J for Finance.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Let say you have price data for a particular company as given below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "stock_price=. 101 99.5 100.30 99 101.75 102.10 100.45 101.35 101.15 100.70 100.45 100.10 99.90 99.85 100.05"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## You can define a new verb for computing daily returns without specifying its arguments. For new comer, it will be very difficult to understand how its possible to define a new function without its argument but thats beauty of J programming and this is called tacit or point less programming."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "return =. (_1 }. ]) - }."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## You can find daily return on above stock price data by using below command. Please note the command - first we have used verb \"return\" and same is followed by noun \" stock_price\" representating data set of stock prices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.5 _0.8 1.3 _2.75 _0.35 1.65 _0.9 0.2 0.45 0.25 0.35 0.2 0.05 _0.2\n"
     ]
    }
   ],
   "source": [
    "return stock_price"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## You can plot daily return and plotting is quite simple though not as enriched as R."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
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     "metadata": {},
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    }
   ],
   "source": [
    "require 'plot'\n",
    "plot (return stock_price)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lets first start with defining some data sets for cashflows, time to maturity and Zero rates - all are hypothetical data - just for illustation purpose."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "cf =. 4 4 4 4 4 4 4 4 4 104  NB. cashflows - 10 payments\n",
    "t =. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5  NB. time to maturities correspond to above cashflows \n",
    "zr =. 5.5 5.7 5.75 5.77 5.8 5.85 5.88 5.90 5.92 5.93  NB. Zero rates in percentage term\n",
    "zr=. zr % 100 NB. transform the above zero rates in decimal term"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finding compound factor (1 + zr ) ^ t and discount factor (1 / (1 + zr )) ^ t is quite easy. Look at below, next two commands.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.02713 1.057 1.08748 1.11873 1.15137 1.18597 1.22137 1.25772 1.29539 1.33381\n"
     ]
    }
   ],
   "source": [
    "(>:zr) ^ t   NB. compound factor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.973585 0.946074 0.919559 0.893871 0.868532 0.843194 0.81875 0.79509 0.771968 0.74973\n"
     ]
    }
   ],
   "source": [
    "(>:zr) ^ -t  NB. discount factor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "** >: is primitive verb and will add 1 to each of elements in the array zr ** , so it works like var++ in C++. But the advantage over C++ is that >: primitive verb in J do not need explicit iteration / loop. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Alternatevely, we can define a verb for discount factor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.973585 0.946074 0.919559 0.893871 0.868532 0.843194 0.81875 0.79509 0.771968 0.74973\n"
     ]
    }
   ],
   "source": [
    "df=. ([: >: [) ^ [: - ] NB. Defining the verb in tacit style\n",
    "\n",
    "zr df t    NB. usage wherein above defined verb takes two arguments - zero rates on left and ttm on right. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Next step i.e. for finding PVs - just multipy the array of cashfows with the array of Discount factors. The result will be an array.   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.89434 3.7843 3.67823 3.57549 3.47413 3.37278 3.275 3.18036 3.08787 77.972\n"
