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Lecture Four Time Value of Money and Its Applications

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Main Idea about Time Value of Money Money that the firm or an individual has in its possession today is more valuable than future payments because the money it now can be invested and earn positive returns. “A bird in hand worth more than two in the bush”

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Why Money has Time Value? The existence of interest rates in the economy results in money with its time value. Sacrificing present ownership requires possibility of having more future ownership. Inflation in economy is one of the major cause. Risk or uncertainty of favorable outcome.

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Major Importance of Time Value of Money It is required for accounting accuracy for certain transactions such as loan amortization, lease payments, and bond interest. In order to design systems that optimize the firm’s cash flows. For better planning about cash collections and disbursements in a way that will enable the firm to get the greatest value from its money.

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Major Importance of Time Value of Money Funding for new programs, products, and projects can be justified financially using time –value-of-money techniques. Investments in new equipment, in inventory, and in production quantities are affected by time-value-of-money techniques.

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Major Concepts Here Simple Interest: Interest paid (or earned) on the original amount, or principal borrowed (or lent). Example-1: Assume that you deposit $100 in a savings account paying 8% simple interest and keep it there for 10 years. What is your total interest at the end of ten years?

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Major Concepts Here (Cont.) o Compound Interest: Interest paid or earned on any previous interest earned as well as on the principal amount. Future Value: The value of a present amount at a future date, found by applying compound interest over a specified time period. Compounding: The process of going from present values to future values is called compounding.

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Major Concepts Here (Cont.) Computational tools for compounding: Financial tables, financial calculators and computer spreadsheets are used for computations of both the future and present values. The following formula is used to calculate the future value of an amount. FV n = PV x (1+i) n The later portion of the equation (1+i) n is called the FVIF i,n, or the Future Value Interest Factor. Thus, when one uses financial table the equation becomes : FV n = PV x (FVIF i,n )

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Major Concepts Here (Cont.) Example-2 Mr. Tushar deposits Tk in a savings account paying 12% interest compounded annually. What is the future value of his fund at the end of 5 th year? FV n = PV x (1+i) n = PV x (FVIF 12%,5 ) = Tk x = Tk. 1762

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Major Concepts Here (Cont.) Example -3 The Rule of 70: You can find approximately how long (no. of years) does it take your fixed deposit to becomes double by just simply dividing 70 with the rate of interest Example: If the compound interest rate is 7%, how many years does it take your savings to become double? 70 years 7 = 10 years

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Major Concepts Here (cont.) Present Value: The current value of a future amount of money or a series of payments. Formula to Calculate Present Value: PV = FV n x 1/(1+i) n The term 1/(1+i) n is called (PVIF i,n ), Present Value Interest Factor. Its value is always less than one as it is a discounting factor.

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Major Concepts Here (cont.) Discounting Process: The process of finding the present value of a payment or a series of future cash flows, which is reverse of compounding. Example-3: Mr. Arman has an opportunity to receive Tk.5000 five years from now. If he can earn 9% on his investments in the normal course of events, what is the most he should pay now for this opportunity?

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Self-Test Problem Practice Q 1: Suppose you will receive $2000 after 10 years and now the interest rate is 8%, calculate the present value of this amount. Practice Q 2: You have $1500 to invest today at 9% interest compounded semi-annually. Find how much you will have accumulated in the account at the end of 6 years.

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Self Test Problems: Solving for interest rate (i) & period (n) Practice Q.3 Suppose you can buy a security at a price of Tk78.35 that will pay you Tk100 after five years. What will be the rate of return, if you purchase the security? Practice Q.4 Suppose you know that a security will provide a 10% return per year, its price is Tk68.30 and you will receive Tk.100 at maturity. How many years does the security take to mature?

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Major Concepts Here (cont.) Annuity: A series of equal payments or receipts of money at fixed intervals for a specified number of periods. Types of Annuity: 1) Ordinary (deferred) annuity: Payment or receipts occurring at the end of each period. Installment payment on a loan. 2) Annuity Due: Payment or receipts occur at the beginning of each period. Example, insurance payment.

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Example for Annuity: Example-4: Mr. Hamid is choosing which of two annuities to receive. Both are 5-year, $1000 annuities; annuity A is an ordinary annuity, and annuity B is an annuity due. Which is the better option for him if he considered the future value? (The market interest rate is 7%). Note: FV & PV of annuity due are always greater than those of an ordinary annuity.

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Example for Annuity (cont.) Solution: For ordinary annuity FVA n = PMT [{(1+i) n -1}/i] = Tk 1000 x [{(1+0.07) 5 -1}/0.07] = Tk For annuity due FVA n = PMT [{(1+i) n -1}/i x (1+i)] = Tk 1000 x [{(1+0.07) 5 -1}/0.07 X (1+0.07)] = Tk

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Example for Annuity (cont.) Example-5: (P. 166) Cute Baby Company, a small producer of plastic toys, wants to determine the most it should pay to purchase a particular ordinary annuity. Find the present value if the annuity consists of cash flows of $700 at the end of each year for 5 years. The firm requires the annuity to provide a minimum return of 8%. For an ordinary annuity, PVA n = PMT x [1-(1+i) -n ]/i Using the table PVA is PVA n = PMT x (PVIFA i,n ) and the FVA is FVA n = PMT x (FVIFA i,n ) For annuity due, PVA n = PMT x [{1-(1+i) -n ]/I}(1+i)]

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Perpetuity # A stream of equal payments expected to continue forever. Formula: Payment PMT PV (Perpetuity) = = Interest Rate i The present value of this special type of annuity will be required when we value perpetual bonds and preferred stock.

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Effective Annual Interest Rate The effective annual interest rate is the interest rate compounded annually that provides the same annual interest as the nominal rate does when compounded m times per year. EAR = (1+ i simple /m) m – 1 where, m is the number of compounding period per year. Example-6 Nominal (annual) interest rate = 8%, compounded quarterly on a one-year investment. Calculate the effective rate.

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Example-7 Mr. Mahin has Tk that he can deposit any of three savings accounts for 3-year period. Bank A compounds interest on an annual basis, bank B twice each year, and bank C each quarter. All 3 banks have a stated annual interest rate of 4%. a. What amount would Mr. Mahin have at the end of the third year in each bank? b. What effective annual rate (EAR) would he earn in each of the banks? c. On the basis of your findings in a and b, which bank should Mr. Mahin deal with? Why?

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Example-8 A municipal savings bond can be converted to $100 at maturity 6 years from purchase. If this state bonds are to be competitive with B.D Government savings bonds, which pay 8% annual interest (compounded annually), at what price must the state sell its bonds? Assume no cash payments on savings bond prior to redemption.

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