An investor wants to save money to purchase real estate. He buys an annuity with quarterly payments that earn $$ 5.9 \% $$ interest, compounded quarterly. Payments will be made at the end of each quarter. Find the total value of the annuity in 15 years if each quarterly payment is $$ \$ 222 $$. Do not round any intermediate computations, and round your final answer to the nearest cent. If necessary, refer to the list of financial formulas.

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UpStudy AI · Accepted Answer

To find the total value of the annuity after 15 years with quarterly payments, we can use the future value of an ordinary annuity formula:

$$
FV = P \times \frac{(1 + r)^n - 1}{r}
$$

Where:
- $$ FV $$ = future value of the annuity
- $$ P $$ = payment amount per period
- $$ r $$ = interest rate per period
- $$ n $$ = total number of payments

### Step 1: Identify the known values
- Quarterly payment $$ P = 222 $$
- Annual interest rate = $$ 5.9\% $$
- Quarterly interest rate $$ r = \frac{5.9\%}{4} = \frac{0.059}{4} = 0.01475 $$
- Total number of years = $$ 15 $$
- Total number of payments $$ n = 15 \times 4 = 60 $$

### Step 2: Substitute the values into the formula
Now we can substitute the values into the future value formula:

$$
FV = 222 \times \frac{(1 + 0.01475)^{60} - 1}{0.01475}
$$

### Step 3: Calculate the future value
Let's perform the calculation.

First, we need to calculate $$ (1 + 0.01475)^{60} $$:

$$
(1 + 0.01475)^{60}
$$

Then we will substitute this value back into the formula to find $$ FV $$.

I'll perform these calculations now.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+0.01475\right)^{60}$$
- step1: Add the numbers:
  $$1.01475^{60}$$
- step2: Convert the expressions:
  $$\left(\frac{4059}{4000}\right)^{60}$$
- step3: Use the properties of exponents:
  $$\frac{4059^{60}}{4000^{60}}$$
Calculate or simplify the expression $$ 222 * ((1 + 0.01475)^{60} - 1) / 0.01475 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{222\left(\left(1+0.01475\right)^{60}-1\right)}{0.01475}$$
- step1: Add the numbers:
  $$\frac{222\left(1.01475^{60}-1\right)}{0.01475}$$
- step2: Convert the expressions:
  $$\frac{222\left(\left(\frac{4059}{4000}\right)^{60}-1\right)}{0.01475}$$
- step3: Subtract the numbers:
  $$\frac{222\times \frac{4059^{60}-4000^{60}}{4000^{60}}}{0.01475}$$
- step4: Multiply the numbers:
  $$\frac{\frac{111\times 4059^{60}-111\times 4000^{60}}{2^{299}\times 125^{60}}}{0.01475}$$
- step5: Convert the expressions:
  $$\frac{\frac{111\times 4059^{60}-111\times 4000^{60}}{2^{299}\times 125^{60}}}{\frac{59}{4000}}$$
- step6: Multiply by the reciprocal:
  $$\frac{111\times 4059^{60}-111\times 4000^{60}}{2^{299}\times 125^{60}}\times \frac{4000}{59}$$
- step7: Rewrite the expression:
  $$\frac{111\times 4059^{60}-111\times 4000^{60}}{2^{299}\times 125^{60}}\times \frac{32\times 125}{59}$$
- step8: Rewrite the expression:
  $$\frac{111\times 4059^{60}-111\times 4000^{60}}{2^{299}\times 125^{60}}\times \frac{2^{5}\times 125}{59}$$
- step9: Reduce the numbers:
  $$\frac{111\times 4059^{60}-111\times 4000^{60}}{2^{294}\times 125^{59}}\times \frac{1}{59}$$
- step10: Multiply the fractions:
  $$\frac{111\times 4059^{60}-111\times 4000^{60}}{2^{294}\times 125^{59}\times 59}$$
- step11: Multiply:
  $$\frac{111\times 4059^{60}-111\times 4000^{60}}{59\times 2^{294}\times 125^{59}}$$
### Step 4: Calculate the values

1. **Calculate $$ (1 + 0.01475)^{60} $$**:
   $$
   (1 + 0.01475)^{60} \approx 2.407374
   $$

2. **Substituting into the future value formula**:
   $$
   FV = 222 \times \frac{(2.407374 - 1)}{0.01475}
   $$

Now, calculate $$ 2.407374 - 1 $$:
   $$
   2.407374 - 1 \approx 1.407374
   $$

Then, substitute this back into the formula:
   $$
   FV = 222 \times \frac{1.407374}{0.01475}
   $$

### Step 5: Final calculation
Now, let's calculate the final future value $$ FV $$.

I'll perform this calculation now.
Calculate the value by following steps:
- step0: Calculate:
  $$222\left(\frac{1.407374}{0.01475}\right)$$
- step1: Divide the terms:
  $$222\times \frac{703687}{7375}$$
- step2: Multiply:
  $$\frac{222\times 703687}{7375}$$
- step3: Multiply:
  $$\frac{156218514}{7375}$$
### Final Calculation Result

The future value $$ FV $$ of the annuity is approximately:
$$
FV \approx 21182.17139
$$

### Step 6: Rounding the final answer
Rounding this to the nearest cent gives:
$$
FV \approx 21182.17
$$

### Conclusion
The total value of the annuity in 15 years, with quarterly payments of $$ \$ 222 $$ at an interest rate of $$ 5.9\% $$ compounded quarterly, is **\$ 21,182.17**.
The total value of the annuity after 15 years is \$21,182.17.

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