what I want to do in this video is to
give a not too mathy explanation of why
bond prices why bond prices move in the
opposite direction as interest rates so
bond prices versus interest rates so to
start off I'll just start with a fairly
simple bond one that does pay a coupon
and we'll just talk a little bit about
what you'd be willing to pay for that
bond if interest rates moved up or down
so let's start with a bond from some
company
so let me just write this down this
could be company company a it doesn't
just have to be from a company it could
be from a municipality or it could be
from the US government and let's say
it's a bond for $1,000 let's say it has
a two year maturity two year maturity
and let's say that it has a 10% coupon
10% coupon paid semi-annually so this is
semi-annual annual payments so if we
just draw the diagram for this obviously
I ran out of space on the actual bond
certificate but let's draw a diagram of
the payments for this bond so this is
today let me do it in a different color
that's today let me draw a little
timeline right here this is two years in
the future when the bond matures so that
is 24 months in the future halfway is 12
months then this is 18 months and this
right here is 6 months now we went over
a little bit of this in the introduction
to Bond video but it's a 10% coupon paid
semi-annually so it'll pay us ten
percent of the par value per year but
it's gonna break it up into two six
month payments so 10% of a thousand is
$100 so they're gonna give us $50 every
six months they're getting give us half
of our 10% coupon every six months so
we're gonna get $50 here $50 here if
these are going to be our coupon
payments $50 there and
and finally at two years we'll get $50
and we'll also get the par value of our
bond will also get a thousand dollars so
we'll get a thousand plus $50 24 months
from today now the day that this let's
say this is today that we're talking
about the bond is issued and you look at
that you say you know what for a company
like Company A for this risk profile
given where interest rates are right now
I think a 10% coupon is just about
perfect so 10% coupon is just about
perfect you say you know what I think I
will pay $1,000 for it so the price of
the bond the price of that bond right
when it gets issued or on day zero if
you will
you'd be willing to pay $1,000 for it
can you say look you know I'm getting
roughly 10% a year and then I get my
money back 10 percents a good interest
rate for that level of risk now let's
say that the the moment after you buy
that bond just to make things a little
bit obviously things don't interest
rates don't move this quickly but let's
say the moment after you buy that bond
or maybe to be a little bit more
realistic let's say the very next day
interest rates go up if interest rates
go up let me do this in a new color so
let's say that interest interest rates
go up and let's say they go up in such a
way that now that they've moved up for
this type of a company for this type of
risk you could go out in the market and
get 15% a coupon so let's say for this
type of risk you would now expect you
would now expect a 15% interest rate
interest rate obviously for something
less risk you'd expect less interest in
that but for a company just like company
you would now expect a 15% interest rate
so interest rates have gone up now let's
say you need cash and you come to me and
you say hey Sal can you know are you
willing to buy this certificate off of
me I need some cash I need some
liquidity I can't wait for the two years
for me to get my money back how much are
you willing to pay for this bond well
I'll say you know what I'm gonna pay you
less than $1,000 because this bond is
only give
10% I'm expecting 15% so I want to pay
something less than $1,000 that after I
do all of the fancy math in my
spreadsheet it'll come out to be 15% so
I'm gonna pay so the price so in this
situation the price will go down and
I'll actually do the math with a simpler
bond than one that pays coupons right
after this but I just want to give you
the intuitive sense if interest rates go
up someone willing to buy that bond
they'll say gee this only gives a 10%
coupon that's not the 15% coupon I can
get on the open market I'm gonna pay
less than $1,000 for this bond so the
price price will go down or you could
use essentially say that the bond would
be trading at a discount to par bond
would trade at a discount at a discount
to par now let's say the opposite
happens let's say that interest rates go
down let's say that we're in a situation
where interest rates interest rates go
down so now for this type of risk like a
company a people expect 5% people expect
5% rate so how much could you sell this
bond for well if you weren't there and
if I had to just go to companies issuing
their bonds I would have to pay $1,000
or roughly $1,000 for a bond that only
gives me a 5% coupon roughly give or
take I'm not being precise with the math
I really just want to give you the gist
of it so I would pay a thousand dollars
for something and giving a 5% coupon now
this thing is giving me a 10% coupon so
it's clearly better so now the price
would go up so now I would pay more than
par or you would say that this bond is
trading at a premium a premium to par
premium to par so at least in the gut
sense when interest rates went up people
expect more from the bond this bond
isn't giving more so the price will go
down
likewise if interest rates go down
this bond is giving more than what
