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on echo !*******************************************************************! ! ! ! SORITEC Sampler Financial Functions ! ! (Chapter 7) ! ! ! !*******************************************************************! ! ! SORITEC Sampler contains most of the common financial functions ! used in project evaluation and other economic analysis studies. ! These functions are: ! ! (1) Internal Rate of Return ! (2) Present Value ! (3) Loan Amortization ! ! Some examples used to demonstrate financial functions are drawn from ! M. Wohl and B.V. Martin, "Traffic Systems Analysis for Engineers ! and Planners". New York: McGraw-Hill, 1967, p. 215. As the ! examples are not specific to traffic engineering, they are used ! for benchmarking only. ! !*******************************************************************! ! ! ! Loan Amortization ! ! (Section 7.3) ! ! ! !*******************************************************************! ! ! To demonstrate the various features of the AMORT command, assume ! a loan for $100.00 issued for 5 years. Repayment is assumed to ! occur at the end of each period so that if the loan is made at ! t=1, repayment occurs at the end of t=1,2,3,4,5. Interest on the ! loan is 6 percent, expressed at an annual rate. ! ! The AMORT command can be used to calculate annual payments under ! different payback assumptions. For example, if equal or uniform ! payments are made over the 5 year period, or if annual payments ! cover interest charges only with a balloon payment at the end of ! the loan, annual payments can be calculated easily. ! ! Let's set up the problem by defining the value of the loan, the ! interest rate and the loan period. ! SET loan = 100 SET rate = .06 USE 1 5 ! ! Annual payments associated with equal or uniform repayments over ! the five-year loan period are calculated using the command: ! AMORT payment loan rate ! ! The resulting "payment" is returned as a CONSTANT. You may want to ! store these data as a time-series of payments for further analysis. ! Well, we do, anyway, and it is done with a simple assignment statment. ! We'll call the time-series "pay_stream_1'. ! pay_stream_1 = payment PRINT pay_stream_1 ! ! If you want to calculate annual payments using the "Rule of 78", ! the command is: ! AMORT(RULEOF78) payment 100 .06 ! ! If annual payments consist of only interest with repayment of the ! principal in the last period, they are estimated by including the ! BALLOON parameter in the command line, e.g., ! AMORT(BALLOON=100) payment loan rate ! ! We'll save the payments as a time-series in the variable "pay_stream_2" ! for future use. ! pay_stream_2 = payment ! ! Note that we must add the balloon payment in the final year of the loan ! to derive the correct payment stream. ! USE 5 REVISE pay_stream_2 = pay_stream_2 + 100 USE 1 5 PRINT pay_stream_2 ! ! If a loan is scheduled with irregular payments during the life of the ! loan, auxilliary payments time-series can be specified in the command ! line. The auxilliary payments differ from balloon payments in that ! payments to principal are assumed to occur at the same time regular ! payments are made. Thus, periodic payments associated with a balloon ! payment of x at the end of a loan will not be the same as those ! associated with an auxilliary payment schedule with a lump sum payment ! of x in the last payment period. This is best illustrated in an ! example. ! ! For the $100 loan at 6% interest, create an auxilliary time-series ! containing a lump sum payment in year 5, e.g., ! balloon_payment_series = 0 USE 5 5 REVISE balloon_payment_series = 100 USE 1 5 PRINT balloon_payment_series ! ! Now calculate annual payments with the AMORT command. This time, we ! do not enter the "BALLOON=" parameter but instead include the ! "balloon_payment_series" as the last argument in the command line. ! AMORT payment loan rate balloon_payment_series ! ! Note the difference between this amortization schedule and the one with ! the $100 balloon payment. Although inappropriate for estimating ! repayment schedule for typical balloon payment loans, this option is ! particularly useful for loans requiring periodic lump sum payments. ! For example, if half the principal is due at the end of year 2 and half ! in year 5, the command sequence to calculate annual payments is: ! balloon_payment_series = 0 USE 2 REVISE balloon_payment_series = 50 