scenario #3: isaiah is in his 50s and currently does not have a retirement fund. however, he recently read a…

scenario #3: isaiah is in his 50s and currently does not have a retirement fund. however, he recently read a few articles about the insufficient savings of people in retirement and, as a result, he decides he wants to start now. he saves $500 per month for 15 years and earns 7% by investing in the stock market* through an index fund. *note: we use 7% as the return of the stock market (based on historical averages) for simplicity. however, keep in mind that, unlike a savings account, the stock market does not guarantee any specific return rate. stocks rise and fall in price daily, and even throughout the day, so your total earnings ultimately depend on the price at which you buy and sell the shares. using the compound interest calculator, what is the total balance in the account after 15 years? *round your answer to the nearest dollar and input numbers only (no symbols).

scenario #3: isaiah is in his 50s and currently does not have a retirement fund. however, he recently read a few articles about the insufficient savings of people in retirement and, as a result, he decides he wants to start now. he saves $500 per month for 15 years and earns 7% by investing in the stock market* through an index fund. *note: we use 7% as the return of the stock market (based on historical averages) for simplicity. however, keep in mind that, unlike a savings account, the stock market does not guarantee any specific return rate. stocks rise and fall in price daily, and even throughout the day, so your total earnings ultimately depend on the price at which you buy and sell the shares. using the compound interest calculator, what is the total balance in the account after 15 years? *round your answer to the nearest dollar and input numbers only (no symbols).

Answer

Explanation:

Step1: Identify the relevant formula

The formula for the future - value of an ordinary annuity is $FVA = P\times\frac{(1 + r)^{n}-1}{r}$, where $P$ is the payment per period, $r$ is the interest rate per period, and $n$ is the number of periods. The annual interest rate $i = 7%=0.07$. Since the payments are made monthly, the monthly interest rate $r=\frac{0.07}{12}$. The number of years is 15, so the number of periods $n = 15\times12=180$ months, and the payment per period $P = 500$.

Step2: Substitute the values into the formula

$r=\frac{0.07}{12}\approx0.005833$, $n = 180$, $P = 500$ $FVA=500\times\frac{(1 + 0.005833)^{180}-1}{0.005833}$ First, calculate $(1 + 0.005833)^{180}$. Let $x=(1 + 0.005833)^{180}$. Using the formula $a^{b}=e^{b\ln(a)}$, we have $\ln(x)=180\times\ln(1.005833)$. $\ln(1.005833)\approx0.005816$, so $\ln(x)=180\times0.005816 = 1.04688$. Then $x = e^{1.04688}\approx2.8499$. $(1 + 0.005833)^{180}-1\approx2.8499 - 1=1.8499$ $\frac{(1 + 0.005833)^{180}-1}{0.005833}=\frac{1.8499}{0.005833}\approx317.14$ $FVA=500\times317.14 = 158570$

Answer:

158570