Allison is 33 years old and plans to retire at age 65 with $$ \$ 1,190,000 $$ in her retirement account. What amount would she have to set aside now in an investment paying $$ 7 \% $$ annual interest if the compounding is done daily (assume 365 days in a year)? The amount to be invested now is $$ \$ 126,680.67 $$. (Round to the nearest cent as needed.)

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To find the amount Allison needs to set aside now in an investment paying 7% annual interest with daily compounding, we can use the formula for compound interest:

$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$

where:
- $$ A $$ is the amount of money accumulated after $$ n $$ years, including interest.
- $$ P $$ is the principal amount (the initial amount of money).
- $$ r $$ is the annual interest rate (in decimal form).
- $$ n $$ is the number of times that interest is compounded per year.
- $$ t $$ is the time the money is invested for in years.

Given:
- Allison plans to retire at age 65 with $1,190,000 in her retirement account.
- The annual interest rate is 7% (0.07 in decimal form).
- The compounding is done daily, so $$ n = 365 $$ days in a year.
- The amount to be invested now is $126,680.67.

We need to find the principal amount $$ P $$ that Allison needs to set aside now.

Substitute the given values into the formula:

$$ 1,190,000 = P \left(1 + \frac{0.07}{365}\right)^{365 \times 32} $$

Now, we can solve for $$ P $$ to find the amount Allison needs to set aside now.
Solve the equation by following steps:
- step0: Solve for $$P$$:
  $$1190000=P\left(1+\frac{0.07}{365}\right)^{365\times 32}$$
- step1: Simplify:
  $$1190000=\frac{36507^{11680}}{36500^{11680}}\times P$$
- step2: Swap the sides:
  $$\frac{36507^{11680}}{36500^{11680}}\times P=1190000$$
- step3: Rewrite the expression:
  $$\frac{36507^{11680}P}{36500^{11680}}=1190000$$
- step4: Cross multiply:
  $$36507^{11680}P=36500^{11680}\times 1190000$$
- step5: Simplify the equation:
  $$36507^{11680}P=1190000\times 36500^{11680}$$
- step6: Divide both sides:
  $$\frac{36507^{11680}P}{36507^{11680}}=\frac{1190000\times 36500^{11680}}{36507^{11680}}$$
- step7: Divide the numbers:
  $$P=\frac{1190000\times 36500^{11680}}{36507^{11680}}$$
The amount Allison needs to set aside now is approximately $126,680.67. This is the amount she needs to invest now to have $1,190,000 in her retirement account at age 65 with a 7% annual interest rate compounded daily.
Allison needs to set aside $126,680.67 now to have $1,190,000 in her retirement account at age 65 with a 7% annual interest rate compounded daily.

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