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principle number four looks a
day second derivative if you want off bones what's a good principle number
four says he's that the sensitivity offer price of a
bold to change or seen the yield to maturity is increasing at the decreasing
rate with the links in maturity okay that's a pretty complex
sentence so let's read it again the sensitivity if the offer
price over boomed to change or seen the yield to
maturity stop here this is what we saw in principle
number three how prices change with accordance to the yield to
maturity we already saw in principle
number three that prices change with according to deal to
maturity and there are more sensitive
with long term bonds that's fine that's principle number three
but principle number four says okay there are increasing but
there are increasing in the decreasing great when they bones length or time
to maturity is getting bigger and bigger so that's actually a second
derivative of the bones price but we're not going to take
derivatives what we will do we will take that shit will
measure the changes thing the changes how the changes themselves
change over time let's do that here is the same principle the
same good a example us before let's see the numbers in the upper Pernod I just
expanded the same panel but we so the same table that we saw in
principle number three in other words we take a one
thousand dollars par value bold with different here's the
maturity different healed as to maturity hunter Doris coupon
rate hundred dollar school boards are up ten percent
component right inside the table once again I'm
recording the prices all those bones once again not surprisingly you
can see one thousand dollars right here in the middle simply
because it was the yield to maturity of ten percent coupon
rate of ten percent the price is always equal to the par value
regardless of the number of years until maturity I just expanded the table number
three Fromm eggs a principle number
three alright what I'm going to do in
the lower panel is I'm going to measure changes change your singles prices
between two Paris or off to shorter maturity bonds into long
term maturity balls so the gaps between ten and five
years the witchery two bones will be mean the SRO which means hundred seventy one
dollars is the gap the change in the prices between two shorter maturity
boards one seventy one dollars you get by subtropics twelve nineteen from thirteen
ninety seventy one dollars received by
subjecting eleven hundred or three minus eleven hundred
seventy four actually the opposite eleven
seventy four minus eleven all three zero is always the one
thousand minus one thousand minus fifty he's a fifty nine
minus line all nine in minus eighty three seven forty five
minus eight twenty two thirty minus twenty five are the
gaps the differences in prizes between two long term maturity
balls once again the four dollars he received by
sub truck thing seventy nine from seventeen seventy three and so on and so forth so the
lower panel will measure for us the changes in the prices how
changes in prices of core between tool shorter maturity
bonds and too long to maturity and then I will pull today lower
paddled the rolls along the lower panel and this is what we get on the vertical axis you have
here mold prices but differences in prices changes in prizes between two
shorter maturity bald into longer maturity bonds on the
horizontal axis you have the yield to maturity as you can see once again the
slope here represent the sensitivity and we already know from
principle number three the longer maturity bonds their
price is more sensitive to the hill the witchery that's fine that's principle
number three but principle number four reveals or something
you he tells us that that changes
themselves the difference between two shorter maturity
bonds he's actually more sensitive to
to long term injury to both the difference between the prices in
two off to long term maturity bonds so let's stop here let's on this
then what does it mean principle number three says that prices are more sensitive to
yield to maturity for long term a true two boys than short of
witchery that's fine this is very intuitive long term
bonds are more risky prices there are more sensitive short term goals but principle number four tells
us that P. sensitivities decreasing over time which means for too long term
maturity bones the difference between too long
to maturity bosses listen sensitive then the difference between two
shorter maturity why is it important to us the way bondholders choose their
boys to peek in invention lean face these the following first
they decide what kind of investors are we how we were more risk averse
investors or less risk averse investors he who really really hate trace then we will profesor to invest
things shorter maturity bonds rather than long term
untreatable why is that because shorter maturity bonds
are safer they're not the US riskiest loan
to retreat so that's the first thing you
decide which kind of bones do you want to peak short term or
long term you will peak short term if
you're more risk of various you hate to risk you would like to
avoid risk you prefer things that are safer lower risk low return low yield if you prefer to take more risk higher risk higher return how
your yelled you will choose high a long term maturity bolts
that's fine this second stage used to do the
fine tuning school beside Akane now that I
understand that I'm interested in long term or short term
injury towards I need to actually choose do I need to choose the tenured
five year or do I need to choose between thirty years won't or
the twenty five years born what principle number four tells
you is that even the beginning your
