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Calculate the future value of an annuity with semi-annual payments of $1,000,000 at a 5% annual interest rate for 3 and 6 years.

Business/Management

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Answers

3 parts
a
6,387,720.00
b
13,795,520.00
c
25,544,640.00

Step-by-step solution

12 steps

Here's how to calculate the future value of the annuity for each period.

The formula for the future value of an ordinary annuity is:

FV=Pmt×(1+i)n−1iFV = Pmt \times \frac{(1 + i)^n - 1}{i}

Where:

  • FVFV = Future Value
  • PmtPmt = Payment per period = 1,000,000
  • rr = Annual interest rate = 5%=0.055\% = 0.05
  • mm = Number of compounding periods per year = 2 (semi-annually)
  • ii = Interest rate per period = rm=0.052=0.025\frac{r}{m} = \frac{0.05}{2} = 0.025
  • tt = Number of years
  • nn = Total number of periods = m×tm \times t

Part a· 4 steps

3 years

  1. Identify the parameters.

    Pmt=1,000,000Pmt = 1,000,000
    i=0.025i = 0.025
    t=3t = 3 years
    n=2×3=6n = 2 \times 3 = 6 periods

  2. Substitute the values into the formula.

    FV=1,000,000×(1+0.025)6−10.025FV = 1,000,000 \times \frac{(1 + 0.025)^6 - 1}{0.025}

  3. Calculate the term (1.025)6(1.025)^6.

    (1.025)6≈1.159693(1.025)^6 \approx 1.159693

  4. Calculate the future value.

    FV=1,000,000×1.159693−10.025FV = 1,000,000 \times \frac{1.159693 - 1}{0.025} FV=1,000,000×0.1596930.025FV = 1,000,000 \times \frac{0.159693}{0.025} FV=1,000,000×6.38772FV = 1,000,000 \times 6.38772 FV=6,387,720.00FV = \boxed{6,387,720.00}

Part b13,795,520.00

6 years

  1. Identify the parameters.

    Pmt=1,000,000Pmt = 1,000,000
    i=0.025i = 0.025
    t=6t = 6 years
    n=2×6=12n = 2 \times 6 = 12 periods

  2. Substitute the values into the formula.

    FV=1,000,000×(1+0.025)12−10.025FV = 1,000,000 \times \frac{(1 + 0.025)^{12} - 1}{0.025}

  3. Calculate the term (1.025)12(1.025)^{12}.

    (1.025)12≈1.344888(1.025)^{12} \approx 1.344888

  4. Calculate the future value.

    FV=1,000,000×1.344888−10.025FV = 1,000,000 \times \frac{1.344888 - 1}{0.025} FV=1,000,000×0.3448880.025FV = 1,000,000 \times \frac{0.344888}{0.025} FV=1,000,000×13.79552FV = 1,000,000 \times 13.79552 FV=13,795,520.00FV = \boxed{13,795,520.00}

Part c25,544,640.00

10 years

  1. Identify the parameters.

    Pmt=1,000,000Pmt = 1,000,000
    i=0.025i = 0.025
    t=10t = 10 years
    n=2×10=20n = 2 \times 10 = 20 periods

  2. Substitute the values into the formula.

    FV=1,000,000×(1+0.025)20−10.025FV = 1,000,000 \times \frac{(1 + 0.025)^{20} - 1}{0.025}

  3. Calculate the term (1.025)20(1.025)^{20}.

    (1.025)20≈1.638616(1.025)^{20} \approx 1.638616

  4. Calculate the future value.

    FV=1,000,000×1.638616−10.025FV = 1,000,000 \times \frac{1.638616 - 1}{0.025} FV=1,000,000×0.6386160.025FV = 1,000,000 \times \frac{0.638616}{0.025} FV=1,000,000×25.54464FV = 1,000,000 \times 25.54464 FV=25,544,640.00FV = \boxed{25,544,640.00}
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