# Time-Value-of-Money and Amortization Worksheets

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1 2 Time-Value-of-Money and Amortization Worksheets The Time-Value-of-Money and Amortization worksheets are useful in applications where the cash flows are equal, evenly spaced, and either all inflows or all outflows. They help you solve problems involving annuities, loans, mortgages, leases, and savings. You can also generate an amortization schedule. Press # and " to move through each set of variables. Z [ \ ]^, -. / 0 2: TVM and Amortization Worksheets 27

2 TVM and Amortization Worksheet Labels Keys Label Meaning Type of Variable, N Number of periods Enter/compute - I/Y Interest rate per year Enter/compute. PV Present value Enter/compute / PMT Payment Enter/compute 0 FV Future value Enter/compute & [ P/Y Number of payments per year Enter-only # C/Y Number of compounding Enter-only periods per year & ] END End-of-period payments Setting & V BGN Beginning-of-period payments Setting & \ P1 Starting payment Enter-only # P2 Ending payment Enter-only # BAL Balance Auto-compute # PRN Principal paid Auto-compute # INT Interest paid Auto-compute Notes about the TVM and Amortization Worksheets & }! sets N=0, I/Y=0, PV=0, PMT=0, FV=0; P/Y=12, C/Y=12; END (not BGN); P1=1, P2=1; BAL=0, PRN=0, INT=0. & ^ sets N, I/Y, PV, PMT, and FV to zero; does not affect P/Y, C/Y, or the BGN/END setting. & [ & z sets P/Y=12 and C/Y=12. & ] & z sets END (not BGN). & \ & z sets P1=1, P2=1, BAL=0, PRN=0, and INT=0. When solving a problem using only four of the five TVM variables, make sure the unused variable is zero. 28 BA II PLUS Calculator

3 Notes about TVM and Amortization Worksheets (cont.) Enter values for PV, PMT, and FV as negative if they are outflows (cash paid out) or as positive if they are inflows (cash received). To enter a negative value, press S after entering the number. Enter I/Y as the nominal interest rate. The TVM worksheet automatically converts I/Y to a per period rate based on the values for P/Y and C/Y. When you enter a value for P/Y, the same value is automatically entered for C/Y. (You can change C/Y.) The END/BGN setting lets you specify whether the transaction is an ordinary annuity or an annuity due. < In ordinary annuities, the payments occur at the end of each payment period. Most loans are in this category. For ordinary annuities, select END. < In annuities due, payments occur at the beginning of each payment period. Most leases are in this category. For annuities due, select BGN. Pressing % when P1 or P2 is displayed updates P1 and P2 to represent the next range of payments. A computed value for BAL after a specified number of payments may be slightly different than a computed value for FV after the same number of payments. < When solving for BAL, PRN, and INT, the calculator uses the PMT value rounded to the number of decimal places specified by the decimal format. < When solving for FV, the calculator uses the unrounded value for PMT. 2: TVM and Amortization Worksheets 29

4 Entering, Recalling, and Computing TVM Values You enter a TVM value by keying in a value and pressing the appropriate TVM key (,, -,., /, or 0). The value is stored in the TVM variable (N, I/Y, PV, PMT, or FV). You recall a TVM value to the display by pressing J and the TVM key. When you enter or recall a value for any of the five TVM variables (N, I/Y, PV, PMT, or FV), you can be in either standard calculator mode or a worksheet mode. The display responds differently according to the mode you are in. In standard calculator mode (accessed by pressing & U), the variable label, the = sign, and the value you entered or recalled are displayed. In worksheet mode, only the value you entered or recalled is displayed. Any label previously in the display remains. To compute a TVM value, press % and the appropriate TVM key. When you compute a TVM value, you must be in standard calculator mode. Press & U to return to standard calculator mode. Entering a Value for N Using [xp/y] You can use & Z to enter a value for N. Pressing & Z automatically multiplies a displayed number by the value stored in the P/Y variable (number of payments per year). By entering the number of years (for example, 30) and pressing & Z, you can compute the number of payments required to pay off an annuity. Press, to enter that value as the number of payments in a TVM calculation. 30 BA II PLUS Calculator

