Question

Darnell Johnson's employer contributes 4% of Darnell $ 38, 400 salary into his retirement plan. Additionally, the hospital matches the 1% of his salary that Darnell himself puts into his own plan. Find the future value in 10 years assumming (a) funds earn 5% compounded semiannually and (b) funds earn 8% compounded semiannually. (c) Then find the difference between the two. (Hint: Assume all contributions into the plan occur at the end of semiannual periods)

Answer 1

The future value of Darnell Johnson's retirement plan is $21,225.72 assuming a 5% interest rate and $31,601.80 assuming an 8% interest rate, after 10 years. The difference between the two values is $10,376.08.

Step-by-step explanation:

To find the future value of Darnell Johnson's retirement plan, we need to calculate the contributions made by his employer and himself over 10 years, taking into account the interest earned. Let's calculate the future value assuming the funds earn 5% compounded semiannually:

Calculate the contribution made by Darnell's employer each year: 4% of $38,400 = $1,536.

Calculate Darnell's own contribution each year: 1% of $38,400 = $384.

Calculate the future value of the contributions and interest earned after 10 years using the compound interest formula: Future Value = P(1 + r/n)^(nt), where P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.

Here, P = $1,536, r = 5% (or 0.05), n = 2 (semiannually compounded), and t = 10 years. Plugging in these values, we get Future Value = $1,536(1 + 0.05/2)^(2*10) = $21,225.72.

Similarly, we can calculate the future value assuming the funds earn 8% compounded semiannually:

Calculate the contribution made by Darnell's employer each year: $1,536.

Calculate Darnell's own contribution each year: $384.

Calculate the future value using the same formula as before, but with a different interest rate: Future Value = $1,536(1 + 0.08/2)^(2*10) = $31,601.80.

The difference between the two future values is $31,601.80 - $21,225.72 = $10,376.08.

Answer 2

Darnell's total yearly retirement plan contribution, when compounded semiannually, will grow to $3,146.15 at a 5% interest rate and to $4,206.94 at an 8% interest rate over 10 years. The difference between these two future values is $1,060.79.

Step-by-step explanation:

The question involves calculating the future value of a retirement plan with contributions compounded semiannually at different interest rates, and then determining the difference between the two scenarios. To solve it, we'll need to use the future value formula for compound interest:

Future Value = P × (1 + r/n)nt

Where:

• P is the principal amount (initial amount of money)
• r is the annual interest rate (in decimal form)
• n is the number of times the interest is compounded per year
• t is the number of years the money is invested for

Darnell's employer contributes 4% of his salary (4/100 × $38,400 = $1,536) and matches 1% of his contribution, which is also 1% of his salary ($384), so the total contribution per year is $1,536 + $384 = $1,920.

The contributions are compounded semiannually, so n = 2.

For a, with an interest rate of 5% (r = 0.05),

Future Value = $1,920 × (1 + 0.05/2)2×10 = $1,920 × (1.025)20 = $1,920 × 1.63862 = $3,146.15

For b, with an interest rate of 8% (r = 0.08),

Future Value = $1,920 × (1 + 0.08/2)2×10 = $1,920 × (1.04)20 = $1,920 × 2.19112 = $4,206.94

The difference between the two future values is $4,206.94 - $3,146.15 = $1,060.79.