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**Lecture Four – The Time Value of Money (Pt 2)**

Corporate Finance Lecture Four – The Time Value of Money (Pt 2)

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Learning Objectives 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 equation 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, interest-only 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.

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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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**Figure 4.1 The time line of a nest egg**

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**4.1 Future Value of Multiple Payment Streams (continued)**

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?

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**4.1 Future Value of Multiple Payment Streams (Example 1 Answer)**

FV = PV x (1+r)n FV of Cash Flow at T0 = $3,000 x (1.10)3 = $3,000 x 1.331= $3,993.00 FV of Cash Flow at T1 = $5,000 x (1.10)2 = $5,000 x = $6,050.00 FV of Cash Flow at T2 = $7,000 x (1.10)1 = $7,000 x = $7,700.00 FV of Cash Flow at T3 = $9,000 x (1.10)0 = $9,000 x = $9,000.00 Total = $26,743.00

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**4.1 Future Value of Multiple Payment Streams (Example 1 Answer)**

ALTERNATIVE METHOD: Using the Cash Flow (CF) key of the calculator, enter the respective cash flows. CF0=-$3000;CF1=-$5000;CF2=-$7000; CF3=-$9000; Next calculate the NPV using I=10%; NPV=$20,092.41; Finally, using PV=-$20,092.41; n=3; i=10%;PMT=0; CPT FV=$26,743.00

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**4.2 Future Value of an Annuity Stream**

Annuities are equal, periodic outflows/inflows., 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 annuity stream is as follows: FV = PMT * (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 periods involved.

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**4.2 Future Value of an Annuity Stream (continued)**

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?

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**4.2 Future Value of an Annuity Stream (continued)**

Example 2 Answer 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 (continued)**

Example 2 (Answer) FORMULA METHOD FV = PMT * (1+r)n -1 r where, PMT = $2,000; r = 8%; and n=10. FVIFA [((1.08)10 - 1)/.08] = , FV = $2000* $28,973.13 USING A FINANCIAL CALCULATOR N= 10; PMT = -2,000; I = 8; PV=0; CPT FV = 28,973.13

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**4.2 Future Value of an Annuity Stream (continued)**

USING AN EXCEL SPREADSHEET Enter =FV(8%, 10, -2000, 0, 0); Output = $28,973.13 Rate, Nper, Pmt, PV,Type Type is 0 for ordinary annuities and 1 for annuities due USING FVIFA TABLE (A-3) Find the FVIFA in the 8% column and the 10 period row; FVIFA = FV = 2000* = $

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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 (continued)**

Example 3: Present Value of an Annuity. John wants to make sure that he has saved up enough money prior to the year in 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?

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**4.3 Present Value of an Annuity (continued)**

Example 3 Answer Using the following equation: Calculate the PVIFA value for n=4 and r=7% Then, multiply the annuity payment by this factor to get the PV, PV = $40,000 x = $135,488.45

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**4.3 Present Value of an Annuity (continued)**

Example 3 Answer—continued FINANCIAL CALCULATOR METHOD: 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 Rate, Nper, Pmt, FV, Type

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**4.3 Present Value of an Annuity (continued)**

Example 3 Answer—continued PVIFA TABLE (APPENDIX A-4) METHOD For r =7% and n = 4; PVIFA =3.3872 PVA = PMT*PVIFA = 40,000*3.3872 = $135,488 (Notice the slight rounding error!)

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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. Figure 4.5 An ordinary annuity versus an annuity due.

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**4.4 Annuity Due and Perpetuity**

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 BGN for annuity due Mode 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 (continued)**

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?

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**4.4 Annuity Due and Perpetuity (continued)**

Example 4 Answer Given information: PMT = -$3,000; n=20; i= 8%; PV=0; FV ordinary annuity = $3,000 * [((1.08)20 - 1)/.08] = $3,000 * = $137,285.89 FV of annuity due = FV of ordinary annuity * (1+r) FV of annuity due = $137,285.89*(1.08) = $148,268.76

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**4.4 Annuity Due and Perpetuity (continued)**

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 (continued)**

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? Answer: r=10%, PMT = $60; and PV = ($60/.1) = $600 $600 is the most you should pay for the consol.