selected short term maturity bonds you really need to the full
blown off time to select the specific ball that you're
interested because any mistake here between
five year or ten years maturity bonds is going to cost you
dearly you gonna pay through the nose
if you'll make a mistake here and the van something bad will
happen because these big gap the differences in prices can be big big difference in prices between
five year and ten year however even the beginning your selected tool room alter
maturity bones then which one is really not that important the
difference between Toru loco maturity bonds is not that much the difference here is pretty
minor here any small that important principle number four is
important for the fine tuning for the stage that you actually
need to select which kind of bones your interested to
interest that's principle number four principle number five it's a
very very simple principle but in fact is very powerful so let
me explain that principle number five citizen the resilient relationship a
straight line the relationship between the bones quick pull on
the see and you price P. no we're measuring the
relationship the price will double in the quick poem of the
bones see principle number five say there
is a straight line relationship lean their relationship and this
is something very very easy to observe a straight from the
equation all you need is just to take
seed took one and picking the outside of the brackets outside
of the summation because sis fix it's a fixed
number for example six person sixty dollars or whatever ET's
or you need is just to take it outside of the brackets what you're left inside the
brackets is the following the slope will be this summer for one over one
plot one divided by one plus the yield to maturity
to devour fifty this is the slope endanger saved which is
completely unrelated to cede to buckle porn's he's the Parvathi
divided by one plus the into maturity to the poor refrain this is the straight line
relationship the lean their relationship that you can see
the price of the born the ease of the slope that you can see
here after taking sealed sudden the brackets so your life with
the slope time see plus thinker slip that has nothing to do with
the core porn that's why this is a straight
line relationship if your plot data this is what
you will get it full price us a function of the
components a here is the slope of be here
is they intercept where you can see the interpretation of these
is the slope this some more they received recalls the one more
for one plus the ultimate the power of teach while they intersect he simply
means the pendant term that does not contain the scene it has
nothing to do with the quick points which you simply the par
value discounted to present time or divided by one plus the
little maturity to the power of pain this principle is very very
simple they principle to understand and also to pray to
me practice that it is also very powerful simply
because we can do money with us in the following example we're
going to try and make money using these principles let's see how to do an arbitrage
using bowls but before we do that let's
understand first of all what's the meaning of the word
arbitrage arbitrage it's a finance corn
soup that says the risk free money risk free return it's basically a free lunch
everybody likes to do that if you see an opportunity to see
arbitrage you need to do two things first of all give me a call so
our world so more about it and so I can do it as quickly as you
can with us much money as you can because it's free lunch you're
making money out of thing air out of nothing you're taking
absolutely no respect you create money you make positive return
appropriate out of nothing everybody likes to do that in
usually arbitrage opportunities are disappeared in a matter of
seconds why because all kinds of hedge funds
and financial institutions then Providence us or is also are running computer algorithms
always search for arbitrage importantes
whenever they need to take to the other reason arbitrage
opportunities they jump on it they don't walk they run they
jump on it like crazy because that's free lunch this is making
money out of nothing and because of the market forces
the market forces will well make the arbitrage opportunities
disappear they willingly make them in is going to take a
matter of seconds maybe want to maybe maybe three seconds in
that seat when everybody see something
like that everybody jump on that in the market forces will
eliminate that we're going to see a couple of
examples for arbitrage in discourse induces a the first
one usually arbitrage she's done by
closing the circle or deal you buy something and then you
need to sell something it's local it's not enough to take
half a deal could do only half of the grid
the ing the deal here you don't just buy something in the in the
weeks for that the price will go up or sell something in the wish
that the price will come down you need simultaneously to buy
something since something of that
replicates the same cash flows all of the things that you
bought but if you sell it at the higher
price than the price that you paid when you bought then the you end up with a
positive cashflow right now wait times zero in all future councils will
disappear this is usually the way to do
arbitrage so let's see this the following
example and let's learn how to do arbitrage using boards let's assume that we have three
bones so let's call them a B. and C. all of them have the same par
value of one thousand dollars or of them have the same time
horizon in this example I would like to
keep the time horizon fake it really doesn't matter their own
your practice questions you have at least couple of prop problems
to practice these with specific time horizons that you can
