Thomas is planning to withdraw 

​$80008000 

from a savings account at the end of each quarter for 

fourfour 

years. If the payments are deferred for 

sevenseven 

years and interest is 

5.68 %5.68% 

compounded​ semi-annually, what amount has to be invested now into the savings​ account?

To determine the amount that needs to be invested now, we can use the concept of the present value of an annuity deferred. Here is the step-by-step solution:

### Step 1: Determine the interest rate per period
Since the interest is compounded semi-annually (twice a year), we first need to find the semi-annual interest rate and then adjust it to a quarterly rate.

1. **Annual interest rate**: $ 5.68\% $
2. **Semi-annual interest rate**: $ \frac{5.68\%}{2} = 2.84\% $

### Step 2: Convert semi-annual rate to an effective quarterly rate
To convert the semi-annual rate to a quarterly rate, we use the formula for converting an annual rate to a different compounding period.

\[
(1 + i_{sa})^{\frac{1}{2}} = (1 + i_{q})^{\frac{1}{4}}
\]

Where $ i_{sa} $ is the semi-annual interest rate and $ i_{q} $ is the quarterly interest rate.

\[
(1 + 0.0284)^{\frac{1}{2}} = (1 + i_{q})^{\frac{1}{4}}
\]

\[
1.0284^{0.5} = (1 + i_{q})^{0.25}
\]

\[
1.0140724 \approx 1 + i_{q}
\]

\[
i_{q} \approx 0.0140724
\]

So, the effective quarterly interest rate is approximately $ 1.40724\% $.

### Step 3: Find the present value of the annuity
Thomas plans to withdraw \$80,000 at the end of each quarter for 4 years. This is a total of $ 4 \times 4 = 16 $ quarters.

Using the present value formula for an ordinary annuity:

\[
PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}
\]

Where:
- $ PMT $ is the quarterly payment = \$80,000
- $ i $ is the quarterly interest rate = 0.0140724
- $ n $ is the total number of payments = 16

\[
PV = 80000 \times \frac{1 - (1 + 0.0140724)^{-16}}{0.0140724}
\]

\[
PV = 80000 \times \frac{1 - (1 + 0.0140724)^{-16}}{0.0140724}
\]

### Step 4: Calculate the present value

First, compute the denominator:

\[
1 + 0.0140724 = 1.0140724
\]

Then the exponentiation:

\[
1.0140724^{-16} \approx 0.798940
\]

Now, the numerator:

\[
1 - 0.798940 = 0.201060
\]

Finally, the present value:

\[
PV = 80000 \times \frac{0.201060}{0.0140724} \approx 80000 \times 14.287 = 1,142,960
\]

### Step 5: Account for the deferred period
The annuity payments are deferred for 7 years, which is 28 quarters. We need to discount the present value found back 28 quarters to find the amount that needs to be invested now.

\[
PV_{now} = PV \times (1 + i)^{-28}
\]

Where:
- $ PV $ is the present value of the annuity calculated above = \$1,142,960
- $ i $ is the quarterly interest rate = 0.0140724

\[
PV_{now} = 1,142,960 \times (1.0140724)^{-28}
\]

First, compute the exponentiation:

\[
(1.0140724)^{-28} \approx 0.682492
\]

Finally, the present value now:

\[
PV_{now} = 1,142,960 \times 0.682492 \approx 780,000
\]

Therefore, the amount that needs to be invested now into the savings account is approximately \$780,000.

Would you like more details on any part of this calculation or have any other questions?

### Related Questions
1. How is the present value of an annuity different from the future value?
2. What is the formula for converting an annual interest rate to a different compounding period?
3. How does the deferral of payments affect the present value calculation?
4. Can you explain the concept of compound interest in more detail?
5. How would the calculation change if the payments were made monthly instead of quarterly?
6. What are the steps to find the effective interest rate for different compounding periods?
7. How does the length of the deferral period impact the present value of an annuity?
8. What is the impact of changing the interest rate on the present value of an annuity?

### Tip
Always double-check the interest rate conversion and the number of periods when dealing with different compounding frequencies to ensure accurate calculations.

Learn how to calculate the present value of a deferred annuity where $80,000 is withdrawn quarterly for four years after a seven-year deferral period. Interest is compounded semi-annually at 5.68%. Detailed step-by-step solution provided.

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