ruthann is 28 years old and is retiring at the age of 65. when she retires, she estimates that she will need…

Explanation:

Step1: Calculate Contribution Amount

RuthAnn's annual income is $36,278.13, and she contributes 11% of it. So, contribution ( C = 0.11\times36278.13 ). ( C = 3990.5943 ) dollars per year.

Step2: Calculate Time to Retirement

She is 28, retiring at 65, so number of years ( n = 65 - 28 = 37 ) years.

Step3: Calculate Future Value of Annuity

The formula for future value of an ordinary annuity is ( FV = C\times\frac{(1 + r)^n - 1}{r} ), where ( r = 0.071 ), ( C = 3990.5943 ), ( n = 37 ).

First, calculate ( (1 + 0.071)^{37} ). Let's compute that: ( 1.071^{37}\approx 10.644 ) (using calculator or approximation).

Then, ( (1.071^{37}- 1)\approx 9.644 ).

Now, ( \frac{9.644}{0.071}\approx 135.83 ).

Then, ( FV = 3990.5943\times135.83\approx 542,147.93 ) (this is the total amount saved at retirement).

Step4: Calculate Annual Income from Savings

She needs income for 30 years, so we use the present value of annuity formula to find the annual withdrawal ( A ), where ( PV = 542147.93 ), ( n = 30 ), ( r = 0.071 ).

The formula for present value of ordinary annuity is ( PV = A\times\frac{1 - (1 + r)^{-n}}{r} ).

We solve for ( A ): ( A = PV\times\frac{r}{1 - (1 + r)^{-n}} ).

Calculate ( (1 + 0.071)^{-30}\approx 0.129 ).

Then, ( 1 - 0.129 = 0.871 ).

( \frac{0.071}{0.871}\approx 0.0815 ).

Then, ( A = 542147.93\times0.0815\approx 44,205.06 )? Wait, no, wait, maybe I messed up. Wait, no, the future value is the total savings, then to get annual income, we can also think of it as the amount she can withdraw each year for 30 years, so using the formula for the present value of an annuity due? No, ordinary annuity. Wait, maybe I made a mistake in step 3. Wait, no, let's recalculate the future value correctly.

Wait, let's use the exact formula for future value:

( FV = 3990.5943\times\frac{(1.071^{37}-1)}{0.071} ).

Let's compute ( 1.071^{37} ) more accurately. Using a calculator, ( 1.071^{37} ):

We can compute step by step:

Year 1: 1.071

Year 2: 1.071*1.071=1.147

Year 5: ~1.071^5≈1.405

Year 10: (1.405)^2≈1.974

Year 20: (1.974)^2≈3.897

Year 30: 3.897*1.405≈5.475

Year 37: 5.4751.071^7≈5.4751.616≈8.848 (wait, earlier approximation was wrong). Wait, maybe better to use a calculator for ( 1.071^{37} ). Let's use the formula ( \ln(1.071) \approx 0.0686 ), so ( 37\times0.0686 \approx 2.538 ), ( e^{2.538}\approx 12.64 ). So ( 1.071^{37}\approx 12.64 ). Then, ( 12.64 - 1 = 11.64 ), divided by 0.071: ( 11.64 / 0.071 ≈ 163.94 ). Then, ( FV = 3990.5943 * 163.94 ≈ 3990.5943 * 163.94 ≈ 654,200 ) (approx). Wait, maybe my initial calculation was wrong. Let's use a financial calculator approach.

Alternatively, use the future value of annuity formula correctly. Let's compute ( (1 + 0.071)^{37} ):

Using a calculator, ( 1.071^{37} = e^{37 \times \ln(1.071)} \approx e^{37 \times 0.06857} \approx e^{2.537} \approx 12.64 ) (more accurately, using a calculator: 1.071^37 ≈ 12.644).

So, ( (12.644 - 1) = 11.644 ). Then, ( 11.644 / 0.071 ≈ 163.99 ). Then, ( FV = 3990.5943 * 163.99 ≈ 3990.5943 * 163.99 ≈ 654,300 ) (approx).

Now, present value of annuity for 30 years, rate 7.1%:

( PV = A * \frac{1 - (1 + 0.071)^{-30}}{0.071} )

We know ( PV = 654300 ), so solve for A:

( A = 654300 * \frac{0.071}{1 - (1.071)^{-30}} )

Calculate ( (1.071)^{-30} = 1 / 1.071^{30} ). ( 1.071^{30} \approx e^{30 * 0.06857} \approx e^{2.057} \approx 7.82 ), so ( (1.071)^{-30} \approx 0.128 ).

Then, ( 1 - 0.128 = 0.872 ). ( 0.071 / 0.872 ≈ 0.0814 ). Then, ( A = 654300 * 0.0814 ≈ 53,260 )? Wait, no, this is confusing. Wait, maybe the correct approach is to calculate the future value of the contributions, then divide by the present value annuity factor for 30 years to get the annual income.

Wait, let's use the correct formula for future value of annuity:

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

Where ( C = 0.11 * 36278.13 = 3990.5943 ), ( r = 0.071 ), ( n = 65 - 28 = 37 ) years.

Calculating ( (1 + 0.071)^{37} ):

Using a calculator, ( 1.071^{37} \approx 12.644 ) (as before).

So, ( (12.644 - 1) = 11.644 ). Then, ( 11.644 / 0.071 ≈ 163.99 ).

Then, ( FV = 3990.5943 * 163.99 ≈ 3990.5943 * 163.99 ≈ 654,200 ) (approx).