5 Compound Interest Many lending institutions add the interest you earn to the principal. The interest you earn from the previous compounding period becomes part of the principal for the next compounding period. Compound interest enables you to earn a greater amount of interest on your initial investment. In order to earn compound interest, the interest must remain with the principal. For example, if you invest \$100 at an annual interest rate of 10% compounded annually, you earn \$10 interest after one year. At the end of the second year, the interest is calculated on \$110 (\$100 principal plus \$10 accumulated interest), so you earn \$11 in the second year. As additional interest accumulates, your interest earnings increase each year. 2: TVM and Amortization Worksheets 31

6 Time-Line Diagrams A time-line diagram can help you visualize cash flows by showing the amounts paid or received (cash outflows or cash inflows) at various points in time. Cash flows received are shown with arrows pointing up, as with the loan amount at the left. Cash flows invested have arrows pointing down, as with the 35 regular payments and the balloon payment at the right. Loan amount 23,000 Enter inflows as positive. Time Pmt 2 Pmt 3 Pmt 34 Pmt 35-15,000 Balloon Pmt Entering Inflows and Outflows The calculator follows the established convention of treating inflows of cash (cash received) as positive and outflows of cash (cash paid out) as negative. You must enter inflows as positive values and outflows as negative values. The calculator displays computed inflows as positive values and computed outflows as negative values. 32 BA II PLUS Calculator

7 Procedure: Using the TVM Worksheet The worksheet stores the values and settings you enter until you clear the worksheet or change the values or settings. Therefore, you may not need to do all the steps in the procedure every time you work a TVM problem. Press & }! to reset all variables to their defaults (N=0, I/Y=0, PV=0, PMT=0, FV=0; P/Y=12, C/Y=12; END; P1=1, P2=1; BAL=0, PRN=0, INT=0). If P/Y (payments per year) should not be 12, press & [, key in the number of payments per year, and press!. Press #. C/Y= (compounding periods per year) and its current value are displayed. When you enter a value for P/Y, the same value is automatically entered for C/Y. If the compounding periods per year is different than the payments per year, key in the value and press!. If you want beginning-of-period payments (END is the default setting), press & ] & V. If beginning-of-period payments are selected, the BGN indicator is displayed; no indicator is displayed for end-ofperiod payments. Press & U to return to standard calculator mode. Enter values for the four known variables. Key in each value and press the appropriate key (,, -,., /, or 0). The value of the unknown variable is 0. Press % and the key for the unknown variable to compute its value. 2: TVM and Amortization Worksheets 33

8 Procedure: Generating an Amortization Schedule The worksheet for amortization calculations uses the values you entered and computed in the TVM worksheet to compute amortization data. The procedures on these pages give you two ways to generate an amortization schedule. Press & }! to reset all variables to their defaults (N=0, I/Y=0, PV=0, PMT=0, FV=0; P/Y=12, C/Y=12; END; P1=1, P2=1; BAL=0, PRN=0, INT=0). Press & \. P1= and its current value are displayed. Specify the range of payments. < To enter a value for P1 (the first payment in the range), key in a value and press!. < To enter a value for P2 (the last payment in the range), press #, enter a value, and press!. Press # repeatedly to display the automatically computed values: < BAL the remaining balance after payment P2 < PRN the principal < INT the interest paid over the specified range Press & \ or, if INT is displayed, press # to display P1 again. Repeat steps 2 and 3 for each range of payments to generate an amortization schedule. 34 BA II PLUS Calculator

9 Procedure: Automatically Generating a Schedule After you enter the initial values for P1 and P2, as described above, you can automatically compute an amortization schedule. Press & \ or, if INT is displayed, press # to display P1= and its current value. Press %. This automatically updates both P1 and P2 to represent the next range of payments. The calculator computes the next range of payments using the same number of periods as in the previous range of payments. For example, if the previous range was 1 through 12 (12 payments), pressing % updates the range to 13 through 24 (12 payments). Press # to display P2. < If you pressed % when P1 was displayed, a new value for P2 is automatically displayed. (You can still enter a new value for P2, if necessary.) < If you did not press % when P1 was displayed, you can press % when P2 is displayed to enter values for both P1 and P2 for the next range of payments. Press # as needed to display the automatically computed values for BAL, PRN, and INT for the next range of payments. Repeat steps 1 through 4 until the schedule is complete. 2: TVM and Amortization Worksheets 35

