ฀ ฀ ฀฀ ฀ (1+ r ) ฀ ฀฀ r

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฀ ฀ 1 ฀฀ ฀ 1− ฀ n ฀฀ ฀฀ ฀฀ (1+ r ) ฀฀฀฀ The last portion of the equation, , is the Present Value Interest Factor of r an Annuity (PVIFA). The values of various PVIFAs are displayed in Appendix A-4 for different combinations of discrete interest or discount rates (r) and the number of payments (n). A practical application of such a calculation would be in calculating how much to have saved up in an account prior to a child attending college or prior to retirement so as to be able to withdraw equal annual amounts each year over a required number of years. Example#3:#Present#Value#of#an#Annuity# John wants to make sure that he has saved up enough money prior to the year before which his daughter begins college. Based on current estimates, he figures that college expenses will amount to $40,000 per year for 4 years (ignoring any inflation or tuition increases during the 4 years of college). How much money will John need to have accumulated in an account that earns 7% per year, just prior to the year that his daughter starts college? Here the known variables are: n=4; PMT=$40,000; and r = 7%. What we have to solve for is the PV of the annuity series, which can be done by any one of 4 methods, i.e. formula, financial calculator, spreadsheet, or PVIFA factor using Table A-4. FORMULA(METHOD:( Using the following equation:

฀ ฀ 1 ฀฀ ฀ 1−฀ n ฀฀ ฀฀ ฀฀ (1+ r ) ฀฀ ฀฀ PV = PMT × r 1. Calculate the PVIFA value for n=4 and r=7%.

฀ ฀ ฀฀ 1 ฀ 1− ฀ ฀ [1− (0.762895)] ฀ 4 ฀฀ ฀฀ (1+ 0.07) ฀฀฀฀ 0.07 = =3.387211 0.07 2. Then, multiply the annuity payment by this factor to get the PV, PV = $40,000 × 3.387211 = $135,488.45 FINANCIAL(CALCULATOR(METHOD:( It is important to remind students that the calculator must be in END mode so that the payments are treated as an ordinary annuity. Set the calculator for an ordinary annuity (END mode) and then enter: N= 4; PMT = 40,000; I = 7; FV=0; CPT PV = 135,488.45 SPREADSHEET(METHOD:( Enter =PV(7%, 4, 40,000, 0, 0); Output = $135,488.45

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Rate, Nper, Pmt, FV,Type PVIFA(TABLE((APPENDIX(AE4)(METHOD( Since the rate of interest (7%) and the number of withdrawals (4) are both discrete values, we could easily solve this problem by obtaining the PVIFA value from the table in Appendix A-4 i.e. 3.3872 and multiplying it by the PMT ($40,000) to get a PV of $135488. Notice the slight rounding error!

4.4(Annuity(Due(and(Perpetuity(

(Slides(4E20(to(4E25)(

Annuity(due:( Certain types of financial transactions such as rent, lease, and insurance payments involve equal periodic cash flows that begin right away or at the beginning of each time interval. This type of annuity is known as an annuity due. Figure 4.5 in the text shows both types of annuities on a time line. An annuity due stream is scaled back for one period as shown by the arrows. Note that when calculating the PV of an annuity due stream one less period of interest would be required for each payment, since the first cash flow begins right away. Likewise, when calculating the FV of the cash flow stream at the end of 4 periods, an additional period of interest would apply to each periodic cash flow, since the 4th payment occurs at the beginning of the 4th year. Figure(4.5( Ordinary(Annuity(versus(Annuity(Due(( T0

#1

#2

T1

$100

$100

$100 $100

T2

T3

$100

$100

$100

T4

$100 Ordinary Annuity

Annuity Due

For problems involving an annuity due, the equations used to calculate the PV and FV of an ordinary annuity can simply be adjusted by multiplying them by the term (1+r). That is,

฀ ฀ ฀ 1 ฀ ฀ 1− ฀ n ฀ ฀ ฀฀ (1+ r ) ฀ PV = ฀ PMT × r ฀ ฀฀ ฀

฀฀ ฀฀ ฀฀ ฀฀

฀ ฀ ฀ ฀ ฀ ฀ ฀ ฀× (1+ r ) ฀ ฀฀ ฀

Or PV annuity due = PV ordinary annuity × (1+r) AND

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n ฀ 1 + r ) −1 ฀ ( FV = ฀ PMT × ฀ ×( 1+ r ) ฀ ฀ r ฀ ฀

4.6

Or FV annuity due = FV ordinary annuity × (1+r) When using a financial calculator we must set the mode to BGN for an annuity due or END for an ordinary annuity and proceed just as we would any other PV or FV problem. In the case of a spreadsheet, the “Type” of the cash flow, within the =PV or =FV functions is set to “0” or omitted for an ordinary annuity and “1” for an annuity due. Example#4:#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? Given information: PMT = -$3,000; n=20; i= 8%; PV=0; n ฀ 1+ r ) − 1 ฀ ( Future value of an ordinary annuity = FV = ฀ PMT × ฀ ฀ ฀ r ฀ ฀

