# Module 1 – Lecture 4 MONEY TIME RELATIONSHIP Prof. Dr. M.F. El-Refaie.

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Module 1 – Lecture 4 MONEY TIME RELATIONSHIP Prof. Dr. M.F. El-Refaie

Based on the concept of time value of money and bound to the mechanism of compound interest, it is often required to interrelate some of the following quantities: P is the value or sum of money at a time denoted as the present, m.u. F is the value or sum of money at a future time, m.u. A is the series of consecutive, equal, end-of-period amounts of money, m.u per unit time. n is the number of interest periods. i is the interest rate. The transformation multipliers (or time-value conversions) are called the “Compound Interest Factors" or the “Engineering Economy Factors". The names of these factors together with their standard abbreviations and notations are given in Table 4.1. Compound interest factors:

Table 4.1: Standard factor notations Standard notationFactor name Single-payment present worth (SPPWF) Single-payment compound amount (SPCAF) Uniform series present worth (USPWF) Capital recovery (CRF) Sinking fund (SFF) Uniform series compound amount (USCAF)

Table 4.2: Computations using standard notation formulafactorgivenTo find Fp pF Ap PA FA AF

SINGLE-PAYMENT COMPOUND AMOUNT FACTOR (SPCAF) This factor was implicitly derived previously.

If you have L.E.2000 now and invest it at 10%, how much will it be worth in 8 years? Example 4.1

Given: P = L.E. 2000, i = 10% per year, and n = 8 years Find: F Using a calculator: Use a calculator to evaluate the (1+i) n term F = L.E.2000 (1 + 0.10) 8 = L.E.4287.18 Solution Using a spreadsheet package as Excel, the future worth calculation could be determined from FV Function: FV returns the future value of an investment based on periodic, constant payments and a constant interest rate. Syntax FV (rate, nper, pmt, pv, type)

Where Rate is the interest rate per period. nper is the total number of payment periods in an annuity. Pmt is the payment made each period; it cannot change over the life of the annuity. Typically, pmt contains principal and interest but no other fees or taxes. Pv is the present value, or the lump-sum amount that a series of future payments is worth right now. Type is the number 0 or 1 and indicates when payments are due. 0 or omitted at the end of the period, 1 at the beginning of the period F = FV (10%, 8,, 2000, 0)

The following sums are to be invested at 5 % interest rate: 6x10 5 m.u. now, 3x10 5 m.u. two years from now and 4x10 5 m.u. five years from now. How much will be accumulated ten years from now? Example 4.2

F = P 1 (F/P, 5, 10) + P 2 (F/P, 5, 8) + P 3 (F/P, 5, 5) = 6 × 10 5 × (1.05) 10 + 3 × 10 5 × (1.05) 8 + 4 × 10 5 × (1.05) 5 = 1.9311 × 10 6 m.u. Solution

P' = 6 × 10 5 (F/P, 5, 5) + 3 × 10 5 (F/P, 5, 3) + 4 × 10 5 = 6 × 10 5 × (1.05) 5 + 3 × 10 5 × (1.05) 3 + 4 × 10 5 = 15.1306 ×10 5 m.u. F = P' (F/P, 5, 5) = 15.1306 × 10 5 × (1.05) 5 = 19.311 × 10 5 Alternative solution

Solution of the Excel sheet is shown in figure where we can see in the equation bar that: F = FV (5%, 10,, 600000, 0) + FV (5%, 8,, 300000, 0) + FV (5%, 5,, 400000, 0) Solution

SINGLE-PAYMENT PRESENT WORTH FACTOR (SPPWF)

Suppose that L.E.1000 is to be received after 5 years. At an annual interest rate of 12%, What is the present worth of this amount? Example 4.3

Solution This example could be solved by one of three ways: 1-Using a calculator (Mathematical Formulas) P = 1000 × (1 + 0.12) -5 = 1000 × (0.5674) = L.E. 567.40 In order To have L.E.1000 in your savings account at the end of 5 years, you must deposit L.E.567.40 now.

2. Using interest tables Interest Table For Discrete Compounding; i = 12 % Present Worth Factor Compound Amount Factor Present Worth Factor Compound Amount Factor Periods P/AF/AP/FF/Pn 3.03734.77930.63551.57354 3.60486.35280.56741.76235 4.11148.11520.50661.97386 Note: this method might be obsolete as it requires lots of data tables. These tables are formed using the simple formulae developed here in this lecture

3. Using Spreadsheet as Excel, the present value calculation is determined from the function PV: PV functions returns the present value of an investment. Syntax PV (rate, nper,pmt, fv, type) Rate is the interest rate per period Nper is the total number of payment periods in an annuity. Pmt is the payment made each period and cannot change over the life of the annuity. Fv is the future value, or a cash balance you want to attain after the last payment is made. If fv is omitted, it is assumed to be 0 Type is the number 0 or 1 and indicates when payments are due. 0 or omitted at the end of the period, 1 at the beginning of the period.

