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[Kyle0819]

1. Noel made the following deposits: Present – P210K, After 3 months – P80K, After 1.5 years the present – P90K, After 2.5 years the present – P120K and After 4 years the present – P310K. Ten years from now, Noel plans use the money to start a business. How much can he equally withdraw for 5 quarters if at 12% compounded semiannually? The 5 equal withdrawals will be made at the end of the 30th quarterly period?

 

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To solve this problem, we need to calculate the future value of the deposits made by Noel and then determine the equal withdrawals he can make.

First, let's calculate the future value of the deposits:
1. Present deposit: P210,000
2. Deposit after 3 months: P80,000
3. Deposit after 1.5 years: P90,000
4. Deposit after 2.5 years: P120,000
5. Deposit after 4 years: P310,000

To calculate the future value, we will use the compound interest formula:

FV = PV * (1 + r/n)^(n*t)

Where:
FV = future value
PV = present value
r = interest rate
n = number of compounding periods per year
t = number of years

In this case, the interest rate is 12% compounded semiannually, so r = 0.12 and n = 2.

1. Future value of the present deposit:
FV1 = P210,000 * (1 + 0.12/2)^(2*10)
FV1 = P210,000 * (1 + 0.06)^20
FV1 = P210,000 * 1.06^20
FV1 ≈ P537,438.16

2. Future value of the deposit after 3 months:
FV2 = P80,000 * (1 + 0.12/2)^(2*9.75)
FV2 = P80,000 * (1 + 0.06)^(19.5)
FV2 ≈ P184,392.72

3. Future value of the deposit after 1.5 years:
FV3 = P90,000 * (1 + 0.12/2)^(2*8.5)
FV3 = P90,000 * (1 + 0.06)^(17)
FV3 ≈ P173,970.47

4. Future value of the deposit after 2.5 years:
FV4 = P120,000 * (1 + 0.12/2)^(2*7.5)
FV4 = P120,000 * (1 + 0.06)^(15)
FV4 ≈ P203,661.53

5. Future value of the deposit after 4 years:
FV5 = P310,000 * (1 + 0.12/2)^(2*6)
FV5 = P310,000 * (1 + 0.06)^(12)
FV5 ≈ P546,537.66

Now, let's calculate the equal withdrawals Noel can make for 5 quarters (end of 30th quarterly period):

Since the compounding period is semiannually, the interest rate per quarter is 0.12/2 = 0.06.
The number of quarters in 10 years is 10 * 4 = 40.

Using the annuity formula to calculate the equal withdrawals:

PVA = PMT * [(1 - (1 + r)^(-n))/r]

Where:
PVA = present value of the annuity (total amount to be withdrawn)
PMT = equal payments to be withdrawn
r = interest rate per period
n = number of periods

Let's calculate the equal withdrawals:

PVA = PMT * [(1 - (1 + 0.06)^(-40))/0.06]

PVA = PMT * [1 - (1.06)^(-40)] / 0.06

PVA = PMT * (1 - 0.376889) / 0.06

PVA = PMT * 10.22148

PMT = PVA / 10.22148

The total amount to be withdrawn (PVA) is the sum of the future values of the deposits:

PVA = FV1 + FV2 + FV3 + FV4 + FV5

PVA = P537,438.16 + P184,392.72 + P173,970.47 + P203,661.53 + P546,537.66

PVA ≈ P1,645,000.54

Now, let's calculate the equal withdrawals (PMT):

PMT = P1,645,000.54 / 10.22148

PMT ≈ P160,860.64

Therefore, Noel can withdraw approximately P160,860.64 for 5 quarters (end of the 30th quarterly period) in equal amounts.