Your grandmother is gifting you $100 per month for four years while you pursue your bachelor's degree. With an APR of 5.5%, what is the present value of these payments on the day you enter college?

To calculate the present value of the monthly payments from your grandmother, you would use the formula for the present value of an annuity due to the payments being made at the beginning of each period (month). The formula for the present value of an annuity is:

PV = Pmt * [(1 - (1 + r)^-n) / r]

Where: - PV is the present value of the annuity. - Pmt is the amount of the individual annuity payment. - r is the periodic interest rate (APR divided by the number of periods per year). - n is the total number of payments.

In your case, the monthly payment Pmt is $100, the APR is 5.5% or 0.055 when we convert it into a decimal, and the number of payments n is 4 years times 12 months per year, which equals 48.

First, we need to calculate the monthly periodic interest rate by dividing the APR by 12 (the number of periods per year):

Monthly periodic rate r = APR / 12 = 0.055 / 12 ≈ 0.00458333

Now, let's use the annuity formula to calculate the present value:

PV = $100 * [(1 - (1 + 0.00458333)^-48) / 0.00458333]

Next, calculate the exponent and subtract it from 1:

(1 + 0.00458333)^-48 ≈ 0.7963 1 - 0.7963 ≈ 0.2037

Now, divide this result by the monthly rate:

0.2037 / 0.00458333 ≈ 44.4749

Finally, multiply this by the monthly payment amount:

PV ≈ $100 * 44.4749 ≈ $4447.49

Therefore, the present value of the $100 monthly payments over four years, given a 5.5% annual interest rate compounded monthly, is approximately $4447.49 on the day you enter college.

Extra: The concept of present value is important in finance and economics because it allows individuals to determine the current worth of a stream of future payments given a certain rate of return (interest rate). This is useful in comparing investment opportunities and making financial decisions. The idea is that a dollar today is worth more than a dollar tomorrow due to its potential earning capacity. The interest rate in these calculations reflects the cost of capital or the opportunity cost of money; it represents what you could earn if you invested the money elsewhere at a given rate of return.

For college students, understanding this concept can help with decision-making regarding student loans, future investments, and the time value of money in general. The present value is calculated under the assumption that money can earn interest, and therefore each payment in the future is worth less than an equivalent payment today. The formula accounts for this by discounting each future payment back to its value in today's dollars. This is why we have to apply the annuity formula using the rate per period (in this case, monthly) and the total number of periods payments are made.