The Appropriate Formula To Find The Value Of The Annuity2 Fi

The Appropriate Formula To Find The Value Of The Annuity2 Find

1. Use the appropriate formula to find the value of the annuity.

2. Find the interest. Periodic Deposit Rate Time: $120 at the end of every six months, 3.5% compounded semiannually, over a period of 10 years. The value of the annuity is ​$ ​(Do not round until the final answer. Then round to the nearest dollar as​ needed.) b.b. The interest is $

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Calculating the present value of an annuity is a fundamental task in financial mathematics, particularly for investment analysis and retirement planning. This problem involves determining the current value of a series of periodic deposits, specifically $120 made at the end of every six months over ten years, with an interest rate compounded semiannually at 3.5%. Additionally, the task includes calculating the total interest accumulated over the period. The following discussion elaborates on the formulas used to compute these values and demonstrates the necessary calculations in detail.

Understanding the Annuity and Relevant Parameters

The problem describes an ordinary annuity, where payments are made at the end of each period. The periodic payment (PMT) is $120, with payments occurring semiannually (twice a year) over ten years. To analyze this, we identify key variables:

  • Periodic payment (PMT) = $120
  • Interest rate per period (i) = annual nominal rate / 2 = 3.5% / 2 = 1.75% or 0.0175
  • Total number of periods (n) = 10 years * 2 = 20 periods

Formula for the Present Value of an Annuity

The present value (PV) of an ordinary annuity can be calculated using the formula:

PV = PMT × [(1 - (1 + i)^(-n)) / i]

This formula accounts for the discounting of each payment to its present value, considering compound interest over each period.

Calculating the Present Value

Substituting the known values into the formula:

  • PMT = 120
  • i = 0.0175
  • n = 20

Calculating (1 + i)^(-n):

(1 + 0.0175)^(-20) = (1.0175)^(-20) ≈ 0.7040

Now, computing the numerator:

1 - 0.7040 = 0.2960

Finally, computing the present value:

PV = 120 × (0.2960 / 0.0175) ≈ 120 × 16.914 ≈ 2,029.68

Rounding to the nearest dollar:

PV ≈ $2,030

Calculating the Total Amount of Deposits

The total amount deposited over ten years is:

Total deposits = PMT × n = 120 × 20 = $2,400

Determining the Total Interest Earned

The total interest earned is the difference between the future accumulated value and the total deposits made. To find the future value (FV) of the annuity, we use the formula:

FV = PMT × [((1 + i)^n - 1) / i]

Substituting values:

FV = 120 × [((1.0175)^20 - 1) / 0.0175]

Calculating (1.0175)^20 ≈ 1.399

So, (1.399 - 1) = 0.399

Then, FV = 120 × (0.399 / 0.0175) ≈ 120 × 22.8 ≈ 2,736

Note this is the future value of the annuity at the end of 10 years. The interest earned is therefore:

Interest = FV - Total deposits = 2,736 - 2,400 = $336

Rounding to the nearest dollar, the interest earned is approximately:

$336

Summary of Results

The present value of the annuity, representing the current worth of all future deposits discounted at the semiannual interest rate, is approximately $2,030. Over the ten-year period, the total interest accrued on these deposits amounts to about $336. These calculations demonstrate the core principles of time value of money, illustrating how periodic investments grow under compound interest and how their present value can be assessed using appropriate formulas.

References

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