17. How much money should be deposited today in an account that earns $$ 2 \% $$ compounded semiannually so that it will accumulate to $$ \$ 10,000 $$ in three years?

Question

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Let the amount to be deposited today be $$ P $$. The future value of the deposit after $$ t = 3 $$ years at an annual interest rate of $$ 2\% $$ compounded semiannually is given by

$$
FV = P \left(1 + \frac{0.02}{2}\right)^{2 \times 3}
$$

Since the account needs to accumulate to $$ \$10,\!000 $$, we have

$$
10,\!000 = P \left(1 + 0.01\right)^6
$$

Simplify the expression:

$$
10,\!000 = P \left(1.01\right)^6
$$

To solve for $$ P $$, divide both sides by $$ (1.01)^6 $$:

$$
P = \frac{10,\!000}{(1.01)^6}
$$

Now, calculate $$ (1.01)^6 $$:

$$
(1.01)^6 \approx 1.06152015
$$

Therefore,

$$
P \approx \frac{10,\!000}{1.06152015} \approx 9410.47
$$

So, the amount that should be deposited today is approximately $$ \$9,\!410.47 $$.
You should deposit approximately \$9,410.47 today.

How much money should be deposited
today in an account that earns compounded
semiannually so that it will accumulate to
in three years?

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How much money should be deposited today in an account that earns compounded semiannually so that it will accumulate to in three years?