# Q about interest rates and payoffs

Is the amount I could save at payment-per-month X at interest Y over time period Z the same as the size of a loan I could pay off at interest rate Y over time period Z using payments per month of X?

Is there a good “rule of thumb” for figuring out, for example,

–How much payments-per-month will be at a certain interest rate, for a certain sized loan, over a certain loan-payoff period?
–How much total interest will be paid on a loan of a certain size, at a certain interest rate, extended over a certain period of time?
–Other sorts of calculations that might be relevant to this kind of thing?

-Kris

My gut feeling was no, so I set up two excel formulas, one to put \$100 per month into a savings account paying 10% per year, compounded monthly ( =a1*(1+0.1/12))+100) and one to pay off a 10% loan that is compounded monthly by making \$100/mo payments. (=c1*(1+0.1/12))-100)

``````

save it		pay it
\$0.00		dec	\$1,256.56
\$100.00		jan	\$1,167.03
\$200.83		feb	\$1,076.76
\$302.51		mar	\$985.73
\$405.03		apr	\$893.94
\$508.40		may	\$801.39
\$612.64		jun	\$708.07
\$717.75		jul	\$613.97
\$823.73		aug	\$519.09
\$930.59		sep	\$423.41
\$1,038.35	oct	\$326.94
\$1,147.00	nov	\$229.67
\$1,256.56	dec	\$131.58

``````

after 1 year, your savings account has \$1256.56 in it, but if you had taken a loan for that same amount, you wouldn’t have paid it off by the end of the year. This is, of course, because the interest is working against you, instead of for you.

I’m not positive I understand your initial question, but I think I do.

On one hand, you deposit, say, \$1000 at the end of every month into an account earning 5% compounded monthly for 30 years. How much will you have saved after 30 years?

On the other hand, if you finance a loan where you pay \$1000 at the end of every month for 30 years at 5% compounded monthly. How much was the original loan for?

Are you asking if these two amounts are the same?

The answer is no, but they are related.

If you’re making regular deposits and want to know how much you’ll save, the formula is (and I hope this is clear, it’s a bit difficult to type clearly):

``````

FV = pmt*[(1 + i/p)[sup]N[/sup] - 1]
-----------------------
i/p

``````

FV = total amount saved up at the end

pmt = amount of each deposit

i = annual interest rate

p = number of times the interest is compounded per year = number of deposits/payments per year

N = total number of deposits made

So if you deposit \$1000 at the end of every month for 30 years, and it’s earning 5% compounded monthly, you will have saved up a total of \$832,258.61 after the full 30 years.

On the other hand, for a loan the equation is:

``````

PV = pmt*[1 - (1 + i/p)[sup]-N[/sup]]
-----------------------
i/p

``````

PV = size of original loan

(all other variables same as before).

So if you’re paying off a loan at 5% compounded monthly, end of the month payments of \$1000 each for 30 years, the original loan must have been for \$186,281.62.

So they’re not the same but they are related by compounded interest:

832,258.61 = 186,281.62(1 + .05/12)[sup]360[/sup]

This is exact, perhaps only a bit off due to rounding errors.

In the first case, if you want to know how much interest your investment has earned, simply realize that you’ve deposited \$1000 on 360 separate occasions (a total of \$360,000), while you actually have \$832,258.61 in the account after the 30 years. The difference came from the interest earned: 832,258.61 - 360,000 = \$472,258.61 in interest has been earned.

Similary, for the loan, the interest you’ve paid is 360,000 - 186,281.62 = \$173,718.38.

Just let me know if any of this isn’t clear.

That was all perfectly clear, thanks!

I was hoping there was a much simpler way to just estimate the kinds of things I asked in my later questions, though.

-Kris

I imagine there probably are “standard rules of thumb” for estimating these things, but unfortunately I’m not aware of any specific ones. Maybe someone else here is, though.

On the other hand, if you’re just looking for a tool that will do the work for you, try searching the internet for “annuity calculator”. There should be several to choose from.

Answer to your first question is pretty simple. Your regular monthly payments is an ordinary annuity. The size of the loan is the just the present value of this annuity. For your other questions, the formulas can be found in any basic personal finance book, or you can do the calcs in a spreadsheet. Oh, OK, here’s how you figure out the total interest paid over the life of the loan: multiple the monthly payment by the total number of payments, then subtract the original loan amount. But be forewarned, this will be a scary number!

Thanks for the replies.

I was hoping to find something I could do in my head while strolling to class and stuff like that. But it looks like you just have to sit there with a calculator.

Thanks again!

-Kris

The only “rule of thumb” I know is that, for practical rates of interest, the number of years that it will take for the investment to double will be approximately 72 divided by i.

So for example, at 7.2% annual interest, it will take about 72/7.2 = 10 years for your investment to double. So \$1000 today will be \$2000 in ten years, \$4000 in twenty years, \$8000 in thirty years, etc. Ah, the power of compound interest.

Using the exact formula, \$1000 at 7.2% compounded annually will be worth \$2004.231 after ten years.

There you go, you can play with this all day while you walk to class.

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