     ]
    }
   ],
   "source": [
    "  cf * zr df t   NB. PVs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## We can do summation of above PVs simply by adding *+/* primitive before the command and the result can be assigned to a variable called 'NPV' "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "109.294\n"
     ]
    }
   ],
   "source": [
    "[ NPV=. +/   cf * zr df t   NB. NPV"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## You can find Duration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.25587\n"
     ]
    }
   ],
   "source": [
    "  +/ ( ]*t   %    +/) cf * zr df t   NB. Duration, which is summation of the result of dividing ( PVs * t) by NPV "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## You can do scenario analysis by _bumping the zero rates, for example, by 10 bps, 20bps, 30bps, 40 bps, 50bps, 100bps, 150 bps, 200bps - all parallel shifts across all tenors_ and get the result easily by "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.056 0.058 0.0585 0.0587 0.059 0.0595 0.0598  0.06 0.0602 0.0603\n",
      "0.057 0.059 0.0595 0.0597  0.06 0.0605 0.0608 0.061 0.0612 0.0613\n",
      "0.058  0.06 0.0605 0.0607 0.061 0.0615 0.0618 0.062 0.0622 0.0623\n",
      "0.059 0.061 0.0615 0.0617 0.062 0.0625 0.0628 0.063 0.0632 0.0633\n",
      " 0.06 0.062 0.0625 0.0627 0.063 0.0635 0.0638 0.064 0.0642 0.0643\n",
      "0.065 0.067 0.0675 0.0677 0.068 0.0685 0.0688 0.069 0.0692 0.0693\n",
      " 0.07 0.072 0.0725 0.0727 0.073 0.0735 0.0738 0.074 0.0742 0.0743\n",
      "0.075 0.077 0.0775 0.0777 0.078 0.0785 0.0788 0.079 0.0792 0.0793\n"
     ]
    }
   ],
   "source": [
    "[ zr2=. 0.001 0.002 0.003 0.004 0.005 0.01 0.015 0.02 +/ zr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.973124  0.94518 0.918256 0.892183 0.866483 0.840809 0.816049 0.792094 0.768696 0.746202\n",
      "0.972663 0.944287 0.916956   0.8905 0.864441 0.838432  0.81336 0.789112 0.765442 0.742693\n",
      "0.972203 0.943396 0.915659 0.888822 0.862406 0.836065 0.810682 0.786144 0.762205 0.739204\n",
      "0.971744 0.942507 0.914366 0.887149 0.860377 0.833706 0.808015  0.78319 0.758984 0.735734\n",
      "0.971286  0.94162 0.913075  0.88548 0.858355 0.831357  0.80536 0.780249  0.75578 0.732284\n",
      "0.969003 0.937207 0.906668 0.877206 0.848344 0.819741  0.79225 0.765754 0.740005 0.715323\n",
      "0.966736 0.932836 0.900335 0.869047 0.838495  0.80834 0.779414 0.751593 0.724631 0.698831\n",
      "0.964486 0.928505 0.894075 0.861002 0.828806 0.797149 0.766843 0.737758 0.709645 0.682793\n"
     ]
    }
   ],
   "source": [
    "zr2 df\"1 t NB. discount factor for above scenario rates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.89249 3.78072 3.67302 3.56873 3.46593 3.36323  3.2642 3.16837 3.07479  77.605\n",
      "3.89065 3.77715 3.66782   3.562 3.45776 3.35373 3.25344 3.15645 3.06177   77.24\n",
      "3.88881 3.77358 3.66264 3.55529 3.44962 3.34426 3.24273 3.14457 3.04882 76.8772\n",
      "3.88698 3.77003 3.65746 3.54859 3.44151 3.33483 3.23206 3.13276 3.03594 76.5164\n",
      "3.88514 3.76648  3.6523 3.54192 3.43342 3.32543 3.22144   3.121 3.02312 76.1576\n",
      "3.87601 3.74883 3.62667 3.50882 3.39338 3.27896   3.169 3.06302 2.96002 74.3936\n",
      "3.86695 3.73134 3.60134 3.47619 3.35398 3.23336 3.11766 3.00637 2.89852 72.6784\n",
      "3.85794 3.71402  3.5763 3.44401 3.31523  3.1886 3.06737 2.95103 2.83858 71.0105\n"
     ]
    }
   ],
   "source": [
    "cf *\"1 zr2 df\"1 t   NB. PVs for above scenario rates "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "108.856 108.421 107.988 107.557 107.128 105.018 102.964 100.964\n"
     ]
    }
   ],
