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Engineering Economics Module No. 06 Time Value of Money By Muhammad Shahid Iqbal The Time Value of Money A fundamental idea in finance that money that one has now is worth more than money one will receive in the future. Because money can earn interest or be invested, it is worth more to an economic actor if it is available immediately. The time value of money is money's potential to grow in value over time. Because of this potential, money that's available in the present is considered more valuable than the same amount in the future. For example, 100 dollars of today's money invested for one year and earning 5 percent interest will be worth 105 dollars after one year. Therefore, 100 dollars paid now or 105 dollars paid exactly one year from now both have the same value to the recipient who assumes 5 percent interest. The Concepts Interest Interest is a charge for borrowing money, usually stated as a percentage of the amount borrowed over a specific period of time. Simple Interest Rate: The total interest earned or charged is linearly proportional to the initial amount of the loan (principal), the interest rate and the number of interest periods for which the principal is committed. I=(P)(i)(N) where P = principal amount lent or borrowed N = number of interest periods ( e.g., years ) i = interest rate per interest period The Concepts Interest Compound Interest Rate: Whenever the interest charge for any interest period is based on the remaining principal amount plus any accumulated interest charges up to the beginning of that period. Compound interest is always assumed in TVM problems. Period Amount Owed Interest Amount Amount Owed at Beginning of the per period @ 10% end of period period 1 1000 100 1100 2 1100 110 1210 3 1210 121 1331 The Time Value of Money If a investor invests a sum of RS. 1000 in a fixed deposit with interest rate 15% compounded annually. The accumulated amount at the end of the year is Year end interest Compounded amount 0 100.00 1 15.00 115.00 2 17.25 132.25 3 19.84 152.09 4 22.81 174.90 5 26.24 201.14 The Time Value of Money Alternatively If we want RS. 100 at the end of 5 years what is the amount that we deposit now at a given interest rate say 15% Year end Present worth Compounded amount 0 100 1 86.96 100 2 75.61 100 3 65.75 100 4 57.18 100 5 49.72 100 Present Value Present Value is an amount today that is equivalent to a future payment, or series of payments, that has been discounted by an appropriate interest rate. The future amount can be a single sum that will be received at the end of the last period, as a series of equally- spaced payments (an annuity), or both. Since money has time value, the present value of a promised future amount is worth less the longer you have to wait to receive it. The time value of money principle says that future dollars are not worth as much as dollars today. Present Value PV = FV/(1 + r)t FV = Future Value PV = Present Value r = the interest rate per period t= the number of compounding periods What is the present value of $8,000 to be paid at the end of three years if the correct (risk adjusted interest rate) is 11%? PV = 8,000/(1.11)3 = 8,000/1.36 = 5,849 I will give you $1000 in 5 years. How much money should you give me now to make it fair to me. You think a good interest rate would be 6% FV= PV ( 1 + i )N 1000 = PV ( 1 + .06)5 1000 = PV (1.338) 1000 / 1.338 = PV 747.38 = PV Future Value Future Value is the amount of money that an investment with a fixed, compounded interest rate will grow to by some future date. The investment can be a single sum deposited at the beginning of the first period, a series of equally-spaced payments (an annuity), or both. Since money has time value, we naturally expect the future value to be greater than the present value. The difference between the two depends on the number of compounding periods involved and the going interest rate. Future Value What is the future value of $34 in 5 years if the interest rate is 5%? (i=.05) FV= PV ( 1 + r )t FV= 34 ( 1+ .05 ) 5 FV= 34 (1.2762815) FV= 43.39. Determine Future Value Compounded Monthly What is the future value of $34 in 5 years if the interest rate is 5%? (i equals .05 divided by 12, because there are 12 months per year. So 0.05/12=.004166, so i=.004166) FV= PV ( 1 + i )N FV= 34 ( 1+ .004166 )60 FV= 34 (1.283307) FV= 43.63. Equal Payment series compounded amount If we want to find the future worth of n equal payments which are made at the end of year of the nth period at an interest rate of i compounded annually. (1 +i)n – 1 FV= A i A person who is now 35 years old is planning for his retired life. He plans to invest an equal sum of Rs. 10,000 at the end of every next 25 years. The bank gives 20% interest rate compounded annually. Find the maturity value of his account when he is 60 years old. Equal Payment series Sinking Fund If we want to find the equivalent amount A that should be deposited at the end of year of the nth period to realize a future sum F at an interest rate of i compounded annually. i A = FA (1 +i)n – 1 A company has to replace the present facility after 15 years of Rs. 500,000. it plans to deposit an equal amount at the end of every year for 15 years at an interest rate of 18% compounded annually. Find the equivalent amount that must be deposited at the end of every year for the next 15 years. Equal Payment Series Present Worth Amount If we want to find the present worth of an equal payment made at the end of year of the nth period at an interest rate of i compounded annually. (1 +i)n – 1 P=A i (1 + i)n A company wants to set up a reserve to an annual equivalent amount of Rs. 10,00,000 for next 20 years. The reserve is assumed to grow at the rate of 15% annually. Find the single payment that must be paid now as the reserve amount. Equal Payment Series Capital Recovery Amount If we want to find the annual equivalent amount A which is to be recovered at the end of year of the nth period for a loan P which is sanctioned now at an interest rate of i compounded annually. i(1 + i)n A= P (1 +i)n – 1 A Bank give a loan to a company worth Rs. 10,00,000 at an interest rate of 18% compounded annually. This amount should be repaid in 15 yearly equal installments. Find the installment amount that company has to pay to the bank. Effective interest Rate Let i be the nominal interest rate compounded annually. But in practice compounding may occur less than a year like monthly, quarterly or semi-annually. The formula to compute effective interest rate is Effective interest rate, R = (1 + i/C)C -1 A person invests Rs. 5,000 in a bank at a nominal interest rate of 12% for 10 years. The compounding is quarterly. Find the maturity amount of the deposit after 10 years. Effective interest rate, R = (1 + i/C)C -1 F = P(1 + R)n