Describe the effect an increase in $i$, the interest rate applied to the present value, has on the monthly payment $P$ in the formula
[tex]$P=P V \cdot \frac{i}{1-(1+i)^{-\pi}}$[/tex]
A. An increase in $i$, the interest rate, will not change $P$, the monthly payment.
B. An increase in $i$, the interest rate, will create an increase in $P$, the monthly payment.
C. An increase in $i$, the interest rate, will create a decrease in $P$, the monthly payment.
D. An increase in $i$, the interest rate, can increase or decrease $P$, the monthly payment, depending on the value of PV.
Answer (1)
The problem provides a formula for monthly payment P in terms of present value P V , interest rate i , and number of payments n .
We analyze the effect of increasing the interest rate i on the monthly payment P .
By examining the derivative d i d P , we determine that it is positive for typical values of P V , n , and i .
Therefore, an increase in the interest rate i will create an increase in the monthly payment P , so the answer is b .
Explanation
Understanding the Problem We are given the formula for the monthly payment P :
P = P V "." 1 − ( 1 + i ) − n i where:
P is the monthly payment.
P V is the present value.
i is the interest rate.
n is the number of payments. We want to determine the effect of an increase in the interest rate i on the monthly payment P .
Analyzing the Derivative To analyze the effect of an increase in i on P , we can examine the derivative of P with respect to i , which is d i d P . If 0"> d i d P > 0 , then an increase in i will increase P . If d i d P < 0 , then an increase in i will decrease P . If d i d P = 0 , then an increase in i will not change P .
Calculating the Derivative Using symbolic differentiation, we find the derivative of P with respect to i :
d i d P = P V ⋅ ( 1 + i ) (( 1 + i ) n − 1 ) 2 − in ( 1 + i ) n + ( 1 + i ) n + 1 (( 1 + i ) n − 1 )
Evaluating the Derivative To determine the sign of d i d P , we can evaluate it for typical values of P V , n , and i . Let's consider an example where P V = 1000 , n = 360 (30-year mortgage), and i = 0.05 (5% interest rate). Substituting these values into the expression for d i d P , we find that d i d P ≈ 999.999619956981 .
Conclusion Since 0"> d i d P > 0 for these typical values, we can conclude that an increase in i will generally lead to an increase in P . This makes intuitive sense because a higher interest rate means that the borrower has to pay more per period to compensate the lender.
Final Answer Therefore, an increase in i , the interest rate, will create an increase in P , the monthly payment.
Selecting the Correct Option The correct answer is (b).
Examples
Consider a home mortgage. If the interest rate increases, the monthly payment also increases, making the mortgage more expensive. This relationship is crucial for understanding the financial impact of interest rate changes on loans and investments. For example, if you are planning to buy a house and interest rates are expected to rise, it would be wise to secure a mortgage soon to avoid higher monthly payments. This concept applies to various financial instruments, including car loans, personal loans, and bonds.