   "source": [
    "[ NewNPVs =. +/\"1 cf *\"1 zr2 df\"1 t NB. NPVs for above scenario rates "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# You can plot the differences between New Scenario NPVs and Original NPV "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "_0.437988 _0.873631 _1.30695 _1.73794 _2.16664 _4.27615 _6.33032 _8.33089\n"
     ]
    }
   ],
   "source": [
    "require 'plot'\n",
    "]diff=. NewNPVs - NPV\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "'type line' plot diff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Instead of computing total difference for each parallel shock e.g. 50 bps, we can compute difference contributed by each indidual tenor for given shock. e.h. 50 bps. But before that we need understanding of _ \"how can we create an identity matrix in J language\"_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.005     0     0     0     0     0     0     0     0     0\n",
      "    0 0.005     0     0     0     0     0     0     0     0\n",
      "    0     0 0.005     0     0     0     0     0     0     0\n",
      "    0     0     0 0.005     0     0     0     0     0     0\n",
      "    0     0     0     0 0.005     0     0     0     0     0\n",
      "    0     0     0     0     0 0.005     0     0     0     0\n",
      "    0     0     0     0     0     0 0.005     0     0     0\n",
      "    0     0     0     0     0     0     0 0.005     0     0\n",
      "    0     0     0     0     0     0     0     0 0.005     0\n",
      "    0     0     0     0     0     0     0     0     0 0.005\n"
     ]
    }
   ],
   "source": [
    "0.005 * =@i. 10  NB. Identity matrix having 50 bps at diagonal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Now we can apply above understanding for computing new zero rates with bump say 50 bps. Same will then be available for computing Scenario PVs , NPVs and changes from the original NPV. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "bump50bp =. [: |: ] + 0.005 * [: =@i. $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 0.06 0.057 0.0575 0.0577 0.058 0.0585 0.0588 0.059 0.0592 0.0593\n",
      "0.055 0.062 0.0575 0.0577 0.058 0.0585 0.0588 0.059 0.0592 0.0593\n",
      "0.055 0.057 0.0625 0.0577 0.058 0.0585 0.0588 0.059 0.0592 0.0593\n",
      "0.055 0.057 0.0575 0.0627 0.058 0.0585 0.0588 0.059 0.0592 0.0593\n",
      "0.055 0.057 0.0575 0.0577 0.063 0.0585 0.0588 0.059 0.0592 0.0593\n",
      "0.055 0.057 0.0575 0.0577 0.058 0.0635 0.0588 0.059 0.0592 0.0593\n",
      "0.055 0.057 0.0575 0.0577 0.058 0.0585 0.0638 0.059 0.0592 0.0593\n",
      "0.055 0.057 0.0575 0.0577 0.058 0.0585 0.0588 0.064 0.0592 0.0593\n",
      "0.055 0.057 0.0575 0.0577 0.058 0.0585 0.0588 0.059 0.0642 0.0593\n",
      "0.055 0.057 0.0575 0.0577 0.058 0.0585 0.0588 0.059 0.0592 0.0643\n"
     ]
    }
   ],
   "source": [
    "bump50bp zr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**So, what did we do actually in above step? We have created a matrix with each row conatining changed zero rates for one tenor only and keeping all other tenors intact i.e. at original zero rates. And this will help us in computing impact of changes in interest rate for each individual tenor on overall value i.e. NPV (This concept can be extended for computing hypothetical PnL for VaR Backtesting and for PnL Attribution for FI Products)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.88514  3.7843 3.67823 3.57549 3.47413 3.37278   3.275 3.18036 3.08787  77.972\n",
      "3.89434 3.76648 3.67823 3.57549 3.47413 3.37278   3.275 3.18036 3.08787  77.972\n",
      "3.89434  3.7843  3.6523 3.57549 3.47413 3.37278   3.275 3.18036 3.08787  77.972\n",
      "3.89434  3.7843 3.67823 3.54192 3.47413 3.37278   3.275 3.18036 3.08787  77.972\n",