10 Basic Loan Calculations Interest Example: Interest Rate You have a 30-year mortgage for \$75,000 and make payments each month of \$ What is the interest rate of your mortgage? Enter number of payments using payment multiplier. 30 & Z, N= Enter loan amount PV= 75, Enter payment amount S / PMT= Compute interest rate. % - I/Y= 8.50 The interest rate is 8.5% per year. 36 BA II PLUS Calculator

11 Basic Loan Calculations Payments Example: Monthly Payment You are considering a 30-year mortgage at 8.5% for \$75,000. How much would the monthly payment be? Enter number of payments using payment multiplier. 30 & Z, N= Enter interest rate I/Y= 8.50 Enter loan amount PV= 75, Compute payment. % / PMT= The monthly payment would be \$ Example: Quarterly Payment (continued from previous example) Your mortgage company also offers an option for a quarterly mortgage with quarterly compounding. How much would your quarterly payment be? (You do not need to enter the loan amount or the interest rate. The compounding periods is automatically reset to equal the payments periods.) Set payments per year to 4. & [ 4! P/Y= 4.00 Return to calculator mode. & U 0.00 Enter number of payments using payment multiplier. 30 & Z, N= Compute payment. % / PMT= -1, The quarterly payment would be \$1, : TVM and Amortization Worksheets 37

12 Future Value of Savings Example: Future Value of Savings You have opened a savings account with \$5,000. The bank pays 5%, compounded at the end of each year. What is the future value of the account after 20 years? Set payments per year to 1. & [ 1! P/Y= 1.00 Return to calculator mode. & U 0.00 Enter number of payments. 20, N= Enter interest rate. 5 - I/Y= 5.00 Enter beginning balance S. PV= -5, Compute future value. % 0 FV= 13, The future value is \$13, BA II PLUS Calculator

13 Present Value of Savings Example: Future Value of Savings You are opening a savings account that you want to be worth \$10,000 in 20 years. The bank pays 5%, compounded at the end of each year. How much do you need to deposit now? Set payments per year to 1. & [ 1! P/Y= 1.00 Return to calculator mode. & U 0.00 Enter number of payments. 20, N= Enter interest rate. 5 - I/Y= 5.00 Enter final balance FV= 10, Compute future value. %. PV= -3, The present value is \$3, This is the amount you need to deposit. 2: TVM and Amortization Worksheets 39

14 Present Value in Present-Value Annuities Example: Present Value of Cost Savings The Furros Company purchased a machine that provides annual savings of \$20,000 per year for the next 10 years. Using an annual discount rate of 10%, compute the present value of the savings using an ordinary annuity and an annuity due. For a present value ordinary annuity: PV =? \$20,000 \$20,000 \$20, N = 10 I/Y = 10 For a present value annuity due for a leasing agreement: PV =? \$20,000 \$20,000 \$20, N = 10 I/Y = BA II PLUS Calculator

15 Example: Present Value of Cost Savings (cont.) Set payments per year to 1. & [ 1! P/Y= 1.00 Return to calculator mode. & U 0.00 Enter number of payments. 10, N= Enter interest rate per payment period I/Y= Enter payment S / PMT= -20, Compute PV for an ordinary annuity. %. PV= 122, Set beginning-of-period payments. & ] & V BGN Return to calculator mode. & U 0.00 Compute PV for annuity due. %. PV= 135, The present value is \$122, with an ordinary annuity and \$135, with an annuity due. 2: TVM and Amortization Worksheets 41

16 Perpetual Annuities A perpetual annuity consists of equal payments that continue indefinitely. An example of a perpetual annuity is a preferred stock that yields a constant dollar dividend. These time-line diagrams represent a perpetual annuity as an ordinary annuity and as an annuity due. For a perpetual ordinary annuity: PV PMT PMT to infinity For a perpetual annuity due: PV PMT PMT PMT to infinity Because the term (1 + I/Y / 100) -N in the present value annuity equations approaches zero as N becomes larger, you can use the following equations to solve for the present value of a perpetual annuity. For a perpetual ordinary annuity: PV = PMT ( I/Y / 100) For a perpetual annuity due: PMT PV = PMT + ( I/Y / 100) 42 BA II PLUS Calculator

