Describe the effect an increase in $n$, the number of payment periods, has on the monthly payment $P$ in the formula
[tex]$P=P V \frac{1}{1-(1+i)^{-4}}$[/tex]
A. An increase in $n$, the number of payment periods, will not change $P$, the monthly payment.
B. An increase in $n$, the number of payment periods, will create an increase in $P$, the monthly payment
C. An increase in $n$, the number of payment periods, will create a decrease in $P$, the monthly payment
D. An increase in $n$, the number of payment periods, can increase or decrease $P$, the monthly payment, depending on the value of PV.
Answer (1)
As the number of payment periods ( n ) increases, the term ( 1 + i ) − n decreases.
This causes the denominator 1 − ( 1 + i ) − n to increase.
Consequently, the fraction 1 − ( 1 + i ) − n i decreases.
Since P = P V ⋅ 1 − ( 1 + i ) − n i , the monthly payment P decreases as n increases. Therefore, the answer is C .
Explanation
Understanding the Formula We are given the formula for the monthly payment P :
P = P V 1 − ( 1 + i ) − n i where P V is the present value, i is the interest rate per period, and n is the number of payment periods. We want to determine the effect of increasing n on P .
Analyzing the Effect of n on (1+i)^(-n) Let's analyze the term 1 − ( 1 + i ) − n i as n increases. Since i is a positive interest rate, 1"> 1 + i > 1 . As n increases, ( 1 + i ) − n decreases because we are raising a number greater than 1 to a larger negative power, making the result smaller (closer to zero).
Analyzing the Effect of n on the Denominator Now, let's consider the denominator 1 − ( 1 + i ) − n . Since ( 1 + i ) − n decreases as n increases, 1 − ( 1 + i ) − n increases as n increases. This is because we are subtracting a smaller number from 1.
Analyzing the Effect of n on the Fraction Finally, let's look at the entire fraction 1 − ( 1 + i ) − n i . Since the numerator i is constant and the denominator 1 − ( 1 + i ) − n increases as n increases, the value of the fraction decreases as n increases.
Conclusion Since P = P V ⋅ 1 − ( 1 + i ) − n i , and P V is constant, P decreases as n increases. Therefore, an increase in n (the number of payment periods) will create a decrease in P (the monthly payment).
Final Answer Therefore, the correct answer is: c. An increase in n , the number of payment periods, will create a decrease in P , the monthly payment
Examples
When you take out a loan, like a mortgage for a house or a car loan, the length of time you have to repay the loan (the number of payment periods) affects your monthly payment. If you choose a longer repayment period, your monthly payments will be smaller, but you'll end up paying more interest overall. Conversely, if you choose a shorter repayment period, your monthly payments will be larger, but you'll pay less interest overall. For example, consider a $200,000 mortgage at a 5% annual interest rate. Over 30 years, the monthly payment is approximately $1,073.64, while over 15 years, the monthly payment is approximately $1,581.60. Choosing the 30-year option reduces the monthly payment but increases the total interest paid.