actually plug in the numbers in the actually calculate how much
you're going to make by doing an arbitrage but let's assume that all of
them were mature the same day and the following characters born they has a coupon rate of
three percent you know the wards it pays three thought a thirty
dollars every year with a five percent yield to maturity wise
equipped five bone be has a coupon rate to
five percent we see yield to maturity or four and a half
percent born to see has a coupon rate or seven percent with the five
percent yield to maturity wise equal to you in a five in other words if you know that
all three boys have the par value of one thousand dollars
you can insert in your financial Concord Eder one thousand
dollars if V. future value face value par value all of them have the signs same
time horizon you can practice that by taking for example time
horizon ten years in the sequel to attain for each and every born to you
know to call poem rates see who he sees it will cool thirty
fifty or seventy dollars so PMT the payments or thirty dollars
fifty dollars of seventy dollars in the why deal to maturity or
ice lish why on your Concordia's will be either five four and a
half four five and then you can ask your
collaborators compute present value compute the price of those
bones then you will understand how
much does it cost you to buy or sell those born on the free
markets so how can we do or be trudge remember that arbitrage you need
to close the circle or deal to buy low sell high you wanna buy
something and sell something at the same time simultaneously but
you want to do it in a way that you buy at the low price sale at
the high price so you will end up making money right now it
times zero it prison time anyone who designed the things
that you buy in the sale so their cash flows future cash
flows will replicate each other will mimic each other would be
identical and therefore once you buy
something and so the other thing they will cancel each other
whenever you buy a bond you are able to receive future cash
flows you pay no but you will receive the quipall than the
future par value whenever you syllable and what
it means is that you become the you sure you Celeborn to all the people
your D. E. shorter you collect money right now but you're
responsible to pay day coupons and paid the par value at the
end so how do you do arbitrage here
what exactly do you need to buy what exactly do you need to sell I'll let you couple of seconds
to think and then want to solve these problems in order to do arbitrage you need to understand that both
born say in the sea both of them have the same yield
to maturity five percent what it means is that if you
want to create a portfolio of loans ANC any kind of
combination between them will also give you five percent
return five percent yield to maturity it really doesn't matter if you
invest hunter person in bold a tenth zero percent in bonds see fifty percent in Bombay fifty
percent in Monsey sixty forty seventy thirteen
nineteen ten eighteen twenty whatever ET's any kind of
combinations between bones ANC or portfolios
between born say NC will give the investor a five percent
return five percent yield much because both of them are focused the first thing seconds the AP is the pay
attention that we're trying to replicate same corporeal Kassebaum be they already had the same or of
them have the same par value but we will try to replicate boomed
breed the same couple Norick of Palm Beach we will try to make a portfolio
between born say and see with the right proportions the
right waits to replicate the same quarry
possible and be here the solution is pretty
straightforward you can see that the couple married a bone these
three percent a coupon rate one C. seven percent so all we need is just to invest fifty percent of our money in
one day fifty percent of our money on buoyancy simply because
five use exactly the average off three and seven in these case if we want to buy fifty percent of our money into
warm day fifty percent of all money too
bossy we will a replicate the same
quarry decibel MTBE you know what I'll ask you the
following question what do you think he's cheap beer deport for you'll off Ponce and
see or born to be and that's the reason that
portfolio between bones ANC is going to be cheaper why is that each for able to replicate the
same cool poem rate sold the born the portfolio of
bones ANC will give you a five percent coupon rate or fifty
dollars per year he cut it has the same par value
of one thousand dollars and the same time horizon bot deport photo boards ANC has a
higher yield to maturity or five percent we already sold out with a
higher yield to maturity holding everything else the same that
price must be lower how your yields lower priced lower yield
higher price so just because boys aliens see
have high your yield to maturity everything else is the same the
combination deport four between bones ANC will give you a lower
price therefore in order to do an
arbitrage here you need to buy port for you'll
off bones AMC was the right proportions fifty percent of
bone they fees to presume Boise you order to replicate the
conglomerate of Palm Beach and at the same time sale boom beat
becomes the shore of warm brie let's understand what happened
here we're buying fifty personal for
my newborn they were buying fifty percent of our money on
buoyancy so we're investing in a poor
foretold bones ANC what's going to happen awesome
result first of all when we are buying
it though the sport for dual worth paying a small amount of
money mortar differently then boom
beat why is that because the port for newborns ANC has a high
your yield to maturity five percent compared to four and a
half percent we said hi your yield to maturity the price must