17 Example: Present Value of Perpetual Annuities The Land of OZ has issued perpetual bonds for replacing bricks in their highway system. The bonds pay \$110 per \$1000 bond. You plan to purchase the bonds if you can earn 15% annually. What price should you pay for the bonds? Clear. & U P P 0.00 Calculate PV for a perpetual ordinary annuity N Calculate PV for a perpetual annuity due. H 110 N You should pay \$ for a perpetual ordinary annuity and \$ for a perpetual annuity due. 2: TVM and Amortization Worksheets 43

18 Variable Cash Flows In annuities, all payments are equal. In variable cash flows, however, the payments are unequal. You can solve for the present value of variable cash flows by treating the cash flows as a series of compound interest payments. The present value of variable cash flows is the value of cash flows occurring at the end of each payment period discounted back to the beginning of the first cash flow period (time zero). PV =? CF 1 CF j-1 Cf j NN1 N Example: Present Value of Annual Savings The ABC Company is purchasing a machine that will save the following end-of-year amounts. Year Amount \$5000 \$7000 \$8000 \$10000 Assuming a discount rate of 10%, does the present value of the cash flows exceed the original cost of \$23,000? PV =? \$5,000 \$7,000 \$8,000 \$10, N=4 44 BA II PLUS Calculator

19 Example: Present Value of Annual Savings Set payments per year to 1. & [ 1! P/Y= 1.00 Return to calculator mode. & U 0.00 Enter interest rate per cash flow period I/Y= Enter 1st cash flow S 0 FV= -5, Enter period number of 1st cash flow. 1, N= 1.00 Compute present value of 1st cash flow. %. PV= 4, Store in M1. D 1 Enter 2nd cash flow S 0 FV= -7, Enter period number. 2, N= 2.00 Compute present value of 2nd cash flow. %. PV= 5, Sum to memory. D H 1 Enter 3rd cash flow S 0 FV= -8, Enter period number. 3, N= 3.00 Compute present value of 3rd cash flow. %. PV= 6, Sum to memory. D H 1 Enter 4th cash flow S 0 FV= -10, Enter period number. 4, N= 4.00 Compute present value of 4th cash flow. %. PV= 6, Sum to memory. D H 1 Recall total present value. J 1 23, Subtract original cost. B N The present value of the cash flows is \$23,171.23, which exceeds the machine s cost by \$ This is a profitable investment for the company. 2: TVM and Amortization Worksheets 45

20 Lease-or-Buy Decision Your business is considering getting a new computer server. If you lease, you would pay \$36,000 per year for five years at the first of each year. You could buy it for \$125,000. The server is expected to save the company \$46,000 per year. It will have no resale value at the end of the five years. The company can borrow at 15% annual interest. You require a 20% annual return on projects and investments of this kind. Ignoring tax effects, should you acquire it, and if so, should you lease or purchase it? Example: Present Value of Cost Savings Set payments per year to 1. & [ 1! P/Y= 1.00 Return to calculator mode. & U 0.00 Enter number of periods. 5, N= 5.00 Enter required annual return rate I/Y= Enter annual savings S / PMT= -46, Compute present value of savings. %. PV= 137, The present value of the annual savings exceeds the purchase price (i.e., the investment will exceed your annual required return rate). Acquiring the server is a good financial move. Should you lease or buy it? 46 BA II PLUS Calculator

21 Example: Present Value of Lease Payments Set payments per year to 1. & [ 1! P/Y= 1.00 Set beginning-of-period payments. & ] & V BGN Return to calculator mode. & U 0.00 Enter number of periods. 5, N= 5.00 Enter periodic interest rate at which your firm can borrow I/Y= Enter annual lease payment S / PMT= -36, Compute present value of lease payments. %. PV= 138, The present value of the lease payments is greater than the purchase price of \$125,000, so it would be best to buy the server outright. 2: TVM and Amortization Worksheets 47

