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Suppose Anita wants to save up for
a down payment on a home.
She opens an annuity account with her bank
that offers 3% annual interest compounded monthly,
and she plans to deposit $800 per month.
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How much will she have saved
for a down payment after five years?
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How much would she had saved
if she instead stashed $800 into a coffee can
burried in her backyard each month?
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Over the course of his life,
Billiam has saved up $1.2 million in a retirement account
that yields 2.5% annual interest compounded monthly.
He’s 67 now, and ready to retire.
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If Billiam anticipates living to be 100-years-old
and wants his saving to last that long,
how much can he afford to withdraw each month?
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Billiam decides that’s not enough money per month
to support his extravagant lifestyle,
so he proposes to withdraw $7000 per month in his retirement instead.
At this rate, how old will Billiam be when he runs out of money?
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Suppose you are contributing $600 a month
into a savings annuity account, intended to be your retirement account,
that pays an annual interest rate of 3.7% compounded monthly.
Recall that the formula relating these values
to the current account balance \(S\) after \(t\) years is
\[S = P\Biggl(\frac{\bigl(1+\frac{r}{12}\bigr)^{12t}-1}{\frac{r}{12}}\Biggr)\,.\]
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Manually calculate the balance of your annuity account
40 years in the future.
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Suppose this amount from the last part
is not enough for you to retire.
You think $1.8 million should be enough to retire on.
You still want to retire in 40 years though.
Instead of $600, how much should you contribute per month?
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Suppose you bought a house.
After making a down-payment,
you apply for a mortgage with an annual 5.99% interest rate
on the remaining balance of $251,200.
Recall that the formula relating these values
to your monthly mortgage payments \(P\) is
\[S = P\Biggl(\frac{1-\bigl(1+\frac{r}{12}\bigr)^{-12t}}{\frac{r}{12}}\Biggr)\,.\]
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Manually calculate what your monthly payments would be
if you got a 15-year mortgage versus a 30-year mortgage.
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Because the balance of the mortgage
accrues interest while you are paying it off,
the total amount you pay will be more
than the initial mortgage amount
since you are paying for the interest too.
Using your answers to the previous question,
calculate how much money you’d be paying total
in the case of a 15-year mortgage and the case of a 30-year mortgage.
(Hint: don’t overthink these calculations.)
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Unable to afford a 15-year mortgage,
you decide to go with the 30-year option.
You can however afford to pay more towards this mortgage
than the minimum required monthly payments that you calculated.
Supposing you pay $300 per month towards your mortgage
in addition to the regular monthly payment,
how long will it be before you’ve paid off the entire mortgage?
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Do some research to figure out
(1) the median price of a house in the town/city
where you plan on working after college,
and (2) the current federal interest rate
from which mortgage rates are determined.
Suppose you take out a mortgage on such a house
for the typical 30-year term.
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What will your monthly mortgage payments be?
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Multiply that monthly mortgage payment by 12
to figure out your annual housing cost.
Now, the classical advice is that
”no more than 28% of a person’s take-home pay
(income after taxes) should go towards housing.”
Adhering to this rule,
what would your annual take-home pay need to be
to afford this mortgage?
What annual income (pre-taxes) would you need?
Challenges
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Recall Billiam’s situation from earlier.
Suppose Billiam is convinced he will live to 100-years-old
and will settle for no less than $7000 per month during his retirement.
The only option then is to defer his retirement and keep working.
If Billiam can contribute $2,200 per month
into his retirement account while he’s still working,
when can he finally retire?