Math Problem Statement

What is the present value of an annuity that pays $74 per year for 5 years and an additional $1,000 with the final payment? Use a nominal rate of 9.65%.

Solution

To calculate the present value (PV) of the annuity, we can break it into two parts:

  1. The annuity part: Payments of $74 per year for 5 years.
  2. The lump sum part: An additional payment of $1,000 at the end of 5 years.

Formula for Present Value of an Annuity

The formula for the present value of an annuity is:

PVannuity=P×(1(1+r)nr)PV_{\text{annuity}} = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)

Where:

  • P=74P = 74 (annual payment),
  • r=9.65%100=0.0965r = \frac{9.65\%}{100} = 0.0965 (annual nominal interest rate),
  • n=5n = 5 (number of years).

Formula for Present Value of a Lump Sum Payment

The present value of a lump sum paid in the future is:

PVlump sum=F(1+r)nPV_{\text{lump sum}} = \frac{F}{(1 + r)^n}

Where:

  • F=1,000F = 1,000 (lump sum payment),
  • r=0.0965r = 0.0965 (interest rate),
  • n=5n = 5 (number of years).

Step-by-Step Calculation

  1. Present value of the annuity:

PVannuity=74×(1(1+0.0965)50.0965)PV_{\text{annuity}} = 74 \times \left( \frac{1 - (1 + 0.0965)^{-5}}{0.0965} \right)

  1. Present value of the lump sum:

PVlump sum=1,000(1+0.0965)5PV_{\text{lump sum}} = \frac{1,000}{(1 + 0.0965)^5}

Let's calculate these values.1. Present value of the annuity: $283.04 2. Present value of the lump sum: $630.89

Total present value of the annuity:

PVtotal=283.04+630.89=913.94PV_{\text{total}} = 283.04 + 630.89 = 913.94

So, the present value of the annuity is $913.94.

Would you like further details on any of these steps or calculations?

Here are some related questions:

  1. How does the discount rate affect the present value of an annuity?
  2. What is the difference between a nominal and effective interest rate?
  3. How would the present value change if payments were made quarterly instead of annually?
  4. How would inflation impact the value of future annuity payments?
  5. What is the future value of this annuity?

Tip: The present value of an annuity decreases as the interest rate increases.

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Math Problem Analysis

Mathematical Concepts

Present Value
Annuities
Lump Sum Payments
Time Value of Money

Formulas

Present Value of Annuity Formula: PV = P * [(1 - (1 + r)^-n) / r]
Present Value of Lump Sum: PV = F / (1 + r)^n

Theorems

Time Value of Money Principle

Suitable Grade Level

Grades 11-12 or Undergraduate Finance