22 Present Value of Lease with Residual Value The Peach Bright Company wants to purchase a machine that it is currently leasing from your company. You offer to sell it for the present value of the lease discounted at an annual interest rate of 22% compounded monthly. The machine has a residual value of \$6500, and 46 monthly payments of \$1200 remain on the lease. If the payments are due at the beginning of each month, how much should you charge for the machine? The total value of the machine is the present value of the residual value plus the present value of the lease payments. Set payments per year to 1. & [ 1! P/Y= 1.00 Set beginning-of-period payments. & ] & V BGN Return to calculator mode. & U 0.00 Enter number of payments. 46, N= Calculate and enter periodic interest rate N - I/Y= 1.83 Enter residual value of asset S 0 FV= -6, Compute present value of residual. %. PV= 2, Enter amount of lease payment S / PMT= -1, Compute present value of lease payments. %. PV= 40, Peach Bright should pay your company \$40, for the machine. 48 BA II PLUS Calculator

23 Monthly Payments You are planning to purchase a new small desk and chair set that is sale priced at \$525. You can finance your purchase at 20% APR, compounded monthly, for two years. How much is the monthly payment? PV = \$525 FV = \$0 PMT =? PMT =? PMT =? N = 24 I/Y = Example: Monthly Payments Enter number of payments using payment multiplier. 2 & Z, N= Enter interest rate I/Y= Enter loan amount. 525 S. PV= Compute payment. % / PMT= Your monthly payment is \$ : TVM and Amortization Worksheets 49

24 Yield to Maturity on Bond Purchased on Interest Date A 9% \$1,000 semiannual commercial bond has 13 remaining coupon payments. You can purchase the bond for \$ (ignoring commissions). At this price, what is your yield to maturity and the annual effective rate? Example: Yield to Maturity Set payments per year to 2. & [ 2! P/Y= 2.00 Return to calculator mode. & U 0.00 Enter number of remaining coupon payments. 13, N= Enter bond price S. PV= Calculate the coupon payment < 1000 N / PMT= Enter bond redemption value FV= 1, Compute annual yield. % - I/Y= Store in memory. D 1 Example: Effective Annual Interest (continued from previous example) Use the Interest Conversion worksheet (Chapter 7) to calculate the effective annual interest rate. Select and clear Interest Conversion worksheet. & v & z NOM= 0.00 Recall rate from memory. J 1! NOM= Enter compounding periods. # # 2! C/Y= 2.00 Compute annual effective rate. " % EFF= The annual yield to maturity is 12.37% with semiannual compounding. The equivalent annual effective rate is 12.75%. 50 BA II PLUS Calculator

25 Saving for the Future by Making Monthly Deposits Accounts with payments made at the beginning of the period are referred to as annuity due accounts. Interest on annuity due accounts starts accumulating earlier and produces slightly higher yields. An individual has decided to invest \$200 at the beginning of each month in a retirement plan. What will the account balance be at the end of 20 years if the fund earns an annual interest of 7.5 % compounded monthly, assuming beginning-of-period payments? Interest = 7.5% Number of payments (N) = Future value(fv) =? Payment amount (PMT) = \$200 Example: Regular Deposits (Annuity Due) Set beginning-of-period payments. & ] & V BGN Return to calculator mode. & U 0.00 Enter number of payments using payment multiplier. 20 & Z, N= Enter interest rate I/Y= 7.50 Enter amount of payment. 200 S / PMT= Compute future value. % 0 FV= 111, Depositing \$200 at the beginning of each month for 20 years results in a future amount of \$111, : TVM and Amortization Worksheets 51

26 Amount to Borrow and Down Payment You want to buy a car that sells for \$5,100. The finance company charges 13.51% APR, compounded monthly, on a 48-month loan. If you can afford a monthly payment of \$125, how much can you borrow? How much do you need for the down payment? Example: Loan Amount and Down Payment Calculate the loan amount. Then subtract it from the cost of the car to find the down payment. PV =? FV = \$0 \$125 \$125 \$ N=48 I/Y = Enter number of payments using payment multiplier. 4 & Z, N= Enter monthly interest rate I/Y= Enter payment. 125 S / PMT= Compute loan amount. %. PV= 4, Calculate down payment. H 5100 S N To buy the car, you can borrow \$4, and make a down payment of \$ BA II PLUS Calculator