Then the Solution of the above example is. P = PV (12%, 5, 0, 1000, 0).

Suppose you buy a share for L.E.10 and sell it for L.E.20, your profit is L.E.10. If that happens within a year, your rate of return is an impressive 100% (L.E.10 / L.E.10 = 1). If that takes 5 years, what would be the average annual rate of return on your investment? Example 4.4

Solution This example could be solved by either of the following methods: 1. F = P (1 + i) n 20 = 10(1 + i) 5 Then i = 2 (1/5) – 1 = 14.87 % 2. Using spreadsheet application such as Excel. A financial function such as RATE (n, 0, P, F) allows us to calculate an unknown interest rate. The precise command statement would be as follows: i = RATE (5, 0, -10, 20) = 14.87%

A series of equal installments paid at the ends of consecutive interest periods will accumulate to a certain future value after n periods. UNIFORM-SERIES COMPOUND- AMOUNT FACTOR (USCAF)

Accumulate to Payment at end of period No. 1 2 3...... n-1 n A A series of equal payments Dealing with each payment separately we can deduce the formula for determining the future value F for any series of equal installments as shown in table.

The future compound amount is the summation of the right-hand column. Thus, Subtracting

Suppose you make 15 equal annual deposits of 1000 L.E. each into a bank account paying 5% interest per year. The first deposit will be made one year from today. How much money can be withdrawn from this bank account immediately after the 15th deposit? Example 4.5

The value of A is L.E.1000, n = 15 years, and i = 5% per year. Immediately after the 15th payment, the future equivalent amount is. F = l000 (F/A, 5%, 15) = 1000 (1.05 15 – 1) / 0.05 = 1000 × (21.5786) = L.E. 21578.60 Solution:

Suppose you make an annual contribution of L.E.3000 to your saving account at the end of each for 10 years. If your saving account earns 7% interest annually how much can be withdrawn at the end of ten years Example 4.6

Given: A = L.E.3000, n = 10 years, and i = 7% per year. Find: F F = 3000(F/A, 7%, 10) = L.E.3000 (13.8164) = L.E.41449.20. To obtain the future value of the annuity on Excel, we may use the following financial command: = FV (7%, 10, 3000, 0, 0) Solution

You have an opportunity to purchase a piece of vacant land for m.u.30000 cash. If you bought it, you would plan to hold the property for 15 years and then sell it at a profit. During this period, you would have to pay annual property taxes of m.u.600. You would have no income from the property. Assuming that you want a 10 % rate of return from the investment, at what net price would you have to sell it 15 years hence? Example 4.7

F = 30000 (F/P,10%,15) + 600 (F/A,10%,15) = 30000 (1.08) 15 + 600 ((1.08 15 – 1) / 0.08) = m.u. 144380.93 Using Spreadsheets (Excel) the above example could be solved as, FV= (10%,15, 600, 30000, 0) Solution

SINKING FUND FACTOR (SFF)

An enterprising student is planning to have personal savings totaling L.E.1000000 when she retires at age 65. She is now 20 years old. If the annual interest rate will average 7% over the next 45 years on her savings account, what equal end-of-year amount must she save to accomplish her goal? Example 4.8

A = 1000000 (A/ F, 7%, 45) A= 1000000 (0.07/((1+0.07) 45 –1)) A= 1000000(0.0035) = L.E.3500 The future amount, F, is L.E.1000000. The equal annual amount this student must place in a sinking fund that grows to L.E.1000000 in 45 years at 7% annual interest Solution

Also the above example could be solved by spreadsheet as Excel using function called PMT, PMT function Calculates the payment for a loan based on constant payments and a constant interest rate. Syntax PMT (Rate, Nper, Pv, Fv, Type) Description of the arguments in PMT function, Rate is the interest rate for the loan. Nper is the total number of payments for the loan. Pv is the present value, or the total amount that a series of future payments is worth now; also known as the principal. Fv is the future value, or a cash balance you want to attain after the last payment is made. If Fv is omitted, it is assumed to be 0 (zero), that is, the future value of a loan is 0. Type is the number 0 (zero) or 1 and indicates when payments are due.

The Excel solution then is, PMT (7%, 45, 0, 1000000, 0)

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Engineering Economy Module 1: Lectures 1-8 Dr. Mohamed F. El-Refaie Dr. Mohamed F. El-Refaie (Sunday and Tuesday, L3,L4,L5 and L6) (Sunday and Tuesday, L3,L4,L5 and L6) Dr. Sayed Kaseb Dr. Sayed Kaseb (Saturday and Wednesday, L1,L2,L7 and L8) (Saturday and Wednesday, L1,L2,L7 and L8) Module 1 Teaching Team

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