people's expectations are so people are
willing to pay more for that bond now
let's actually do it with an actual if
actually do the math to figure out the
actual price that someone a rational
person would be willing to pay for a
bond given what happens to interest
rates and to do this I'm gonna do what's
called a zero coupon bond I'm gonna show
you a zero coupon bond and actually the
math is much simpler on this because you
don't have to do it for all of the
different coupons you just have to look
at the final payment so a zero coupon
bond is literally a bond that just
agrees to pay the holder of the bond the
face value so let's say the face value
the par value is $1,000 two years from
today two years from today
there's no coupon so if I were to draw a
payout diagram it would just look like
this this is today this is one year this
is two years you just get $1,000 now
let's say on day one interest rates for
a company like Company A this is Company
A's bonds people so this is starting off
so day one day one let's say people's
expectations for this type of bond is
they want 10 percent 10 percent per year
per year interest so given that how much
would they be willing to pay for
something that's going to pay them back
a thousand dollars in two years so the
way to think about it is let's pee in
this I'm gonna do a little bit of math
now but hopefully we'll be too bad so
let's say P is the price that someone is
willing to pay for a bond so whatever
price that is if you compound it by 10
percent for two years so I do 1.10
that's one plus ten percent so after one
year if I compound it by ten percent
will be P times this and then after
another year I'll out multiply it by one
point one o again this essentially is
how much I should get after two years if
I'm getting 10 percent on my initial
payment or the initial amount that I'm
paying for my bond so this should be
equal to this should be equal to
the thousand dollars so let me just be
very clear here let's P is what someone
who expects 10% per year for this type
of risk would be willing to pay for this
bond so when you compound their payment
by 10 percent for two years that should
be equal to a thousand dollars so if you
do the math here where you get you get P
times 1.1 squared is equal to 1,000 or P
is equal to one thousand divided by 1.1
squared another way to think about it is
the price that someone would willing to
be willing to pay if they expect a 10%
return it's the present value of one
thousand dollars in two years discounted
by ten percent this is one point 100 or
1 plus 10% so what is this number right
here I think it's gonna calculator out
the calculator out and so if we have
1,000 divided by 1.1 squared that's
equal to 826 dollars and well I'll just
round down 826 dollars so this is 826
dollars so if you were to pay 826
dollars today for this bond and in two
years that company would give you back a
thousand dollars you will have
essentially have gotten a 10 percent
annual compounded interest rate on your
money now what happens if the interest
rate goes up let's say the very next day
and I'm not gonna be very specific I'm
gonna assume it's always two years out
you know it's one day less but that's
not going to change the math
dramatically let's say it's the very
next second that interest rates were to
go up so let's say second one so it
doesn't affect our math in any dramatic
way so let's say interest rates go up so
now all of a sudden so interest people
expect more interest goes up the new
expectation is to have a 15% return on
and it on a on a loan to a company like
company a so now what's the price we're
willing to pay well use the same formula
the price is going to be equal to
thousand dollars divided by instead of
discounting it by 10% we're gonna
discount it by 15% over two years so one
plus 15% compounded over two years
bring out the calculator bring out the
calculator I think you have a sense we
have a larger number now in the
denominator so the price is gonna go
down so let's actually calculate the
math one thousand dollars divided by one
point one five squared is equal to seven
hundred and fifty six dollars give or
take a little bit so now the price has
gone down the price is now seven hundred
and fifty six dollars this is how much
someone's willing to pay in order for
them to get a fifteen percent return and
get a thousand dollars two years or get
a thousand dollars in two years and
essentially for it to be a fifteen
percent return now just to finish up the
argument what happens if interest rates
go down so if let's say interest the
expected interest rate on this type of
risk goes down and let's say it's now
five percent what is someone willing to
pay for this zero coupon bond well the
price is if you compound it two years by
1.05 that should be equal to thousand or
the price is equal to a thousand divided
by two years of compounding at five
percent
and get the calculator out again we get
one thousand dollars divided by 1.05
squared is equal to nine hundred and
seven dollars so all of a sudden we're
willing to pay price is now nine hundred
and seven dollars so you see
mathematically when interest rates went
up the price of the bond went from eight
hundred twenty six to seven hundred
fifty six the price went down when
interest rates went down the price went
up and I think it makes sense the more
you expect the higher return you expect
the less you're willing to pay for that
bond anyway hopefully you found that
helpful