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**4.5 Three Loan Payment Methods**

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 Loan Payment Methods (continued)**

Example 6: 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: Pay all of the interest (10% per year) and principal in one lump sum at the end of 5 years; 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; 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 Loan Payment Methods (continued)**

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 Loan Payment Methods (continued)**

Method 2: Interest-Only Loan. Annual Interest Payment (Years 1-4) = $40,000 x 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

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**4.5 Three Loan Payment Methods (continued)**

Method 3: Amortized Loan. n = 5; I = 10%; PV=$40,000; FV = 0;CPT PMT=$10,551.86 Total payments = 5*$10,551.8 = $52,759.31 Interest paid = Total Payments - Loan Amount = $52, $40,000 Interest paid = $12,759.31 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

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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: Compute the amount of each equal periodic payment (PMT). Calculate interest on unpaid balance at the end of each period, minus it from the PMT, reduce the loan balance by the remaining amount, Continue the process for each payment period, until we get a zero loan balance.

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**4.6 Amortization Schedules (continued)**

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

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**4.6 Amortization Schedules (continued)**

Year Beg. Bal Payment Interest Prin. Red End. Bal 1 40,000.00 10,551.89 4,000.00 6,551.89 33,448.11 2 3,344.81 7,207.08 26,241.03 3 2,264.10 7,927.79 18,313.24 4 1,831.32 8,720.57 9,592.67 5 959.27 The loan payoff amount at the end of 2 years is $26,241.03

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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 (continued)**

Example 8: 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 and Interest Rates for Annuities (continued)**

Example 8 Answer Method 1: Using a financial calculator INPUT ? TVM KEYS N I/Y PV PMT FV Compute Method 2: Using an Excel spreadsheet Using the “=NPER” function we enter the following: Rate = 8%; Pmt = -8000; PV = 0; FV = ; Type = 0 or omitted; i.e. =NPER(8%,-8000,0,100000,0) The cell displays

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**4.8 Solving a Lottery Problem**

In the case of lottery winnings, 2 choices Annual lottery payment for fixed number of years, OR 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 Solving a Lottery Problem (continued)**

Example 9: 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?

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**4.8 Solving a Lottery Problem (continued)**

Example 9 Answer Using the TVM keys of a financial calculator, enter: PV=26,000,000; FV=0; N=30; PMT = -1,625,000; CPT I = % % = rate of interest used to determine the 30-year annuity of $1,625,000 versus the $26,000,000 lump sum pay out. Choice: If you can earn an annual after-tax rate of return higher than 4.65% over the next 30 years, go with the lump sum. Otherwise, take the annuity option.

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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 (continued)**

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 (continued)**

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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**Additional Problems with Answers Problem 1**

Present Value of an Annuity Due. Julie has just been accepted into Harvard and her father is debating whether he should make monthly lease payments of $5,000 at the beginning of each month, on her flashy apartment or to prepay the rent with a one-time payment of $56, 662.If Julie’s father earns1% per month on his savings should he pay by month or take the discount by making the single annual payment?

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**Additional Problems with Answers Problem 1 (Answer)**

P/Y = 12; C/Y = 12; MODE = BGN INPUT ,662 5,000 0 TVM KEYS N I/Y PV PMT FV OUTPUT % Monthly rate = 12.7%/12 = % If he can get 1% interest per month...then his annual rate is 12% and he can generate $4, per month with the $56,662 it would take to pay off the rent. He is ahead $15.49 per month by making the one time payment. INPUT ,662 0 TVM KEYS N I/Y PV PMT FV OUTPUT 4,984.51

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**Additional Problems with Answers Problem 2**

Future Value of Uneven cash flows. If Mary deposits $4000 a year for three years, starting a year from today, followed by 3 annual deposits of $5000, into an account that earns 8% per year, how much money will she have accumulated in her account at the end of 10 years?

