{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Investment comparison calculator\n",
    "\n",
    "## Created by Nancy Aggarwal on Jul 12, 2020"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.optimize import minimize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Pay regular premium, gain fixed interest"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Frequency of Compounding interest = frequency of payments"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Say one pays an amount $x$ regularly for $n$ time-intervals. Say one wants to get one's investement out after $N$ intervals ($N>=n$). Say the compound rate of interest (calculated at the same frequency) is $i$.\n",
    "\n",
    "Now, the amount after $n$ intervals is:\n",
    "\n",
    "$Y_n = x(1+i)\\sum_{j=0}^{n-1}(1+i)^j$\n",
    "\n",
    "Finally, the return after $N$ intervals is:\n",
    "\n",
    "$Z_N = Y_n*(1+i)^{N-n}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Frequency of compounding interest > frequency of payments"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Say one pays an amount $x$ regularly for $n$ time-intervals. Say one wants to get one's investement out after $N$ intervals ($N>=n$). Say the compound rate of interest (calculated at the a frequency m) is $i$.\n",
    "\n",
    "Now, the amount after $n$ intervals is:\n",
    "\n",
    "$Y_n = x(1+i)^m\\sum_{j=0}^{n-1}(1+i)^{mj}$\n",
    "\n",
    "Finally, the return after $N$ intervals is:\n",
    "\n",
    "$Z_N = Y_n*(1+i)^{m(N-n)}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calcTotalReturn(i,x,n,N,m):\n",
    "    # n is years of premium in some time unit (say year or quarter)\n",
    "    # N is years of maturity in the same time uint\n",
    "    # i is interest in percent per that time unit\n",
    "    # x is premium per that time unit\n",
    "    # m is the number of times interest is compounded between two payments\n",
    "#     print(\"interest = {}, premium = {}, years of premium = {}, years of maturity = {}\".format(i,x,n,N))\n",
    "    i = i/100\n",
    "    seriesarray = [(1+i)**(j*m) for j in range(n)]\n",
    "    Yn = x*((1+i)**m)*sum(seriesarray)\n",
    "    ZN = Yn*(1+i)**(m*(N-n))\n",
    "    return ZN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example\n",
    "\n",
    "$8\\%$ yearly interest, premium of every 6 months at the rate of 55/yr for 16 years, interest compounded every month. Policy matured after 25 years."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "intervalfactor = 2 #(convert years to semesters)\n",
    "compoundingfactor = 6 #months in a semester\n",
    "\n",
    "IntervalsOfPremium = 16*intervalfactor\n",
    "IntervalsOfMaturity = 25*intervalfactor\n",
    "RateofInterest = 8/(intervalfactor*compoundingfactor) #percent\n",
    "IntervalPremium = 55/intervalfactor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3722.659639968529"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calcTotalReturn(RateofInterest,IntervalPremium,IntervalsOfPremium,IntervalsOfMaturity,compoundingfactor)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Now back-calculate interest given final sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "FinalSum = 75*25 + 7.5e2 + 1e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3625.0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FinalSum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def err(i,tup):\n",
    "    Model = calcTotalReturn(i[0],tup[0],tup[1],tup[2],tup[3])\n",
    "    Meas = tup[4]\n",
    "    error = abs((Model-Meas)/Meas)\n",
    "#     print(i,Model,Meas,error)\n",
    "    return error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "minimizeResult=minimize(err,1,[IntervalPremium,IntervalsOfPremium,IntervalsOfMaturity,compoundingfactor,FinalSum],method=\"Powell\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "inferredInterest=minimizeResult.x*intervalfactor*compoundingfactor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.858347053111679"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "inferredInterest"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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