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

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

Where:

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

The principal amount ($P$) is $2,000.00, the interest rate ($r$) is 5.2% or 0.052 in decimal, and the time ($t$) is 13 years.

(a) Compounded annually:

For annual compounding, $n = 1$.

$A = 2000 \left(1 + \frac{0.052}{1}\right)^{1 \times 13}$

(b) Compounded semiannually:

For semiannual compounding, $n = 2$.

$A = 2000 \left(1 + \frac{0.052}{2}\right)^{2 \times 13}$

(c) Compounded quarterly:

For quarterly compounding, $n = 4$.

$A = 2000 \left(1 + \frac{0.052}{4}\right)^{4 \times 13}$

(d) Compounded monthly:

For monthly compounding, $n = 12$.

$A = 2000 \left(1 + \frac{0.052}{12}\right)^{12 \times 13}$

(e) Compounded daily (assuming 365 days in a year):

For daily compounding, $n = 365$.

$A = 2000 \left(1 + \frac{0.052}{365}\right)^{365 \times 13}$

I'll calculate the values now.