USE 5 REVISE balloon_payment_series = 50 USE 1 5 PRINT balloon_payment_series AMORT payment loan rate balloon_payment_series ! ! If the interest associated with a loan or investment is quoted in a ! periodicity other than payment schedule, the AMORT command will convert ! the periodicity of the interest rate to that of the current USE period. ! Either SIMPLE or COMPOUND conversion is permitted. These latter keywords ! are only relevant when periodicity conversion is specified. ! ! To illustrate this option consider a one-year CD for $1000 with an ! effective yield of 10.5% compounded daily purchased, say, in January ! of this year. Interest payments on the CD are calculated monthly. ! We can calculate the monthly yield with the following set of commands. ! ! Set up the parameters for the problem. ! SET daily_interest = .105/365 SET CD_amount = 1000 USE 1985m1 1985m12 AMORT(PERIOD=D,COMPOUND,BALLOON=1000) monthly_yield CD_amount daily_interest PRINT monthly_yield ! ! The total you get from your investment can then be calculated directly. ! SET total_yield = CD_amount + monthly_yield * 12 PRINT total_yield !*******************************************************************! ! ! ! Present Value ! ! (Section 7.2) ! ! ! !*******************************************************************! ! ! The PV command calculates the new present value of a stream of ! net benefits or costs associated with a financial venture given ! an interest rate or time-series of interest rates. Optional ! modifiers in the command line allow you to convert the periodicity ! of the interest rate to conform to the net income stream and ! to change interest calculation from the default, COMPOUND, to ! SIMPLE. The "net income stream" may also be specified as two ! time-series, "benefits" and "costs" - SORITEC Sampler will calculate ! the difference for you. ! ! Let's take a look at the Present Value estimates associated with ! the payment streams of some of the loans we analyzed in the previous ! section. ! ! This is also a good check of the consistency between the AMORT and ! PV commands since normally, the present value of the loan at the end of ! the period should equal the amount of the loan. Calculate ! the present value of the loan with uniform payments with the command: ! USE 1 5 PV present_value_1 pay_stream_1 rate ! ! Note that the amount of the loan is less than the present value. Upon ! inspection, the relation between the two variables is: ! ! present_value_1/(1 + rate) = loan ! ! The reason for this is that the AMORT command assumes the loan is ! dispensed at the beginning of period t=1 and repayments are made ! at the end of periods t=1,2,...,n. This makes SORITEC Sampler's ! amortization estimates consistent with most amortization tables. ! The PV command, on the other hand, assumes the loan is dispensed in ! period t=0 and repayments are made at the beginning of periods ! t=1,2,...,n. To make the estimates consistent, simply divide the ! present value estimate by (1 + rate), e.g., ! present_value_1 = present_value_1/(1 + rate) PRINT present_value_1 ! ! A similar result is obtained from the payment stream associated with ! the balloon payment amortization schedule, i.e., ! PV present_value_2 pay_stream_2 rate present_value_2 = present_value_2/(1 + rate) PRINT present_value_2 ! ! Let's now calculate, for each loan schedule, the present values of ! the ith-year payments on the $100 loan. While we're at it, let's ! calculate the interest and principal owed at the end of each period. ! ! We can do this in a DOT-loop with a nested DO-loop. ! Some preliminary variable assignments and format statement ! specifications are required. ! SET indicator = 0 200 format(///7x,'Repayment Schedule for a 5-year Loan of $100 at 6% Interest') 201 format(21x,'Uniform Payments over 5 Years'/) 202 format(7x,'Annual Interest Payments Plus Principal at End of 5th Year'/) 203 format(3x,' Interest owed at Principal owed at') 204 format(' end of kth year end of kth year Total payment at', & ' Present value of') 205 format(' k (prior to payment) (prior to payment)',2x, & 'end of kth year kth year payment') 206 format(' - ------------------ ------------------',2x, & '--------------- ----------------') 207 format(f3.0,5x,f6.2,12x,f6.2,13x,f6.2,10x,f6.2) USE 1 5 fill k 1 2 3 4 5 DOT pay_stream_1 pay_stream_2 SET indicator = indicator + 1 ! ! Initialize the time-series in which we will store the