Percent Value and Inflation

[LaAishaElliott]

Suppose that your parents are willing to lend you $20,000 for part of the cost of your college education and living expenses. They want you to repay them $20,000, without any interest, in a lump sum 15 years after you graduate, when they will be about to retire and move. Meanwhile, you will be busy repaying federally guaranteed loans for the first 10 years after graduation. But you realize that you can't repay the lump sum without saving up. So you decide that you will put aside money in an interest-bearing account every month for the five years before the payment is due. You feel comfortable with the idea of putting aside $275 a month (the amount of the payment on your government loans). How high an annual nominal interest rate on savings do you need to accumulate the $20,000 in 60 months, if interest is compounded monthly? Enter into a spreadsheet the values d=275, r=0.05 (annual rate), and n=60, and the savings formula with r replaced by r/12 (the monthly interest rate). You will find that the amount accumulated is not enough. Change r to 0.09; it's more than enough. Try other values until you determine r to two decimal places.


The equation is A=d[((1+i)^n-1)/i]. 20000=275[((1+r/12)^60-1)/(r/12)]

I substituted the values into the eqn, but I cannot figure out how to solve for r.

 

[JeffM]

Suppose that your parents are willing to lend you $20,000 for part of the cost of your college education and living expenses. They want you to repay them $20,000, without any interest, in a lump sum 15 years after you graduate, when they will be about to retire and move. Meanwhile, you will be busy repaying federally guaranteed loans for the first 10 years after graduation. But you realize that you can't repay the lump sum without saving up. So you decide that you will put aside money in an interest-bearing account every month for the five years before the payment is due. You feel comfortable with the idea of putting aside $275 a month (the amount of the payment on your government loans). How high an annual nominal interest rate on savings do you need to accumulate the $20,000 in 60 months, if interest is compounded monthly? Enter into a spreadsheet the values d=275, r=0.05 (annual rate), and n=60, and the savings formula with r replaced by r/12 (the monthly interest rate). You will find that the amount accumulated is not enough. Change r to 0.09; it's more than enough. Try other values until you determine r to two decimal places.


The equation is A=d[((1+i)^n-1)/i]. 20000=275[((1+r/12)^60-1)/(r/12)]

I substituted the values into the eqn, but I cannot figure out how to solve for r.

You are not asked to solve the equation directly. You are asked to solve it by successive approximations using a spread sheet.

\(\displaystyle 275 * \dfrac{(1 + \frac{0.05}{12})^{60} - 1}{\frac{0.05}{12}} = 18,701.67.\) You can confirm this using a calculator.

\(\displaystyle 275 * \dfrac{(1 + \frac{0.09}{12})^{60} - 1}{\frac{0.09}{12}} = 20,741.64.\) You can also confirm this using a calculator.

So the minimum required rate of interest must be above 5% and below 9%. And it must be closer to 9% than to 5%. So try 8%. Keep on trying until you get it.

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