= $3,000 × [((1.08)20 - 1)/.08] = $3,000 × 45.76196 = $137,285.89 Future value of an annuity due

n ฀ 1+ r ) − 1 ฀ ( = FV = ฀ PMT × ฀× (1+ r ) ฀ ฀ r ฀ ฀

= $137,285.89 × (1.08) = $148,268.76 Note: If we set the mode in the calculator to END; and enter: PMT = -$3,000; n=20; i= 8%; PV=0; and CPT FV; we get FV =$137,285.89 Likewise, if we set the mode to BGN; we get FV = $148,268.76 Similarly, using a spreadsheet and setting the Type to 0 we get FV= $137,285.89 i.e. the PV of an ordinary annuity and if we set the Type to 1; we get FV = 148,268.76 i.e. FV of an annuity due

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Perpetuity( A Perpetuity is an equal periodic cash flow stream that will never cease. One example of such a stream is a British consol, which is a bond issued by the British government which promised to pay a specified rate of interest forever, without ever repaying the principal. The PV of a perpetuity is calculated by using the following equation:

PV =

PMT r

Example#5:#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? Here, r=10%, PMT = $60; and PV = ($60/.1) = $600 is the most you should pay for the consol. Remind students that r should be in decimals.

4.5(Three(Loan(Payment(Methods(

(Slides(4E26(to(4E30)(

Depending on the terms agreed upon at the time of issue, borrowers can typically pay off a loan in one of 3 ways: 1. They can pay off the principal (the original loan amount that you borrowed) and all the interest (the amount the lender charges you for borrowing the money) at one time at the maturity date of the loan. This is called a 2. They can make periodic interest payments and then pay the principal and final interest payment at the maturity date. This is called an 3. They can pay both principal and interest as they go by making equal payments each period. This is called an Example#6:#Discount#versus#InterestPonly#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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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 × (1 + r)n FV5 = $40,000 × (1+0.10)5 = $40,000 × 1.61051 = $64, 420.40 Interest paid = Total payment - Loan amount Interest paid = $64,420.40 - $40,000 = $24,420.40 Method(2:(InterestEOnly(Loan( Annual Interest Payment (Years 1-4) = $40,000 × 0.10 = $4,000 Year 5 payment = Annual interest payment + Principal payment = $4,000 + $40,000 = $44,000 Total payment = $16,000 + $44,000 = $60,000 Interest paid = $20,000 Method(3:(Amortized(Loan( To calculate the annual payment of principal an interest we can use the PV of an ordinary annuity equation and solve for the PMT value using n = 5; I = 10%; PV = $40,000, and FV = 0. That is:

PMT =

PMT =

PV ฀฀ 1 ฀฀ ฀฀1 − n ฀฀ (1 + r) ฀฀ ฀฀ r

4.9

$40,000 ฀฀ 1 ฀฀ ฀฀1 − 5 ฀฀ (1 + 0.10 ) ฀฀ ฀฀ 0. 1

or

$40,000 = $10,551.86 3.7909 Total payments = 5 × $10,551.8 = $52,759.31 Interest paid = Total Payments - Loan Amount = $52,759.31-$40,000 Interest paid = $12,759.31 PMT =

Comparison of Total payments and interest paid under each method Loan Type Total Payment Interest Paid Discount Loan $64,420.40 $24,420.40 Interest-only Loan $60,000.00 $20,000.00 Amortized Loan $52,759.31 $12,759.31 So, the amortized loan is the one with the lowest interest expense, since it requires a higher annual payment, part of which reduces the unpaid balance on the loan and thus results in less interest being charged over the 5-year term.

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4.6(Amortization(Schedules(

(Slides(4E31(to(4E33)( (

An amortization schedule is a tabular listing of the allocation of each loan payment towards interest and principal reduction, which can help borrowers and lenders figure out the payoff balance on an outstanding loan. To prepare an amortization schedule, we must first compute the amount of each equal periodic payment (PMT) using the PVIFA equation or the appropriate financial calculator keys. Next, we calculate the amount of interest that would be charged on the unpaid balance at the end of each period, minus it from the PMT, reduce the loan balance by the remaining amount, and continue the process for each payment period, until we get a zero loan balance. Example#7:#Loan#amortization#schedule# Prepare a loan amortization schedule for the amortized loan option given in Example 6 above. What is the loan payoff amount at the end of 2 years? PV = $40,000; n=5; i=10%; FV=0; CPT PMT = $10,551.89 Year Beg. Bal. Payment Interest 1 40,000.00 10,551.89 4,000.00 2 33,448.11 10,551.89 3,344.8...