27 Regular Deposits for a Specified Future Amount You plan to open a savings account and deposit the same amount of money at the beginning of each month. In 10 years, you want to have \$25,000 in the account. How much should you deposit if the annual interest rate is 7% with quarterly compounding? C/Y (compounding periods per year) is automatically set to equal P/Y (payments per year), so you need to set C/Y. Example: Monthly Deposits Compounded Quarterly Set compounding periods to 4. & [ # 4! Set beginning-of-period payments. & ] & V P/Y= C/Y= BGN Return to calculator mode. & U 0.00 Enter number of deposits using payment multiplier. 10 & Z, N= Enter interest rate. 7 - I/Y= 7.00 Enter future value FV= 25, Compute deposit amount. % / PMT= You need to make monthly deposits of \$ : TVM and Amortization Worksheets 53

28 Time Value of Money/Amortization Schedule This two-part example shows you how to use the TVM and Amortization worksheets to compute the monthly payment on a 30-year loan and then generate an amortization schedule for the first three years of the loan. Example: Mortgage Payment Using the TVM worksheet, determine the monthly payment on a 30-year mortgage with a loan amount of \$120,000 and an annual percentage rate of 9.125%. Enter number of payments using payment multiplier. 30 & Z, N= Enter interest rate I/Y= 9.13 Enter loan amount PV= 120, Compute payment. % / PMT= Notice that the interest rate displays as 9.13, even though you entered The calculator uses for I/Y to make the calculation. To see the stored value of I/Y displayed with three digits, press & 3! & U J I/Y. The computed monthly payment is \$ Because PMT is money paid out, it is displayed as a negative number. 54 BA II PLUS Calculator

29 Example: Loan Amortization (continued from previous example) Use the Amortization worksheet to generate an amortization schedule for the first three years of the loan. Assume that the first payment is in April; therefore, the first year has 9 payment periods. There are 12 payment periods per year thereafter. Select the Amortization worksheet. & \ P1= 1.00 Set ending period to 9. # 9! P2= 9.00 Display first year amortization data. # # # BAL= PRN= INT= 119, , Change beginning period to 10. # 10! P1= Change ending period to 21. # 21! P2= Display second year amortization data. Move to P1 and press % to enter next range of payments. # # # BAL= PRN= INT= 118, , # % P1= Display P2. # P2= Display third year amortization data. # # # BAL= PRN= INT= 117, , Note that the principal and interest are displayed as negative because they are outflows. 2: TVM and Amortization Worksheets 55

30 Interest and Loan Balance after Specified Payment To evaluate the financial advisability of financing all or some of the sale price of a property, a seller must know the amount of interest that will be received and the remaining balance at the end of the term (balloon payment). A seller is asked to finance \$82,000 at 10% annual interest, amortized over a 30-year term but with a balloon payment due after five years. The seller wants to know: The amount of the monthly payment. The amount of interest he will receive. The amount of the balloon payment. Example: Compute Monthly Payment Enter number of payments using payment multiplier. 30 & Z, N= Enter interest rate I/Y= Enter loan amount PV= 82, Compute payment. % / PMT= Example: Compute Amortization Schedule (continued from previous example) Select Amortization worksheet. & \ P1= 1.00 Enter end period after five years. # 5 & Z! P2= View computed balance due after five years. # BAL= 79, View computed interest paid after five years. # # INT= -40, If the seller financed the sale, he would receive: \$ each month for five years. \$40, in interest over the five-year term. \$79, as the balloon payment. 56 BA II PLUS Calculator

31 Canadian Mortgages Canadian mortgages typically require the borrower to make monthly payments, although interest is compounded semiannually. Additionally, mortgages are usually refinanced at the end of a fixed period of time, such as five years. A home buyer borrows \$60,000 for 20 years at an annual interest rate of 13 % compounded semiannually. How much are the monthly payments and the amount necessary to pay off the mortgage after five years? Example: Payment Leave payments per year at 12. & [ P/Y= Set compounding periods per year to 2. # 2! C/Y= 2.00 Return to calculator mode. & U 0.00 Enter number of payments using payment multiplier. 20 & Z, N= Enter interest rate per year I/Y= Enter present value PV= 60, Compute payment. % / PMT= Example: Amortization Schedule (continued from previous example) Select Amortization worksheet. & \ P1= 1.00 Enter number of payments as P2 using payment multiplier. # 5 & Z! P2= View balance after five years. # BAL= 55, The monthly payment is \$688.52, and \$55, is required to pay off the mortgage after five years. 2: TVM and Amortization Worksheets 57

32 58 BA II PLUS Calculator

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