The Term Structure of Interest Rates Spot, Par, and Forward Curves (2024 CFA® Level I Exam – FI 9)
Summary
TLDRThe video script provides an in-depth exploration of fixed income securities, focusing on the complexities of bond pricing in relation to the term structure of interest rates. It begins by revisiting the basic model of bond pricing, which assumes a constant yield to maturity, and then transitions into a more sophisticated model that accounts for fluctuating interest rates. This involves understanding and calculating spot rates, par rates, and forward rates, which are essential for constructing an arbitrage-free bond price. The presenter uses various examples to illustrate how these rates are derived and how they interact with each other. The importance of these concepts is emphasized through their impact on the shape of the yield curve, which can be upward sloping, downward sloping, or flat. The script concludes with practical advice for students preparing for the CFA exam, suggesting that while calculating par rates might not be a common exam question, understanding the relationship between spot rates, par rates, and forward rates is crucial. The presenter encourages students to practice these calculations to ensure they can apply them effectively in different scenarios.
Takeaways
- 📈 **Spot Rates**: The yield on zero-coupon bonds, used to price bonds by discounting each cash flow at the appropriate rate for its term.
- 🔄 **Par Rates**: The yield to maturity that makes a bond sell at its par value; slightly different from spot rates due to mathematical adjustments.
- 🔗 **Forward Rates**: The expected interest rate for borrowing or lending at a future time period, derived from spot rates to prevent arbitrage.
- ↗️ **Upward Sloping Curve**: When short-term rates are lower than long-term rates, spot rates and par rates align closely with forward rates being higher.
- ↘️ **Downward Sloping Curve**: When short-term rates are higher than long-term rates, the par curve is slightly above the spot curve, and forward rates are lower.
- 🔲 **Flat Curve**: When all rates are equal across different maturities, spot, par, and forward rates converge.
- 💡 **Arbitrage-Free Pricing**: Using spot, par, and forward rates ensures that bond pricing is accurate and prevents arbitrage opportunities.
- 📚 **CFA Level Model**: A more complex model than the basic 'kindergarten' model, allowing interest rates to change over the life of a bond.
- 🧮 **Calculation Complexity**: Computing bond prices using spot and forward rates is more complex and requires algebraic manipulation.
- 📊 **Yield Curve Shapes**: Understanding the shape of the yield curve is crucial for interpreting spot, par, and forward rates.
- 📈 **Increasing Short-Term Yields**: An increase in short-term yields with a smaller increase in long-term yields can lead to a flattening of the yield curve.
- ⏱️ **Time Value of Money**: The concept of time value of money is fundamental in calculating present values of future cash flows at different rates.
Q & A
What is the basic assumption made in the kindergarten model of bond pricing?
-The basic assumption made in the kindergarten model of bond pricing is that the yield to maturity on the bond is fixed over the life of the bond, meaning the term structure of interest rates is unchanging.
How does the CFA level model differ from the kindergarten model in terms of interest rates?
-The CFA level model allows for interest rates to change over the life of the bond, as opposed to the kindergarten model which assumes a fixed yield to maturity.
What is a spot rate?
-A spot rate, also known as a zero rate, is the yield on a zero-coupon bond that provides a specific cash flow for a single period. It is used to discount a cash flow back to its present value at a specific point in time.
What is the significance of forming a spot curve, par curve, and forward curve?
-Forming these curves allows for a more accurate and comprehensive understanding of bond pricing. The spot curve represents the yields of zero-coupon bonds, the par curve shows the yields of bonds priced at par, and the forward curve indicates the expected interest rates for future periods.
How does arbitrage play a role in the pricing of bonds?
-Arbitrage opportunities arise when there are inconsistencies in bond pricing. An arbitrage-free bond price ensures that a bond cannot be bought and sold in a way that generates a risk-free profit, thus maintaining market efficiency.
What is the relationship between spot rates and par rates?
-Par rates are the yields to maturity that make a bond sell at its par value. They are derived from spot rates and are generally slightly lower than the corresponding spot rates due to the mathematics involved in their calculation.
