I'm trying to model a certain problem in loan lending, and was hoping to get some inputs:
A lender lends amount P to a borrower. The borrower has to repay this loan in $t$ equal monthly installments (EMI) at some annual interest rate $I$.
The EMI is calculated as:
$$E = \dfrac{P \times I}{12(1-(1+\frac{I}{12})^{-t})}$$
Now, there is a chance that the borrower may default payments, so taking this into consideration, the EMI for a month $i$ would be defined as:
$$E(i) = \begin{cases} 0,& \Pr(default|i) > 0.5\\ E,& \Pr(default|i) <=0.5 \end{cases} $$
For the lender, the value of each successive EMI decreases over time due to inflation. The value $v_i$ for a month $i$ is defined as:
$$v_i = \dfrac{E(i)}{(1+d)^{i}}$$, where $d \in (0,1)$ is fixed.
By setting $t$ and $I$, the lender wishes to maximize net value, which is defined as:
$$NV = \sum_{i=1}^{t} v_i$$
subject to the constraints: $$t \in \{24,36,48,60\}\\ I \in (0.13,0.19)$$
My questions are:
Is there a term for these kind of optimization problems?
How should the lender choose $t$ and $I$ so as to maximize for $NV$ subject to the above constraints? I'm very unfamiliar with optimization problems, so any help is appreciated.
Assuming you have some data about the borrower, what are some starting points for estimating the probability in $E(i)$?