results. These ! are a series of present values, an interest paid series and a series ! comprising the remaining principal on the loan. ! USE 1 5 pres_val_series = 0 interest_series = 0 principal_series = 0 ! ! Initialize other constants that will be used in calculations. ! SET principal = loan SET sum_present_value = 0 DO i = 1 TO 5 ! ! Calculate the interest on the principal remaining in the time period. ! SET interest = principal * rate USE i i ! ! Calculate the principal repaid with the total payment, which is the ! difference between the total payment paid in the period and the ! interest paid in the period. ! SET payment = : SET principal_paid = payment - interest ! ! Assign the principal remaining in the current period to the ! appropriate element in a time-series of remaining principal at t. ! REVISE principal_series = principal ! ! Calculate the remaining principal on the loan at the start of the ! next period (equal to the remaining principal at t minus principal ! paid during t). ! SET principal = principal - principal_paid ! ! Now estimate the present value of payment stream. ! ! ! First, set the time period. ! USE 1 i PV present_value : rate SET present_value = (present_value/(1+rate)) - sum_present_value SET sum_present_value = sum_present_value + present_value ! ! Assign the present value, interest and principal estimates to the ! appropriate element of the "pres_val_series", "interest_series" and ! "principal_series" time-series. ! USE i i REVISE pres_val_series = present_value REVISE interest_series = interest ! ! That's it for the DO-loop. Wait while Sampler grinds through the ! loop. Each time it executes the PV command, Sampler will print out ! the unadjusted present value estimates (before dividing by 1+rate). ! END ! ! Print out the resulting series ! USE 1 5 ! ! Write out the series in a table. ! OFF ECHO WRITE(200) IF indicator > 1; THEN; WRITE(202); ELSE ; WRITE(201) WRITE(203) WRITE(204) WRITE(205) WRITE(206) WRITE(207) (k interest_series principal_series : pres_val_series) ON ECHO ENDDOT !*******************************************************************! ! ! ! Internal Rate of Return ! ! (Section 7.1) ! ! ! !*******************************************************************! ! ! The internal rate of return of an investment is calculated by ! SORITEC Sampler with the IRR command. IRR calculations use ! a discounted income stream procedure to calculate the rate of ! return. Thus, the PV and IRR commands are consistent. We ! illustrate the IRR command with an example comparison of two ! mutually exclusive project alternatives with the following ! characteristics: ! ! Project A Project B ! Project Service Life 10 years 30 years ! Capital Outlays in Year 0 $400,000 $1,000,000 ! Annual Project Earnings $56,951 $76,577 ! over Service Life ! ! Initialize the data for each alternative and calculate ! the internal rate of return for each. ! ! Project A: ! USE 1 10 project_a_earnings = 56951 project_a_capital_cost = 400000 IRR(CAPITAL=400000,TOLB=.0000001,MAXIT=50) rate_of_return_a project_a_earnings ! ! To demonstrate the consistency between the IRR and PV commands, ! calculate the net present value of the project earnings stream ! using the rate of return estimated by the IRR command. ! PV net_present_value project_a_earnings rate_of_return_a ! ! Note that instead of using the CAPITAL modifier in the IRR command, ! we could have subtracted the initial capital cost from the annual ! earnings in the first period, i.e., USE 1 1 REVISE project_a_earnings = project_a_earnings - project_a_capital_cost USE 1 10 IRR rate_of_return_a project_a_earnings ! ! Project B: ! USE 1 30 project_b_earnings = 76577 project_b_capital_cost = 1000000 IRR(CAPITAL=1000000,TOLB=.001,MAXIT=50) rate_of_return_b & project_b_earnings ! ! Note that the project earnings does not have to be specified as ! as "net" in the IRR command line. For example, we could define ! a cost stream for project B as a time series, e.g., ! project_b_costs = 0 USE 1 REVISE project_b_costs = 1000000 USE 1 30 ! ! and then calculate the internal rate of return without the ! "CAPITAL=" modifier in the IRR command line. ! IRR(TOLB=.001,MAXIT=50) rate_of_return_b & project_b_earnings project_b_costs ! ! Obviously, this capability is meant for more complicated ! earnings and cost streams, but the example is illustrative. ! ! That's it for SORITEC Sampler's financial functions! QUIT