259 lines
48 KiB
Plaintext
259 lines
48 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"printyellow (generic function with 1 method)"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"using Printf, Roots\n",
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"\n",
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"include(\"jlFiles/printmat.jl\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"using Plots\n",
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"\n",
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"#pyplot(size=(600,400))\n",
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"gr(size=(480,320))\n",
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"default(fmt = :svg)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"# Yield to Maturity"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The yield to maturity (ytm) is the $\\theta$ that solves\n",
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"\n",
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"$P = \\sum_{k=1}^{K} \\frac{cf_{k}}{(1+\\theta) ^{m_{k}}}$,\n",
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"\n",
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"where $cf_k$ is the cash flow from the bond (portfolio) $k$ periods ahead and $P$ is the current price of the bond (portfolio).\n",
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"\n",
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"We typically have to find $\\theta$ by a numerical method."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Task 1\n",
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"\n",
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"Starting by coding a function that calculates a price of the bond as a function of the cash flows ($cf$, a vector), times to the cash flows ($k$, also a vector) and the discount rate ($\\theta$, a single number).\n",
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"\n",
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"Assume that the bond portfolio pays 0.2 each year for 10 years (1 year from now, 2 years from now,...). (It pays no face value. This is an annuity.) \n",
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"\n",
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"(1) What is a fair price if the discount rate $\\theta$ is 0.05?\n",
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"(2) Plot the fair price as a function of $\\theta$ (use θ = -0.02:0.005:0.1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"(1) Price at θ=0.05: 1.544\n"
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]
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},
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{
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"data": {
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||
},
|
||
"execution_count": 18,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# calculate the formula above\n",
|
||
"function PV(θ, cf, k) #cf is a vector of all cash flows at times k[1],k[2],...\n",
|
||
" cdisc = cf./((1.0.+θ).^k) \n",
|
||
" P = sum(cdisc) \n",
|
||
" return P\n",
|
||
"end\n",
|
||
"\n",
|
||
"θ = -0.02:0.005:0.1\n",
|
||
"k = 1:10\n",
|
||
"cf = 0.2*ones(10)\n",
|
||
"\n",
|
||
"printlnPs(\"(1) Price at θ=0.05: \",PV(0.05,cf,k))\n",
|
||
"\n",
|
||
"n = length(θ)\n",
|
||
"Pfair = fill(NaN, n)\n",
|
||
"for i = 1:n\n",
|
||
" Pfair[i] = PV(θ[i], cf, k)\n",
|
||
"end\n",
|
||
"\n",
|
||
"# Pfair = PV.(θ, Ref(cf), Ref(k))\n",
|
||
"\n",
|
||
"plot(θ, Pfair, title=\"Present value of cash flow\", legend=false, xlabel=\"θ\")\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Task 2\n",
|
||
"\n",
|
||
"Assume the price this bond (portfolio) is 1.54. Solve for the ytm. Compare with the previous figure. Repeat with a bond price of 1.71.\n",
|
||
"\n",
|
||
"Hint: the `find_zero()`command of the `Roots.jl` package."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 21,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"The price and yield to maturity is \n",
|
||
" P θ\n",
|
||
" 1.540 0.051\n",
|
||
" 1.710 0.030\n",
|
||
"\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"P1 = 1.54\n",
|
||
"P2 = 1.71\n",
|
||
"\n",
|
||
"ytm1 = find_zero(θ->PV(θ,cf,k)-P1,(-0.1,0.1)) \n",
|
||
"ytm2 = find_zero(θ->PV(θ,cf,k)-P2,(-0.1,0.1)) \n",
|
||
"\n",
|
||
"printlnPs(\"The price and yield to maturity is \")\n",
|
||
"printmat([P1,P2],[ytm1,ytm2],colNames=[\"P\",\"θ\"])\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Task 3a\n",
|
||
"\n",
|
||
"Now interpret `cf` as the cash flow to a company where `cf[1]` also incorporates the initial investment and `m[1]=0`. (Alternatively, let `P` represent the initial investment.)\n",
|
||
"\n",
|
||
"The cash flow process is as follows. Find the IRR (internal rate of return).\n",
|
||
"\n",
|
||
"Hint: `findzero(fn,0)` would start searching at 0."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 23,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"0.2349421128831387"
|
||
]
|
||
},
|
||
"execution_count": 23,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"cf = [-150,100,0,130]\n",
|
||
"m = [0,1,2,3]\n",
|
||
"\n",
|
||
"iir = find_zero(θ->PV(θ, cf, m), 0)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Task 3b\n",
|
||
"\n",
|
||
"Change the cash flow as in the cell below. Find the IRR. Could there be several solutions?\n",
|
||
"\n",
|
||
"(Hint: `find_zeros()`)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 32,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"2-element Vector{Float64}:\n",
|
||
" 0.04650143593633046\n",
|
||
" 0.9160439083362444"
|
||
]
|
||
},
|
||
"execution_count": 32,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"cf = [-140,320,0,-190]\n",
|
||
"\n",
|
||
"iir = find_zeros(θ->PV(θ, cf, m),-1,1)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": null,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": []
|
||
}
|
||
],
|
||
"metadata": {
|
||
"@webio": {
|
||
"lastCommId": null,
|
||
"lastKernelId": null
|
||
},
|
||
"anaconda-cloud": {},
|
||
"kernelspec": {
|
||
"display_name": "Julia 1.7.0-rc1",
|
||
"language": "julia",
|
||
"name": "julia-1.7"
|
||
},
|
||
"language_info": {
|
||
"file_extension": ".jl",
|
||
"mimetype": "application/julia",
|
||
"name": "julia",
|
||
"version": "1.7.0"
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 1
|
||
}
|