To calculate the future value of an investment with daily compounding interest, we use the formula:

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

Where:

• $A$ = the future value of the investment/loan, including interest
• $P$ = the principal investment amount (initial deposit) = $1,000.00
• $r$ = annual interest rate (decimal) = $9.07\% = 0.0907$
• $n$ = number of times the interest is compounded per year = 365
• $t$ = the time the money is invested for, in years = 13

Substituting the values:

$A = 1000 \times \left(1 + \frac{0.0907}{365}\right)^{365 \cdot 13}$

Now, let's calculate this step by step.

1. $\frac{0.0907}{365} = 0.00024849315$
2. $1 + 0.00024849315 = 1.00024849315$
3. $365 \cdot 13 = 4745$
4. $A = 1000 \times (1.00024849315)^{4745}$

We will now compute $(1.00024849315)^{4745}$.ating the future value with daily compounding interest principal = 1000 # Initial deposit annual_rate = 0.0907 # Annual interest rate as a decimal compounds_per_year = 365 # Compounding daily years = 13 # Time period in years

Daily compounding formula

future_value = principal * (1 + annual_rate / compounds_per_year) ** (compounds_per_year * yearsThe amount in the account after 13 years, with daily compounding at an annual interest rate of 9.07%, will be approximately $3,250.97.

Would you like a detailed breakdown of the calculations or additional information? Here are some related questions you might explore:

1. How does the future value change with different compounding frequencies (e.g., monthly or annually)?
2. What is the total interest earned over the 13 years?
3. How much would the amount differ with a slightly higher or lower interest rate?
4. How does the time period (e.g., 10 years vs. 15 years) affect the future value?
5. How does compounding daily compare to compounding continuously?

Tip: Daily compounding can significantly increase returns over long periods compared to annual or simple interest. Always check the compounding frequency when comparing investment options.