Dr. Acosta has been secretly depositing $10,500 in her savings account every December starting in 1994. Her account earns 6 percent compounded annually. How much did she have in December 2006? To answer this question you need to calculate for a) the future value. b) the future value of an annuity due. c) the present value. d) the future value of an ordinary annuity.
The future value (FV) of an investment is a crucial financial concept that quantifies how much money an investment will grow to over time, taking into account a given interest rate. In Dr. Acosta’s case, she has been consistently depositing $10,500 into her savings account every December since 1994, and her account earns an annual interest rate of 6 percent, compounded annually. To calculate the future value, we employ the compound interest formula:
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Where FV is the future value, PV is the present value (the initial deposit), r is the annual interest rate (expressed as a decimal), and n is the number of years the money is invested.
For Dr. Acosta’s scenario, the initial deposit (PV) is $10,500, the annual interest rate (r) is 0.06 (6 percent expressed as a decimal), and the number of years (n) is 12, accounting for the period from 1994 to 2006. Utilizing this formula, we can calculate the future value to determine how much Dr. Acosta had in her account in December 2006.
An annuity due is characterized by a series of equal payments made at the beginning of each period. In Dr. Acosta’s case, she makes $10,500 deposits at the beginning of each December. To ascertain the future value of an annuity due, we employ the formula:
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Here, FVAD is the future value of an annuity due, PMT represents the periodic payment (the annual deposit), r is the annual interest rate (expressed as a decimal), and n signifies the number of periods (in years).
For Dr. Acosta’s scenario, the annual deposit (PMT) is $10,500, the annual interest rate (r) is 0.06, and the number of years (n) is 12. Using this formula, we can calculate the future value of the annuity due to determine the total amount she had in her account by December 2006.
c) Present Value:
The present value (PV) is the current worth of a series of future cash flows, discounted at a specific interest rate. In this case, we aim to determine the present value of Dr. Acosta’s savings, representing the amount she had in her account in 1994. We utilize the formula for the present value of a single sum:
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In this formula, PV signifies the present value, FV is the future value (which we calculated in part a), r denotes the annual interest rate (expressed as a decimal), and n stands for the number of years.
For Dr. Acosta’s scenario, FV is the amount we calculated in part (a), the annual interest rate (r) is 0.06, and the number of years (n) is 12. Employing this formula, we can establish the initial amount Dr. Acosta deposited in 1994.
An ordinary annuity is characterized by a series of equal payments made at the end of each period. Dr. Acosta’s deposits, in this case, are made at the end of each December. To determine the future value of an ordinary annuity, we apply the formula:
FVOA = PMT \times \frac((1 + r)^n – 1}{r}
In this formula, FVOA represents the future value of an ordinary annuity, PMT symbolizes the periodic payment (the annual deposit), r signifies the annual interest rate (expressed as a decimal), and n signifies the number of periods (in years).
For Dr. Acosta’s scenario, the annual deposit (PMT) is $10,500, the annual interest rate (r) is 0.06, and the number of years (n) is 12. Using this formula, we can determine the future value of the ordinary annuity.
By conducting these calculations and using the appropriate formulas, we can ascertain how much Dr. Acosta had in her savings account in December 2006, taking into consideration the compounding interest and the timing of her deposits. These financial concepts provide a comprehensive picture of her savings growth over the years, allowing her to make informed financial decisions.
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