How are forward rates determined?
-Forward rates are determined by ensuring that there is no arbitrage opportunity between a single longer-term bond and a series of shorter-term bonds that match the time horizon of the longer-term bond. They represent the expected interest rate for a future period.
What is the typical maximum number of years for which the CFA exam might ask to compute bond prices using spot rates?
-The typical maximum number of years for which the CFA exam might ask to compute bond prices using spot rates is four years, as computations for longer periods become excessively complex.
What is the implication of an upward sloping yield curve?
-An upward sloping yield curve implies that long-term interest rates are higher than short-term rates, which can indicate a healthy economy with expected growth and inflation.
What does a downward sloping yield curve suggest about future interest rates?
-A downward sloping yield curve, also known as an inverted yield curve, suggests that investors expect future interest rates to decrease, which can be a signal of a potential economic downturn.
How are forward rates used in calculating the price of a bond?
-Forward rates are used to discount the expected cash flows of a bond at each future period. This allows for the calculation of a bond's price based on the expected interest rates for each period, rather than a single yield to maturity.
What is the relationship between the spot curve and the forward curve on a graph?
-On a graph, the spot curve represents the yields of zero-coupon bonds, while the forward curve represents the expected interest rates for future periods. The forward rates are typically higher than both the spot and par rates in an upward sloping yield curve scenario, but can be lower in a downward sloping yield curve scenario.
Outlines
📚 Introduction to Fixed Income and Interest Rate Structures
Jim introduces the topic of fixed income within the CFA program, focusing on the term structure of interest rates, spot, par, and forward curves. He contrasts the simplicity of the 'kindergarten model' used in previous modules, which assumes a fixed yield to maturity, with the more complex, 'CFA level' model that allows for changing interest rates. The importance of understanding and calculating spot rates, par rates, and forward rates is emphasized, as they form the basis for an arbitrage-free bond pricing model.
📈 Understanding Spot Rates and the Arbitrage-Free Bond Pricing
The concept of spot rates is explored, which are the yields on default risk-free zero-coupon bonds across different maturities. An example of a three-year bond with a 10% coupon rate is used to illustrate how each cash flow is treated as a zero-coupon bond and discounted at different spot rates. The arbitrage-free bond pricing model is introduced, ensuring that bonds are priced accurately without the possibility of arbitrage opportunities. An upward sloping spot curve is discussed, along with the impact of changes in interest rates on the curve.
🔢 Calculating Par Rates from Spot Rates
Par rates are defined as the yields to maturity that make a bond sell at its par value of $100. The process of calculating par rates from spot rates using algebraic manipulation is explained. An example is given for a government bond with a 1% coupon rate and various spot rates, showing how to solve for the coupon rate (PMT) that results in a bond price of $100. The relationship between par rates and spot rates is highlighted, noting that par rates are slightly less than the corresponding spot rates.
🔄 Forward Rates and Their Relationship with Spot Rates
The concept of forward rates is introduced, explaining how they represent the expected interest rate for a future period, derived from the comparison between a longer-term security and a series of shorter-term securities. An example calculation is provided for a three-year spot rate and a four-year spot rate, illustrating how to find the one-year forward rate three years from now. The process of calculating bond prices using forward rates is also discussed, with an example showing how to discount future cash flows at the respective forward rates.
📉 Analyzing the Relationship Between Spot, Par, and Forward Curves
The relationship between spot, par, and forward curves is analyzed in the context of different yield curve scenarios. It is explained that in an upward sloping yield curve, spot rates and par rates align closely with forward rates being higher. In a downward sloping yield curve, par rates are slightly above spot rates, and forward rates are below them. When the yield curve is flat, all three curves converge. The importance of understanding these relationships for accurate bond pricing and financial analysis is emphasized.
Mindmap
Keywords
💡Spot Rates
💡Par Rates
💡Forward Rates
💡Yield Curve
💡Arbitrage Free
💡Coupon Rate
💡Par Value
💡Zero Coupon Bond
💡Yield to Maturity
💡Macroeconomic Conditions
Highlights
Introduction to fixed-income and the term structure of interest rates, including spot, par, and forward curves.
