Slide 1:Time Value of Money Relevant reading: Chapter 2 Topic objectives: Appreciate the concept of interest. Explain the concept of the time value of money. Perform economic equivalence calculations. Know cash flow diagrams and cash flow patterns.


Interest :Interest Money is a commodity. Hence, money has a cost. The cost of money is interest. The cost of money measured by an interest rate (a percentage). Based on these definitions, interest can be defined as the cost of having money available for use.


The Time Value of Money :The Time Value of Money Suppose you have with you right now 100,000 riyals. You want to purchase a car for 100,000 riyals. You thought that investing the money for one your at 6% return would give you 106,000 and you can purchase the car and you will have 6,000 leftover. If the price of the car increases by 8% per year due to inflation, you will not have enough money to purchase the car a year from now. Because of inflation, your purchasing power will decrease as you delay your purchase.


Slide 4:To make up this future loss in purchasing power, the investment return rate should be larger than the inflation rate. Money has a time value because it can earn more money over time (earning power). Money has a time value because its purchasing power changes over time (inflation).


Elements of Transactions Involving Interest :Elements of Transactions Involving Interest The initial amount of money invested or borrowed is called the principal (P). The interest rate (i) measures the cost or price of money and it is expressed as a percentage per time period. A period of time called the interest period (n) determines how frequently interest is calculated. The number of interest periods (N) specifies the length of time marking the duration of the transaction. A plan for receipts or disbursements (An) yields a particular cash flow pattern over a specified length of time. A future amount of money (F) results from the cumulative effects of the interest rate over a number of interest periods.


Example :Example A company borrows 30,000 riyals from a bank at a 9% annual interest. The bank charges SR 300 application fee The bank offers 2 payment plans. P = 30,000 i = 9% n = 1 N = 5 A = 7,712.77 F = 46,158.72


Cash Flow Diagrams :Cash Flow Diagrams The cash flow diagram is a picture that represents time by a horizontal line marked off with the number of interest periods specified. Arrows represent cash flows over time at relevant periods. Receipts are positive flows and represented by upward arrows, and disbursements are negative flows and represented by downward arrows


End-of-Period Convention :End-of-Period Convention Cash flows can occur at any point in time (beginning, middle, end of the interest period). To simplify the calculations, we often apply the end-of-period convention. The end-of-period convention is the practice of placing all cash flow transactions at the end of the interest period.


Methods of Calculating Interests :Methods of Calculating Interests At the end of each interest period, the interest earned on the principal amount is calculated according to a specific interest rate. The two schemes for calculating this earned interest yield either simple interest or compound interest.


Simple Interest :Simple Interest This scheme considers interest earned on only the principal amount during each interest period and interest earned during each interest period does not earn additional interest in the remaining periods. For a deposit of P riyals at a simple interest rate of i for N periods, the total earned interest I is I = (iP) N The total amount, F, available at the end of N periods is F = P + I = P (1 + iN)


Compound Interest :Compound Interest Under this scheme, the interest earned in each period is calculated based on the total amount at the end of the previous period, which includes the principal plus the accumulated interest earned. If you invest (deposit) P riyals at an interest rate i, you will have P + iP = P(1 + i) riyals at the end of one interest period. If the entire amount (principal and interest) are reinvested for another periods, you will have an the end of the second period P(1 + i) + i[P(1 + i)] = P(1 + i)2 After N periods, the total accumulated value F is F = P(1 + i)N


Example :Example See example 2.1 and the solution in the textbook.


Economic Equivalence :Economic Equivalence Which option would you prefer?


Definition of economic equivalence :Definition of economic equivalence The main mechanism in deciding among alternative cash flows involves comparing their economic worth. Calculations for determining the economic effects of one or more cash flows are based on the concept of economic equivalence. Economic equivalence exists between cash flows that have the same economic effect and could be traded for one another. A cash flow can be converted to an equivalent cash flow at any point in time. Remember that the present value of future cash flows is equivalent in value to the future cash flows.


Equivalence Calculations: A Simple Example :Equivalence Calculations: A Simple Example Suppose you want to invest 1,000 riyals at 12% annual interest for 5 years. The future amount F in this case is F = 1,000 (1 + 0.12)5 = 1,762.34 At 12% interest, 1,000 riyals received now is equivalent to 1,762.34 received in five years.


Example :Example See example 2.2 and the solution in the textbook.


Equivalence Calculations Require a Common Time Basis for Comparison :Equivalence Calculations Require a Common Time Basis for Comparison We must convert cash flows to a common base period in order to compare their values. When selecting a point in time at which to compare the values of alternative cash flows, we use either the present time, which yields the present worth, or some point in the future, which yields the future worth.


Example :Example See example 2.3 and the solution in the textbook.


Interest Formulas for Single Cash FlowsCompound-Amount Factor :Interest Formulas for Single Cash FlowsCompound-Amount Factor Given a present sum P invested for N interest periods at interest rate i, the future sum F is F = P(1 + i)N The factor (1 + i)N is called the compound-amount factor. The process of finding F is called the compounding process.


Interest Tables :Interest Tables Tables of compound-interest factors have been developed so that we can easily find the appropriate factor for a given interest rate and the number of interest periods. These tables are included in Appendix B.


Factor Notation :Factor Notation We will used a convenient notation so we can determine precisely which table factor to use. For the future sum formula F = P(1 + i)N, we express the compound-amount factor as (F/P, i, N), which is read as “find F, given P, i, and N”. The future sum formula becomes F = P(F/P, i, N).


Example :Example See example 2.4 and the solution in the textbook.