be lower everything else is the same okay if you buy born say in C. fifty
percent of your money on a fifty percent of your money on C. you will receive a fifty percent
off thirty dollars every year from
the coupons off born they so that's fifteen dollars plus fifty percent off seventy
dollars periodical plausible and see thirty five dollars here you will collect a few year half
full thirty dollars half seventy fifteen plus thirty five dollars
will give you fifty daughter's school boards every year now at the same time your that
you sure of born to be your best seller of blonde beat whenever you are do you sure
your responsible to pay to your investors the coupons the fifty dollars so every year your going to
collect fifty dollars from the poor toward of Bombay and see
and you will correlate and you will have to pay fifty dollars
to your investors seem because your D. shore of Palm Beach
fifty dollars you collect fifty you'll pay
fees to you quite like fifty will pay fifty these cash inflows if with cash
overalls will cancel each other that's nice what about the yield to maturity
I'm sorry what about the par value the powerful you you will
collect fifty percent of the par value of bone they which is one
thousand dollars fifty percent of the par value
of buoyancy was he's also one thousand dollars so fifty dollar
fifty percent here fifty percent here five hundred plus five
hundred the your quarter licked altogether one thousand dollars
Parvan use half of the powerful useful bones the NC you collect par value at the end
but then your day you sure won't be your responsible to pay the
par value of boom beat to your investors so you were paid the
one thousand dollars to your investors once again these cash rules will
cancel cancel each other that part of our news that you
will collect on boards ANC will cancel out with the car
volley that you're responsible to pay to your investors home
won't be as you can see or future cash
flow will cancel each other what it means is that this is a
risk free you don't care off though that
what happens now happens know enough about you don't care
because or future cash flows will cancel each other you they will cancel each other
you don't care and therefore it's risk free investment the beauty here is that you're
making a positive return immediately on the very first
second that you do that once again why is that simply because deport foil
balloons ANC are cheaper then the ball and be won't be easy for able to
replicate the same court on the rate in the right here the
solution is straightforward just split your money evenly between
bonds ANC fifty percent on your money and bought a few someone see then fifty percent of three plus
fifty percent off seven percent coupon rate will give you five
percent coupon rate won't be because the yield to maturity
between any kind of combination of blogs AMC will give you five
percent yield to maturity hi your yield to maturity it
means lower price than bone breed that has a lower yield to
maturity that's why you pay low you'll collect high you bought a little
said hi you right now you make positive cashflow right now in
no future cash will disappear peace means an arbitrage once again when you something
when you see something like that you need not to buy just one
bold don't make all of the safe for to make just saw couple of
dollars you can the replicated the
strategy millions of time simply because their maintenance
of bones he should go on the market traded on the markets you
can replicate these meanings of times and you can make millions
of dollars in a matter of seconds right now to make
positive cash flow in all future cash flows will disappear here in the following slides you
have the formal explanation and that also the formal proof based on principle number five it's really not that important
to understand the the former proof but I do want to emphasize
for those who don't want to go over the form of proof here just the fact that this is
nothing by beyond dogs are brought that we just decent
tangled the price of each and every born so the subluxation
uppercase a BNC our bones a BNC those lower case is a in lower
case is B. here represent the slope when they intersect taken
from principle number five we sold them it slide number fifty six so the slope he's here and they
intercept is here and after that it's nothing
beyond a simple algebra so what do you break apart to deprive so
featured every bone to the slope does look upon plus the
intercept all you need is just to batch
something's into brackets and you can full dog it's fairly
straightforward and nothing beyond that but it's nobody
important more importantly still
understand the concept of how to do arbitrage here which he's
body low sell high why deport photo boards alien ceased
cheaper than a boom be simply because it's called the nation will have
a higher yield to maturity which means a lower price what exactly
do we need to buy what exactly do we need to sail which is
always the same with arbitrage you need to buy low sell high
you make immediate positive cashflow in all future cash
rules will disappear so that's the way to do
arbitrage using bones I would also add one quick
comment here that the new reality obviously you don't have
only three balls you have thousands and thousands
of different forms so what happens in reality is
that many hedge funds and financial institutions and also
propaganda sores their run computer algorithms to search
between those thousands and tens of thousands of different boards search for arbitrage
opportunities so this can trigger you to think
a bald or because of possibilities and the and the desire to look for
patrols opportunities using different computer algorithms

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