Question 1 done

[sinjun]

For what it's worth. I'll post the solutions, so folks can catch my errors, if they're up to it. *grins*

Though I'm too lazy to figure out the HTML ... *wrinkles nose* Besides, that would be just too much procrastination, even for me.


Question 1
You just purchased a new house for $120,000. You were able to make a down payment equal to 25% of the value of the house, and the balance was mortgaged. The rate quoted by the bank was 10% compounded semi-annually. The mortgage has a 20-year amortization period and a five-year term.

a) What will be your monthly payments?

(Note that the basic solution to this type of formula is found in our textbook on page 172, which I used for the following answer.) First we need to find the quoted monthly rate, which requires converting the quoted rate to an EAR. The quoted rate is 10% and m=2, since the interest is compounded semi-annually.

EAR = [1 + Quoted Rate/m]**m – 1
= [1+0.10/2]**2 – 1
= 0.1025 which is 10.25% annually.

Then we find the quoted monthly rate, which means rearranging the EAR formula above, as follows, and finding the Quoted rate/m:

Quoted rate/m = (EAR + 1)**1/m – 1
Quoted rate/12 = (0.1025 + 1)**1/12 – 1
= 0.008165 which is 0.8165% per month.

There are 20 * 12 = 240 payments over the 20 years of amortization. The present value of the mortgage is 120,000 * 0.75 = $90,000, since a 25% down payment was made. To find the monthly payment, we use the annuity present value formula, which can be found on page 158 of our textbook.

Annuity Present Value = PMT * {(1-[1/(1+r)**t])/r}
90000 = PMT * {(1 – [1/(1+0.008165)**240]) / 0.008165
90000 = PMT * (1 – [1/7.040247]) / 0.008165
90000 = PMT * 0.85796 / 0.008165
PMT = 90000 / 105.0777
= $856.51

So, the monthly payment will be $856.51.

b) What will be the balance remaining at the end of the term?

The term is 5 years, which is 5 * 12 = 60 periods. At the end of 5 years there will be 240 – 60 = 180 periods remaining. This is still an annuity, but it is one with 180 periods, instead of 240 periods.

Annuity Present Value = PMT * {(1-[1/(1+r)**t])/r} where t = 180, PMT = 856.51, and r = 0.008165.

So, Annuity Present Value = 856.51 * {(1 – [1/(1+0.008165)**180])/0.008165}
= 856.51 * 94.13704
= 80629.31

At the end of the 5-year term, there will still be a balance of $80,629.31 owing on the mortgage.

c) Assume that 5 years have passed, and the term of the mortgage is up. You must now negotiate a new mortgage for the remaining balance. Interest rates have increased to 12%. You want the new mortgage to have weekly payments and a 15-year amortization period. What will be your new payments?

The remaining balance, as determined in part b) is $80, 629.31. With a 15-year amortization period, there are 15 * 52 = 780 weeks, if we are making weekly payments.

Let’s start again, by determining the EAR.

EAR = [1 + Quoted Rate/m]**m – 1
= [1+0.12/2]**2 – 1
= 0.1236 which is 12.36% annually.

Then we find the quoted weekly rate, which means rearranging the EAR formula above, as follows, and finding the Quoted rate/m:

Quoted rate/m = (EAR + 1)**1/m – 1
Quoted rate/52 = (0.1236 + 1)**1/52 – 1
= 0.002244 which is 0.2244% per month.

This gives us t = 780, present value = 80629.31, and r = 0.002244
Annuity Present Value = PMT * {(1-[1/(1+r)**t])/r}
80629.31 = PMT * {(1 – [1/(1+0.002244)**780]) / 0.002244}
80629.31 = PMT * 368.0663
PMT = 80629.31 / 368.0663
= 219.0619

The weekly payments would be $219.06 each.

And then of course there's

You are an investment advisor who has been approached by a client for help on his financial strategy. He has $250,000 in savings in the bank. He is 55 years old and expects to work for 10 more years, making $100,000 a year. He expects to make a return of 5% on his investments for the foreseeable future. You can ignore taxes.

a) Once he retires 10 years from now, he would like to be able to withdraw $80,000 a year for the following 25 years. His actuary tells him he will live to be 90 years old. How much would he need in the bank 10 years from now to be able to do this?

This is an annuity with PMT = 80,000, t = 25 years, and r = 0.05 and we need to find the present value of this annuity.

Annuity Present Value = PMT * {(1-[1/(1+r)**t])/r}
= 80000 * {(1 – [1 / (1 + 0.05)**25]) / 0.05}
= 1,127,515.57

He’d need $1,127,516 in the bank 10 years from now, in order to be able to do this.

b) How much of his income would he need to save each year for the next 10 years to be able to afford these planned withdrawals after the 10th year?

To begin with, he already has $250,000 in the bank, at 5%, for 10 years.
Future Value of an annuity with a lump sum payment = PMT * (1+r)**t
= 250000 * 1.0510
= 407,223.66

He needs to have $1,127,516 in the bank 10 years from now. If he doesn’t invest anything, he will only have $407,223.66, which is not enough to meet his needs after retirement. Thus he needs to make up the 1,127,515.57 – 407,223.66 = 720,291.91

Annuity Future Value = PMT * [(1+r)**t – 1] / r
720,291.91 = PMT * [(1+0.05)**10 – 1] / 0.05
720,291.91 = PMT * 12.57789
PMT = 57,266.50

In order to do what he wants, he’d have to invest $57,266.50 per year, of his $100,000 salary.

c) Assume that interest rates decline to 4% 10 years from now. How much, if any, would your client have to lower his annual withdrawal, assuming that he still plans to withdraw cash each year for the next 25 years?

This is an annuity with present value = 1,127,515.57, t = 25 years, and r = 0.04 and we need to find the payment of this annuity.

Annuity Present Value = PMT * {(1-[1/(1+r)**t])/r}
1,127,515.57= PMT * {(1 – [1 / (1 + 0.04)**25]) / 0.04}
1,127,515.57= PMT * 15.62208
PMT = 72,174.48

He’d have to lower his withdrawal by 80,000 – 72,174.48 = $7,825.52

Comments

[neftoonzamora.livejournal.com]

Thus proving once again why I went into the medical field. We don't have to know equations and stuff like that. Give me an exploding eyeball over this stuff anyday....

[damara.livejournal.com]

*wrinkles nose*

I don't even want to go near an exploding eyeball...