Present-Worth Factor :Present-Worth Factor The discounting process is to calculate the present worth of a future sum. The value of P for a given F is given by The factor 1/(1 + i)N is called the single-payment present-worth factor, designated as (P/F, i, N). The interest rate i is called the discount factor. The factor P/F is called the discounting factor.


Example :Example See example 2.5 and the solution in the textbook.


Example :Example See example 2.6 and the solution in the textbook.


Example :Example See example 2.7 and the solution in the textbook.


Uneven-Payment Series :Uneven-Payment Series This is an example of uneven-payment cash flow series.


Example :Example In SPIMACO, there is a very expensive machine used for packing drugs in PROTON production line. The machine has been in service for 10 years. The industrial engineer has found that a complete overhaul of the machine would cost SR130,000, and that will keep it running for 5 years more without breaking down. He also has determined that the annual repair cost for the following 5 years are:


Slide 29:If the company can invest their money with interest rate of 6% compounded annually, which option is better in an economic sense? Compute the present worth of the future payments: P = 25,600 + 27,200 (P/F,6%,1) + 28,700 (P/F,6%,2) + 30,100 (P/F,6%,3) + 31,600 (P/F,6%,4) = 127,106 < 130,000 It is better not to overhaul the machine.


Equal-Payment Series :Equal-Payment Series We want to find the equivalent present worth P or future worth F of a uniform cash flow series.


Compound-Amount Factor: Find F, Given A, i, and N :Compound-Amount Factor: Find F, Given A, i, and N The A amount invested at the end of the first period will be worth A(1 + i)N-1 at the end of N periods. The A amount invested at the end of the second period will be worth A(1 + i)N-2 at the end of N periods. The last A amount invested at the end of the Nth period will be worth exactly A at the end of N periods. Hence, the future sum will be F = A(1 + i)N-1 + A(1 + i)N-2 + …+ A(1 + i) + A After simplifying, the future sum is given by


Slide 32:The factor (F/A, i, N) is called the uniform-series compound-amount factor.


Example :Example See example 2.9 and the solution in the textbook.


Example :Example See example 2.10 and the solution in the textbook.


Sinking-Fund Factor: Find A, Given F, i, and N :Sinking-Fund Factor: Find A, Given F, i, and N Given a future sum F, the A amount can be calculated as The factor (A/F, i, N) is called the sinking-fund factor.


Example :Example See example 2.11 and the solution in the book.


Capital-Recovery Factor: Find A, given P, i, and N :Capital-Recovery Factor: Find A, given P, i, and N We want to find the periodic A amount for a given P, i, and N. The A amount is given by See the derivation of this formula in the book. The factor (A/P, i, N) is called the capital-recovery factor.


Example :Example See example 2.12 and the solution in the book.


Example :Example See example 2.13 and the solution in the textbook.


Present-Worth Factor: Find P, given A, i, and N :Present-Worth Factor: Find P, given A, i, and N We want to find the present worth P of future equal cash flow series. The present worth is given by


Example :Example Saudi Arabian Airlines has gone through a restructuring plan ahead of its privatization target. It has offered its employees who have served the company 20 years or more an early retirement plan. The plan will give the employee SR 79,200 per year for 25 years. If you can invest your money in an investment account that will give you 8% interest per year, what is the present worth of these future payments?


Slide 42:The present worth is P = 79,200 (P/A, 8%, 25) = 79,200 (10.6748) = 845,444.16 riyals


Example :Example See example 2.15 and the solution in the textbook.


Capitalized worth :Capitalized worth The present worth of equal cash flows that extend indefinitely is called capitalized worth (CW). If these cash flows are expenses only, the result is called the capitalized cost.


Slide 45:Using the notation, CW is equal to CW = A(P/A, i, ) Using the formula, Therefore, CW = A / i


Example :Example Dhahran Public Library will be built and it has been estimated that the library will require 2 million riyals in real terms per year for electricity bills, water, cleaning, and acquisition of new books. An investment account will be created to deposit an initial amount that will pay 2 million riyals per year for the life the library, which is assumed to be very long. If the investment account can pay 8% interest compounded annually, how much the initial deposit has to be?


Slide 47:The capitalized cost is CE(8%) = 2 / 0.08 = 25 million Hence, 25 million riyals has to be deposited in the investment account that will generate 2 million every year for ever.


Gradient series :Gradient series There are situations which involve periodic payments that increase or decrease by a constant amount G. The first payment in a strict gradient series is zero and subsequent payments increase by G: An = (n-1) G


Slide 49:A linear gradient series involving non-zero first payment must be converted to a zero-first-payment gradient series. The conversion is done by subtracting the first payment from all payments:


Present worth factor – linear gradient: find P, given G, N, and i :Present worth factor – linear gradient: find P, given G, N, and i To find the expression for the present amount P, the single-payment present worth factor is applied to each term in the gradient series: After simplification, the present worth is This factor is called the gradient-series present worth factor.


Example :Example See example 2.16 and the solution in the textbook.


Geometric gradient series :Geometric gradient series The payments in the cash flow increase or decrease by some fixed rate. If g is the percentage change in a payment from one period to the next, the nth payment An is related to the first payment A1 by An = A1(1 + g)n-1


Slide 53:The present worth of any An is given by P = An(1 + i)-n = A1(1 + g)n-1 (1 + i)-n The present worth of all payments is found by summing all the present worths: A compact form of this formula is


Example :Example See example 2.18 and the solution in the textbook.


Composite Cash Flows :Composite Cash Flows Read this section, it might help you solve some problems a little faster.