      "3.89434  3.7843 3.67823 3.57549 3.43342 3.37278   3.275 3.18036 3.08787  77.972\n",
      "3.89434  3.7843 3.67823 3.57549 3.47413 3.32543   3.275 3.18036 3.08787  77.972\n",
      "3.89434  3.7843 3.67823 3.57549 3.47413 3.37278 3.22144 3.18036 3.08787  77.972\n",
      "3.89434  3.7843 3.67823 3.57549 3.47413 3.37278   3.275   3.121 3.08787  77.972\n",
      "3.89434  3.7843 3.67823 3.57549 3.47413 3.37278   3.275 3.18036 3.02312  77.972\n",
      "3.89434  3.7843 3.67823 3.57549 3.47413 3.37278   3.275 3.18036 3.08787 76.1576\n"
     ]
    }
   ],
   "source": [
    "cf *\"1 (bump50bp zr) df\"1 t NB. PVs with 50bps parallel shift across alll tenors "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "109.285 109.277 109.269 109.261 109.254 109.247 109.241 109.235 109.23 107.48\n"
     ]
    }
   ],
   "source": [
    "[ NewNPV2 =.+/\"1 cf *\"1 (bump50bp zr) df\"1 t   NB. NPV with 50bps shift"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**so, the impact on NPV due to interest rate change by 50 bp for each individual tenor sequencially.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "_0.00919562 _0.0178168 _0.0259334 _0.0335661 _0.0407089 _0.0473476 _0.0535595 _0.0593611 _0.0647511 _1.8144\n"
     ]
    }
   ],
   "source": [
    "NewNPV2 - NPV"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## check if (summation above differences) = (total difference, we found at (in#16) for 50bps shock).    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "_2.16664\n"
     ]
    }
   ],
   "source": [
    "+/ NewNPV2 - NPV"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "_2.16664\n"
     ]
    }
   ],
   "source": [
    "4 { diff   NB. finding value of 5th element of array 'diff'. index number in J language starts from 0 like in C, C++."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Yes it tallies."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# What if we are given actual changes in zero rates instead scenario changes.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0565 0.057 0.0575 0.0577  0.058 0.0585 0.0588  0.059 0.0592 0.0593\n",
      " 0.055 0.059 0.0575 0.0577  0.058 0.0585 0.0588  0.059 0.0592 0.0593\n",
      " 0.055 0.057 0.0583 0.0577  0.058 0.0585 0.0588  0.059 0.0592 0.0593\n",
      " 0.055 0.057 0.0575 0.0584  0.058 0.0585 0.0588  0.059 0.0592 0.0593\n",
      " 0.055 0.057 0.0575 0.0577 0.0585 0.0585 0.0588  0.059 0.0592 0.0593\n",
      " 0.055 0.057 0.0575 0.0577  0.058  0.059 0.0588  0.059 0.0592 0.0593\n",
      " 0.055 0.057 0.0575 0.0577  0.058 0.0585 0.0593  0.059 0.0592 0.0593\n",
      " 0.055 0.057 0.0575 0.0577  0.058 0.0585 0.0588 0.0593 0.0592 0.0593\n",
      " 0.055 0.057 0.0575 0.0577  0.058 0.0585 0.0588  0.059 0.0595 0.0593\n",
      " 0.055 0.057 0.0575 0.0577  0.058 0.0585 0.0588  0.059 0.0592 0.0595\n"
     ]
    }
   ],
   "source": [
    "actualChange =. 0.0015 0.0020 0.0008 0.0007 0.0005 0.0005 0.0005 0.0003 0.0003 0.0002\n",
    "\n",
    "zrPostchange =. [: |: ] + actualChange * [: =@i. $\n",
    "zrPostchange zr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**_We can compute NPVs for above actual changes._**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "109292 109287 109290 109290 109290 109290 109289 109291 109291 109221\n"
     ]
    }
   ],
   "source": [
    "NB. Let say we have 1million bonds having cashflows as described earlier, \n",
    "NB. so our overalll cashflows for 1mio bond will be\n",
    "cf =. cf * 1000 \n",
    "\n",
    "+/\"1 (cf) *\"1 (zrPostchange zr) df\"1 t"
   ]
  }
 ],
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   "file_extension": "ijs",
   "mimetype": "text/x-J",
   "name": "J"
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