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Determine the present value of a compound interest retirement account. open an account that helps them supplement their retirement income within 30 years. Determi...

Question

Determine the present value of a compound interest retirement account. open an account that helps them supplement their retirement income within 30 years. Determine the total amount they must deposit now, for each of the following interest rates, so that they have $ 500,000.00 available at retirement. to. a.8% compounded quarterly b. 6% compounded quarterly. c. 8% compounded daily d. 6% compounded daily

Determine the present value of a compound interest retirement account. open an account that helps them supplement their retirement income within 30 years. Determine the total amount they must deposit now, for each of the following interest rates, so that they have $ 500,000.00 available at retirement. to. a.8% compounded quarterly b. 6% compounded quarterly. c. 8% compounded daily d. 6% compounded daily



Answers

A person deposits money into an account, which pays $6 \%$ interest compounded continuously, at a rate of $\$ 1000$ per year for 30 years. Calculate: (a) The balance in the account at the end of the 30 years. (b) The amount of money actually deposited into the account. (c) The interest earned during the 30 years.

So we want to look at the annuity formula which according to previous exercise as the following W equals tee times one plus I. True. Yeah. Mm. Oh CASS. Uh huh. And -1 over I. Mm. So we have here uh um Yeah And we want to choose different values so in this case it's 1200 per year. Um So that's gonna be the p. Value, so 1200 and then be 10 years. So um it's from the point She starts at age 30 Until she retires at 65, 70. is the end value. Mhm. Thank you. And then lastly we have the I. Value which is um the rate, So the interest rate in this case is 8%. So it'll be .08. So I equals 0.08. So that means that she'll end up having $206,780.16. And then how much of that is? Her own money will be 1200. How much you paid in each year times for 35 years that she did it. So only $42,000 of that is her actual money. The rest is all, um, from the investment return.

For this particular question, we are going to deposit $15,000 into a savings account and we're going to leave it there for eight years. And in that savings account it's going to earn 5% interest. So the first thing this question asked us to do it asks us to compound differently each amount. Um And how does the amount of times that we compound the original amount? How does that affect our total investment, the money we're gonna make back? So the first thing we're going to do it asks us to compound annually. So if we are going to compound annually annually, who is just going to be one because there's just one year. So we are going to use the formula are amount is going to be equal to our principal times one plus our rate divided by in which is the number of times compounded to the end times T power. So this is our rate, obviously eight years is our time in Our principle is the 15,000. So here we're gonna start out with 15,000 Times one plus .05 which is our rate. And this first time we're just compounding by one because one would be annually for one year. So one times I ate here. So compounding annually, There would be $22,161.83. And I'm gonna mark this as compounding annually. The second part asks us to compound quarterly. So the only thing that would change if we are compound in quarterly is this bottom number here. So instead of compounding annually we're gonna make that four because there would be four quarters in a dollar. So there's four quarters in a year. So four times eight. And that's going to give us $22,321.96 cents when we compound quarterly. The next part asks us to compound monthly. So the only change again that I'm going to make is to this bottom number. We're gonna change our end in monthly. Well there are 12 months in a year. So we're gonna change that number to a 12. And if we compound monthly we would have 22,000 $358.78 right next to it says what happens if we change that in And we compound daily? So there are 365 days in a year. So I'm going to change that in 2 365. So compounding daily, we would have $22,376.76. The last thing it asks us to do is to change up our formula a little bit because it says, what happens if we compound continuously? Well when we compound continuously we're going to use the formula p times E. To the R. T. Power and that E. Is already a given. So that is going to be on our calculator already. So we are going to have 15,000 times E Times .05 times the eight years. And if we compound continuously it's going to be 22,000 $377.37. And that is continuously. So if we compare all of those amounts, obviously the more that we are compounding, the more is going to increase the final amount that you end up with.

This problem, we are told that at the end of each year, $5,000 is invested into an IRA earning 3% compounded annually for part A. We are asked how much will be in the account at the end of 30 years Using the annuity formula F. Of P I. N. Equals p times the fraction one plus I, to the power of n minus one over I. Where we have the values of P. I. N. Or the meanings of P I. N. And F presented below. So really what we're being asked to find here is f of 5000 0.03 30 Which would just be 5000. But times I'm going to put in rackets here, one plus 0.3 So 1.3 To the power of 30 minus one over 0.03 in a pause and calculate that off screen here. So that is going to come out to about 237,000 877 Now, for part B, were asked to use graphical approximation methods to determine the rate of the interest that would produce $300,000 in the account at the end of 30 years. So down below, I have plotted out half of um I suppose it would be f. 5000 I. 30 so the X axis here is varying the interest rate. The Y axis is the value of F. So we can see that we have a F value of 300,000 When the interest rate is about 0.044.

We have invested $500 at an interest rate of 3.75% which is compounded quarterly. We want to calculate the amount of money will have after one, too. And 10 years. If you plug all of this in the calculator, you'll get for T is equal to one year. A T is equal to $519 and one cent. Two cents. Excuse me. 40 is equal to two years. You get a T secret toe $538 and 75 cents and produce equal to 10 years. You get A T is equal to $726.23.


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