Questions: Here are two ways of investing 20,000 for 20 years. Lump-Sum Deposit Rate Time 20,000 5% compounded annually 20 years Periodic Deposit Rate Time 1000 at the end of each year 5% compounded annually 20 years Use this information and the formulas A=P(1+r)^t and A=P[(1+r)^t-1]/r to complete parts a. and b. below. a. After 20 years, how much more will you have from the lump-sum investment than from the annuity? You will have approximately more from the lump-sum investment than from the annuity. (Round to the nearest dollar as needed.) b. After 20 years, how much more interest will be earned from the lump-sum investment than from the annuity? The interest earned on the lump-sum investment will be approximately more than the interest earned from the annuity.

Here are two ways of investing 20,000 for 20 years. Lump-Sum Deposit Rate Time 20,000 5% compounded annually 20 years Periodic Deposit Rate Time 1000 at the end of each year 5% compounded annually 20 years Use this information and the formulas A=P(1+r)^t and A=P[(1+r)^t-1]/r to complete parts a. and b. below. a. After 20 years, how much more will you have from the lump-sum investment than from the annuity? You will have approximately more from the lump-sum investment than from the annuity. (Round to the nearest dollar as needed.) b. After 20 years, how much more interest will be earned from the lump-sum investment than from the annuity? The interest earned on the lump-sum investment will be approximately more than the interest earned from the annuity.
Transcript text: possible Here are two ways of investing $\$ 20,000$ for 20 years. \begin{tabular}{|l|l|l|} \hline Lump-Sum Deposit & Rate & Time \\ \hline$\$ 20,000$ & $5 \%$ compounded annually & 20 years \\ \hline \end{tabular} \begin{tabular}{|l|l|l|} \hline Periodic Deposit & Rate & Time \\ \hline \begin{tabular}{l} $\$ 1000$ at the end of \\ each year \end{tabular} & $5 \%$ compounded annually & 20 years \\ \hline \end{tabular} Use this information and the formulas $A=P(1+r)^{t}$ and $A=\frac{P\left[(1+r)^{t}-1\right]}{r}$ to complete parts a. and b. below. a. After 20 years, how much more will you have from the lump-sum investment than from the annuity? You will have approximately $\$$ $\square$ more from the lump-sum investment than from the annuity. (Round to the nearest dollar as needed.) b. After 20 years, how much more interest will be earned from the lump-sum investment than from the annuity? The interest earned on the lump-sum investment will be approximately $\$$ $\square$ more than the interest earned from the annuity.
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Solution

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Solution Steps

Step 1: Calculate the Future Value of the Lump-Sum Investment

Using the formula $A_1 = P1(1 + r)^t$, where $P1 = 20000$, $r = 0.05$, and $t = 20$, the future value of the lump-sum investment is $A_1 = 20000(1 + 0.05)^20 = 53066$.

Step 2: Calculate the Future Value of the Annuity

Using the formula $A_2 = \frac{P2[(1 + r)^t - 1]}{r}$, where $P2 = 1000$, $r = 0.05$, and $t = 20$, the future value of the annuity is $A_2 = \frac{1000[(1 + 0.05)^20 - 1]}{0.05} = 33066$.

Step 3: Calculate the Difference in Future Values

The difference in future values between the lump-sum investment and the annuity is $20000$.

Step 4: Calculate the Difference in Interest Earned

The total interest earned from the lump-sum investment is $33066$, and from the annuity is $13066$. Therefore, the difference in interest earned is $20000$.

Final Answer:

The lump-sum investment yields $20000$ more than the annuity in terms of future value, and earns $20000$ more in interest over a period of 20 years.

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