Explanation of the assumption that the yield to maturity on a bond is fixed over its life, which is a simplification.
Transition from a basic model to a more complex CFA level model that allows interest rates to change.
Concept of viewing each cash flow as a zero coupon bond for pricing in the new model.
Importance of spot rates and how they are used to calculate the price of a bond.
Explanation of arbitrage opportunities and how they relate to bond pricing.
Visual representation of an upward sloping spot curve and its implications.
Understanding default risk-free zero coupon bonds and their role in yield and yield spreads.
Equation for achieving a no-arbitrage price using present value calculations.
Quick example of calculating the price of a four-year government bond using spot rates.
Differentiating between spot rates and par rates, and the formula for calculating par rates.
Advice on how to approach exam questions related to par rates and their relationship with spot rates.
Introduction to forward rates, their calculation, and their role in arbitrage-free pricing.
Example of calculating a one-year forward rate three years from today using spot rates.
Comparison between calculating bond prices using spot rates versus forward rates.
Graphical representation and comparison of spot, par, and forward curves in different scenarios.
Practical advice for exam preparation regarding the calculation of bond prices using different rate types.
Summary of the importance of understanding spot, par, and forward curves for fixed-income analysis.
Transcripts
hey it's Jim and this is level one of
the CFA program the topic on fixed
income and the learning module on the
term structure of interest rates spot
par and forward curves I'm hoping that
you recall our conversation a few
learning modules ago about Bond pricing
what we did is we computed the price of
a bond as the present value of the
coupon payments plus the present value
of the par value payment now what we did
is we made a big assumption and that
assumption was that the yield to
maturity on that Bond was fixed over the
life of the bond in the terminology that
we're going to use inside of this
learning module it is the following
we're going to say that the yield curve
is flat or the term structure of
interest rates is
unchanging now in my classes I usually
uh tell tell my students after they
learned it that this was a kindergarten
model because it's so simple now we need
to learn some kind of a college level or
a CFA level model in which we're going
to allow interest rates to change that's
why we need spot par and forward curves
so look at look at these loos's notice
that there's spot par forward in every
one of these so instead of three loos's
here in this learning module we could
just have one and that one losos could
sound something like this hey we want
you to know everything about spot rates
par rates and forward rates and we want
you to be able to form a spot curve a
par curve and a forward curve so let's
go ahead and start with spot rates and
to do this I want you to Envision a
three-year Bond and the timeline that
follows let's suppose that this bond has
a 10% coupon rate and it matures for
$100 so in year one the bond holder
receives $10 in year two the bond holder
receives $10 $10 in year three the bond
holder receives the final $10 plus the
$100 in par value now in that previous
recording we viewed this as one Bond and
we discounted each one of those cash
flows at the yield to maturity on the
bond well we can't do this now that
we're using this as a CFA level model to
determine the price of a bond what we're
going to do is we're going to view each
one of these C cash
flows each one of these coupon payments
and then the final par value payment as
a zero coupon Bond almost like a
standalone Bond so I want you to think
about this we're going to price a bond
today and we're going to get $10 a year
from now we're going to Discount that at
One Rate we're going to call that a spot
rate or a zero rate then we're going to
get $10 two years from now and we're
going to Discount that at a different
rate whether the yield curve is upward
or downward sloping but we're going to
Discount that at the next zero rate and
then we get the 110 at the end of year
three we're going to Discount that at
yet another spot rate or zero rate and
then we're just going to sum those three
uh present values to get the price of a
bond so you see how important this is
spot rates and spot curves are based on
the assumption that a bond is made up of
a bunch of zero coupon bonds and my
simple example there it was a oneyear
zero a 2year zero and a three-year zero
coupon
Bond now I want you to look down at the
bottom left block that's a really
important statement there when we did
that kindergarten model we never
mentioned anything about Arbitrage
opportunities because all we were doing
was talking about um using the five-time
value of money buttons on our calculator
to solve for present value it was really
just an application of time value of
money principles but now when we allow