Now, to find the annual income she can get for 30 years, we use the present value of annuity formula, where the present value is the future value we just calculated (since that's the amount she has at retirement, which is the present value for the 30-year income period).

The formula for the present value of an ordinary annuity is:

( PV = A \times \frac{1 - (1 + r)^{-n}}{r} )

We know ( PV = 654200 ), ( r = 0.071 ), ( n = 30 ). Solve for ( A ):

( A = PV \times \frac{r}{1 - (1 + r)^{-n}} )

Calculate ( (1 + 0.071)^{-30} ):

( (1.071)^{-30} = 1 / (1.071^{30}) ). Let's calculate ( 1.071^{30} ):

Using a calculator, ( 1.071^{30} \approx e^{30 \times \ln(1.071)} \approx e^{30 \times 0.06857} \approx e^{2.057} \approx 7.82 ), so ( (1.071)^{-30} \approx 0.128 ).

Then, ( 1 - 0.128 = 0.872 ). ( 0.071 / 0.872 ≈ 0.0814 ).

Then, ( A = 654200 \times 0.0814 ≈ 53,260 )? Wait, but the goal is $32,523 per year. Wait, that can't be, I must have messed up the formula. Wait, no, the future value is the total amount saved, and then we need to find the annual payment she can withdraw for 30 years, which is an annuity. Wait, maybe the formula is for the present value of the annuity (the amount she needs to have at retirement to get $32,523 per year for 30 years) and compare it to the future value of her savings.

Let's calculate the present value of the income she needs: ( PV_{needed} = 32523 \times \frac{1 - (1 + 0.071)^{-30}}{0.071} ).

Calculate that: ( \frac{1 - (1.071)^{-30}}{0.071} \approx \frac{1 - 0.128}{0.071} \approx \frac{0.872}{0.071} \approx 12.28 ).

So, ( PV_{needed} = 32523 \times 12.28 \approx 399,400 ) (approx). Wait, no, that's not right. Wait, the present value of an annuity due or ordinary? Wait, no, when she retires, the amount she has is the future value of her savings, and she needs to withdraw money for 30 years, so the future value of her savings is the present value for the withdrawal period. So, the formula for the annual withdrawal is ( A = FV \times \frac{r}{(1 + r)^n - 1} ) (for a perpetuity? No, that's for a growing annuity. Wait, no, the correct formula for the annual payment from a lump sum is the present value of annuity formula solved for A: ( A = PV \times \frac{r}{1 - (1 + r)^{-n}} ), where PV is the lump sum at retirement.

Let's recast:

She needs $32,523 per year for 30 years. The present value (at retirement) of that income stream is ( PV_{income} = 32523 \times \frac{1 - (1 + 0.071)^{-30}}{0.071} ).

Calculate ( \frac{1 - (1.071)^{-30}}{0.071} ):

Using a financial calculator, the present value annuity factor (PVAF) for 30 years at 7.1% is ( \frac{1 - (1.071)^{-30}}{0.071} \approx 12.28 ) (as before). So, ( PV_{income} = 32523 \times 12.28 \approx 399,400 ) (approx).

Now, the future value of her savings is ( FV_{savings} = 3990.5943 \times \frac{(1 + 0.071)^{37} - 1}{0.071} ).

Calculate ( (1 + 0.071)^{37} \approx 12.644 ), so ( (12.644 - 1) = 11.644 ), ( 11.644 / 0.071 ≈ 163.99 ), so ( FV_{savings} = 3990.5943 \times 163.99 ≈ 654,200 ) (approx).

Now, compare ( FV_{savings} ) and ( PV_{income} ). Wait, no, actually, the amount she can withdraw annually is ( A = FV_{savings} \times \frac{0.071}{1 - (1.071)^{-30}} ).

Calculate ( \frac{0.071}{1 - (1.071)^{-30}} \approx \frac{0.071}{0.872} \approx 0.0814 ), so ( A = 654200 \times 0.0814 ≈ 53,260 )? But that's way more than $32,523. Wait, this can't be right. I must have made a mistake in the time period. Wait, no, she is contributing for 37 years, then withdrawing for 30 years. Wait, maybe the interest rate is the same for both, but maybe I messed up the formula.

Wait, let's use a different approach. Let's calculate the future value of her contributions:

( FV = 3990.59 \times \frac{(1.071)^{37} - 1}{0.071} )

Using a calculator, ( (1.071)^{37} ):

Let's compute step by step:

Year 1: 1.071

Year 2: 1.071*1.071 = 1.147041

Year 3: 1.147041*1.071 ≈ 1.228481

Year 5: ~1.228481^2 ≈ 1.5091 (year 4: 1.2284811.071≈1.3157, year 5: 1.31571.071≈1.4091)

Year 10: (1.4091)^2 ≈ 1.9856 (year 5: 1.4091, year 6: 1.40911.071≈1.5091, year 7: 1.50911.071≈1.6162, year 8: 1.61621.071≈1.731, year 9: 1.7311.071≈1.854, year 10: 1.854*1.071≈1.985)

Year 20: (1.985)^2 ≈ 3.940

Year 30: 3.940*1.985 ≈ 7.82

Year 37: 7.821.071^7 ≈ 7.821.616 ≈ 12.64 (as before)

So, ( (12.64 - 1) = 11.64 ), divided by 0.071: 11.64 / 0.071 ≈ 163.94

Then, FV = 3990.59 * 163.94 ≈ 3990.59 * 163.