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**Additional Problems with Answers Problem 2 (Answer)**

Future Value in Year 10 = $4000*(1.08)9 + $4000*(1.08)8 + $4000*(1.08)7 + $5000*(1.08)6 + $5000*(1.08)5 + $5000*(1.08)4 =$4000*1.999+$4000* $4000* $5000* $5000* $5000*1.3605 =$7,996+$7,403.6+$6,855.2+ $7,934+ $7, ,802.5 =$44,337.8

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**Additional Problems with Answers Problem 2 (Answer) (continued)**

ALTERNATIVE METHOD: Using the Cash Flow (CF) key of the calculator, enter the respective cash flows. CF0=0;CF1=-$4000;CF2=-$4000;CF3=-$4000; CF4=-$5000; CF5=-$5000; CF6=-$5000 Next calculate the NPV using I=8%; NPV=$20,537.30; Finally, using PV=-$20,537.30; n=10; i=8%; PMT=0; CPT FV$44,338

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**Additional Problems with Answers Problem 3**

Present Value of Uneven Cash Flows: Jane Bryant has just purchased some equipment for her beauty salon. She plans to pay the following amounts at the end of the next five years: $8,250, $8,500, $8,750, $9,000, and $10,500. If she uses a discount rate of 10 percent, what is the cost of the equipment that she purchased today?

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**Additional Problems with Answers Problem 3 (Answer)**

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**Additional Problems with Answers Problem 4**

Computing Annuity Payment: The Corner Bar & Grill is in the process of taking a five-year loan of $50,000 with First Community Bank. The bank offers the restaurant owner his choice of three payment options: Pay all of the interest (8% per year) and principal in one lump sum at the end of 5 years; Pay interest at the rate of 8% per year for 4 years and then a final payment of interest and principal at the end of the 5th year; 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 the owner pay the least interest and why?

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**Additional Problems with Answers Problem 4 (Answer)**

Under option 1: Principal and Interest Due at end Payment at the end of year5 = FVn = PV x (1 + r)n FV5 = $50,000 x (1+0.08)5 = $50,000 x = $73,466.5 Interest paid = Total payment - Loan amount Interest paid = $73, $50,000 = $23,466.50

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**Additional Problems with Answers Problem 4 (Answer) (continued)**

Under option 2: Interest-only Loan Annual Interest Payment (Years 1-4) = $50,000 x 0.08 = $4,000 Year 5 payment = Annual interest payment + Principal payment = $4,000 + $50,000 = $54,000 Total payment = $16,000 + $54,000 = $70,000 Interest paid = $20,000

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**Additional Problems with Answers Problem 4 (Answer) (continued)**

Option 3: Amortized Loan. To calculate the annual payment of principal and interest we can use the PV of an ordinary annuity equation and solve for the PMT value using n = 5; I = 8%; PV=$50,000, and FV = 0. PMT $12,522.82 Total payments = 5*$12, = $62,614.11 Interest paid = Total Payments - Loan Amount = $62, $50,000 Interest paid = $12,614.11

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**Additional Problems with Answers Problem 4 (Answer) (continued)**

Comparison of total payments and interest paid under each method Loan Type Total Payment Interest Paid Discount Loan $73,466.5 $23, Interest-only Loan $70, $20, Amortized Loan $62, $12, 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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**Additional Problems with Answers Problem 5**

Loan amortization. Let’s say that the restaurant owner in Problem 4 above decides to go with the amortized loan option and after having paid 2 payments decides to pay off the balance. Using an amortization schedule calculate his payoff amount. Amount of loan = $50,000; Interest rate = 8%; Term = 5 years; Annual payment = $12,522.82

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**Additional Problems with Answers Problem 5 (Answer)**

AMORTIZATION SCHEDULE Year Beg. Bal Payment Interest Prin. Red End Bal. 1 50, , , , ,477.18 2 41, , , , ,272.53 3 2, , , , ,331.51 4 22, , , , ,595.21 5 11, , , The loan payoff amount at the end of 2 years is $32,272.53

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**Figure 4.2 The time line of a $1,000-per year nest egg.**

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