## Homework 1: Computing IRR
### Goal
In this homework, you need to write a function to compute IRR (internal rate of return) given a vector of cash flow and some related parameters.

### Function prototype

irr=irrFind(cashFlowVec, cashFlowPeriod, compoundPeriod)

- ```irr```: Internal rate of return
- ```cashFlowVec```: vector of cash flow
- ```cashFlowPeriod```: An integer of period (in month) for cash flow
- ```compoundPeriod```: An integer of period (in month) for compounding, which should be a factor of ```cashFlowPeriod```. (For instance, if cashFlowPeriod=12, compoundPeriod could be 1, 2, 3, 4, 6, 12.)

### Usage examples
- ```irrFind(cashFlowVec, 12, 12)```: Yearly payment/collection, yearly compounding
- ```irrFind(cashFlowVec, 12, 3)```: Yearly payment/collection, quarterly compounding
- ```irrFind(cashFlowVec, 12, 1)```: Yearly payment/collection, monthly compounding
- ```irrFind(cashFlowVec, 3, 1)```: quarterly payment/collection, monthly compounding

### Numerical examples
- ```irrFind([-1234,362,548,481], 12, 12)``` returns 0.059616
- ```irrFind([-1234,362,548,481], 12, 1)``` returns 0.058047

### Test Specs
- I/O format
- Input file specs
- Each line is a test case, with no more than 1,000 lines.
- Each line has n+2 integers (n<20) separated by a space, with the first n integers as cash flow vector, and the last two elements as periods for cash flow and compounding, respectively.
- Main program
- [goMain.py](http://mirlab.org/jang/courses/fintech/homework/2024/irr/goMain.py) (Press right button on the link to download it.)
- Usage: ```python goMain.py < inputFile > outputFile```
- Output specs
- Each line contains a single number of IRR in percentage, with four decimal places
- Example files
- Dataset 1 of 18 cases:
- Input file: [input0018open.txt](http://mirlab.org/jang/courses/fintech/homework/2024/irr/input0018open.txt)
- Output file: [output0018open.txt](http://mirlab.org/jang/courses/fintech/homework/2024/irr/output0018open.txt)

### Important facts
- [Judge system for "Intro. to Fintech"](https://hackmd.io/u5mwtsgNR7ewN40T-p5jxg)
- The interest rate is always between -10% and 10%.
- You should try to use the same settings.
- Deadline: [http://mirlab.org/jang/courses/fintech/homework.asp]
- Scoring: Ratio of correctly computed cases in another dataset of 2000 cases.
- Total computing time constraint: 2 sec

### FAQ
- Do I need to upload the main program "goMain.py"?

Ans: No, you only need to upload "irrFind.py" directly, with no compression.

- How to use "goMain.py"?

Ans: The main program takes STDIN as input and STDOUT as output.

- How come the IRR I derived leads to a zero NPV/NFV, but the judge system still thinks the IRR is not correctly?

Ans: There might be several legitimate IRRs (within the right range) for a given test case. As long as your put one of them into the output file, then the case will be considered solved.

- Do you have some hints about solving the NPV/NFV equations?

Ans: You can use either fsolve() or roots().
- fsolve() can only find a root at a time. (Since IRR is usually a small number, you can set the initial guess as 0.)
- roots() can find all the roots simultaneously.

### References
- Root Finding
- [Root Finding in Python](https://pythonnumericalmethods.studentorg.berkeley.edu/notebooks/chapter19.05-Root-Finding-in-Python.html)
- [fsolve](https://blog.csdn.net/weixin_35757704/article/details/128404328)
- Course material on youtube
- [Compounding](https://www.youtube.com/watch?v=sa2RZuXnnqk)
- [IRR](https://www.youtube.com/watch?v=t8kj2bfqfUA)