Question

3.

a)

Victoria invested her savings in a bank at 2.75% compounded monthly. How much money did she invest to enable withdrawals of $3,000 at the beginning of every 6 months from the investment for 8 years, if the first withdrawal is to be made in 12 years? Round to the nearest cent

b)

Answer 1

As HOMEWORKLIB's policy, only one question is answered.

We will equate the FV from the investment at the end of 12 years to the PV of the withdrawals for subsequent 8 years.

Let us assume C in the investment that Victoria makes every month for 12 years

interest rate r = 2.75%

No. of investments = 12 years * 12 months = 144

FV = C*(1+r)1⁴⁴ + C*(1+r)1⁴³ + .... + C*(1+r).

This is a geometric progression a, ar, ar²,....,arⁿ. The sum of geometric progression is a*(rⁿ-1) / (r-1) where a is the first term and r is the common ratio. Applying this formula we get:

first term = C(1+r)¹⁴⁴ , common ratio r = second term / first term = 1/(1+r)

FV = C*(1+r)¹⁴⁴ * (1/(1+r)¹⁴⁴ - 1) / (1/(1+r)-1)

Substitute the value of r

FV = C*(1+2.75%)¹⁴⁴ * (1/(1+2.75%)¹⁴⁴ - 1) /(1/(1+ 2.75%)-1) = 1820.580859 *C

Now let us calculate NPV for the withdrawals that Victoria makes.

Withdrawal W = $3000

interest rate for 6 months = 6 * 2.55% = 16.5%

No. of periods = 8 years *2 = 16 (there will be 2 withdrawals per year as the withdrawal occurs after every 6 months)

NPV = W + W/(1+r) + W/(1+r)² +.... W/(1+r)¹⁵ .

first term = W , common ratio r = second term / first term = 1/(1+r)

NPV = W* (1/(1+r)¹⁶ - 1) / (1/(1+r)-1)

Substitute the value of W and r, we get

NPV = 3000*(1/(1+16.5%)¹⁶ - 1) / (1/(1+16.5%)-1) = 19342.11216

We will equate the FV from the investment at the end of 12 years to the PV of the withdrawals for subsequent 8 years.

FV = NPV

Substituting the values in this equation:

1820.580859 *C = 19342.11216

Hence C = $10.62

Answer: Victoria should invest $10.56 every month