Present AND Future Values OF AN Ordinary AND Annuity DUE Exampes PDF

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Present and future value of annuities. ...

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PRESENT VALUE OF AN ANNUITY An annuity is a series of evenly spaced equal payments made for a certain amount of time. There are two basic types of annuity known as ordinary annuity and annuity due. Ordinary annuity is one in which periodic payments are made at the end of each period. Annuity due is the one in which periodic payments are made at the beginning of each period. The present value an annuity is the sum of the periodic payments each discounted at the given rate of interest to reflect the time value of money. Alternatively defined, the present value of an annuity is the amount which if invested at the start of first period at the given rate of interest will equate the sum of the amount invested and the compound interest earned on the investment with the product of number of the periodic payments and the face value of each payment. Formula Although the present value (PV) of an annuity can be calculated by discounting each periodic payment separately to the starting point and then adding up all the discounted figures, however, it is more convenient to use the 'one step' formulas given below. PV of an Ordinary Annuity = R × PV of an Annuity Due = R ×

1 − (1 + i)-n i 1 − (1 + i)-n × (1 + i) i

Where, i is the interest rate per compounding period; n are the number of compounding periods; and R is the fixed periodic payment. Examples Example 1: Calculate the present value on Jan 1, 2011 of an annuity of $500 paid at the end of each month of the calendar year 2011. The annual interest rate is 12%. Solution We have, Periodic Payment R = $500 Number of Periods n = 12 Interest Rate i = 12%/12 = 1% Present Value PV = $500 × (1-(1+1%)^(-12))/1% = $500 × (1-1.01^-12)/1% ≈ $500 × (1-0.88745)/1% ≈ $500 × 0.11255/1% ≈ $500 × 11.255 ≈ $5,627.54 1

PRESENT VALUE OF AN ANNUITY DUE Example 2: A certain amount was invested on Jan 1, 2010 such that it generated a periodic payment of $1,000 at the beginning of each month of the calendar year 2010. The interest rate on the investment was 13.2%. Calculate the original investment and the interest earned. Solution Periodic Payment R = $1,000 Number of Periods n = 12 Interest Rate i = 13.2%/12 = 1.1% Original Investment = PV of annuity due on Jan 1, 2010 = $1,000 × (1-(1+1.1%)^(-12))/1.1% × (1+1.1%) = $1,000 × (1-1.011^-12)/0.011 × 1.011 ≈ $1,000 × (1-0.876973)/0.011 × 1.011 ≈ $1,000 × 0.123027/0.011 × 1.011 ≈ $1,000 × 11.184289 × 1.011 ≈ $11,307.32 Interest Earned ≈ $1,000 × 12 − $11,307.32 ≈ $692.68

Future Value of Annuity Due

The future value of annuity due formula is used to calculate the ending value of a series of payments or cash flows where the first payment is received immediately. The first cash flow received immediately is what distinguishes an

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annuity due from an ordinary annuity. An annuity due is sometimes referred to as an immediate annuity. The future value of annuity due formula calculates the value at a future date. The use of the future value of annuity due formula in real situations is different than that of the present value for an annuity due. For example, suppose that an individual or company wants to buy an annuity from someone and the first payment is received today. To calculate the price to pay for this particular situation would require use of the present value of annuity due formula. However, if an individual is wanting to calculate what their balance would be after saving for 5 years in an interest bearing account and they choose to put the first cash flow into the account today, the future value of annuity due would be used. Example of Future Value of Annuity Due Formula

To elaborate on the prior example of the future value of an annuity due, suppose that an individual would like to calculate their future balance after 5 years with today being the first deposit. The amount deposited per year is $1,000 and the account has an effective rate of 3% per year. It is important to note that the last cash flow is received one year prior to the end of the 5th year. For this example, we would use the future value of annuity due formula to come to the following equation:

Af t ersol vi ng,t hebal anceaf t er5y ear swoul dbe$5468. 41.

FUTURE VALUE OF AN ORDINARY ANNUITY (1 + i)n − 1 FV of Ordinary Annuity = R × i

P = PMT [((1 + r)n - 1) / r]

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FVoa = PMT [((1 + i)n - 1) / i]

Where: FVoa = Future Value of an Ordinary Annuity PMT = Amount of each payment i = Interest Rate Per Period n = Number of Periods P = The future value of the annuity stream to be paid in the future PMT = The amount of each annuity payment r = The interest rate n = The number of periods over which payments are made This value is the amount that a stream of future payments will grow to, assuming that a certain amount of compounded interest earnings gradually accrue over the measurement period. Usually, the key variable in the equation is the interest rate assumption, which could be severely misstated from the interest rate that is actually experienced in future periods. For example, the treasurer of ABC International expects to invest $100,000 of the firm's funds in a long-term investment vehicle at the end of each year for the next five years. He expects that the company will earn 7% interest that will compound annually. The value that these payments should have at the end of the five-year period is calculated as: P = $100,000 [((1 + .07)5 - 1) / .07] P = $575,074 As another example, what if the interest on the investment compounded monthly instead of annually, and the amount invested were $8,000 at the end of month? The calculation is: P = $8,000 [((1 + .005833)60 - 1) / .005833] P = $572,737 The .005833 interest rate used in the last example is 1/12th of the full 7% annual interest rate.

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