yield curves to be upward or downward
sloping what we're going to do is we're
going to arrive at an Arbitrage free
bond price so of course the kindergarten
model that we did before that gets us in
the bald Park but now when we use this
new model we're going to guarantee that
this is a way more accurate price in
other words we can't chop a three year
bond into its three zero
components and pay less or more for that
bond which would allow us to take the
long or short position and generate an
Arbitrage
profit all right so let's go ahead and
look at a spot curve notice we're going
to draw a picture of the time to
maturity on the horizontal axis and
yield to maturity we can go ahead the
reading calls them spot rates on the
vertical axis notice this is an upward
sloping spot curve and go ahead and read
that bullet point over on the left
default risk-free zero coupon bonds
against each maturity now this is
something that I didn't say earlier but
it's super important in understanding
where we're going um when we consider
talking about yields and yield spreads
in other learning modules so we're going
to start with a default risk-free zeroc
coupon Bond and then here's the picture
of an upward sloping spot
curve now there we go upward sloping
there's downward sloping and then there
is flattening or steeping so if you look
at the red you know we can have a change
in interest rates that that is not
constant notice that in from the
original curve now I'm down on the
bottom uh graph the increase in interest
rates at the far term is just a little
bit right from the original curve all
the way out that's just a little teeny
weeny bit teeny weeny is not a finan
term but down at low at early maturities
look at that change so notice in
flattening of the yield curve we have a
large increase in short-term yields and
a small increase in long-term yields so
we can have flattening or steepening the
the reading does mention something about
a decrease in inflation for flattening
yield curves that's probably something
that I would remember for the exam
now here is this equation that I was
trying to get you to visualize earlier I
didn't put it up earlier because I knew
we were going to get to this but here
we're going to achieve a no Arbitrage
price so all we're doing is the present
value is our payments in the numerator
and then our one plus the interest rate
and so the reading uses the notation Z
for zero rates but remember zero and
spot rates those uh those those mean the
same thing so let's go ahead and do just
a super quick example so here we have a
four-year Government Bond so there it is
risk-free a 1% coupon rate and here are
the spot rates so one year 1.5 then 125
then one so this is a downward sloping
yield curve all right so all we're going
to do is in the numerator under present
value um 1% of 100 is $1 so we're going
to get one and one and one and one at
the end of year one 2 3 and four and
then at the end of year we're going to
get the return of that principal amount
or that par value payment now note what
we're going to do is in the denominator
we're going to use each of those spot
rates right there's the one
1.015
1.125 etc etc and we're going to raise
that to the 1 123 and4th power because
we're discounting them back over 1 2 3
and four years so notice that we get a
present value that's boy what is that
close to 101 what does that tell you the
price of a bond here this is an
Arbitrage free Price what this simply
means is that is that we're willing to
pay almost
$101 for a a a 4-year risk-free
Government Bond in which we're faced
with these zero
rates notice that this is way more
complex than what we did back in that
kindergarten Model A few learning
modules ago because we're using Market
rates of interest at each time period
which is probably reflective of well how
about if I just say different
macroeconomic conditions in year one and
year two and year three and year four in
this case we expect interest rates to
fall so we're willing to pay about $101
for this uh 4-year Government Bond
now I wish I could tell you that there
is a shortcut to Computing that 101 over
on the bottom right but unfortunately
there is not so you're going to have to
go ahead on the exam is compute all of
those things you know if the Institute
gives you a 78-year Government Bond and
gives you 78 spot rates and asked you to
compute the price of that 78-year Bond
I'm going to go ahead and pick tell you
to pick B and move on with your life
because it will take you a super long
time uh to go ahead and compute those my
point is that the Institute probably
four years is the max that it will give
boy that would be stretching it in my
estimation for The Institute to ask you
to do this over five years but it's
really not that big of a deal I think
the likely question is a three-year Bond
so you can clearly clearly do
that all right let's move on to our
second concept here par rates what we're
going to do is we're going to say
something like wait a minute wait a
minute
what what would be the yield to
maturity that forces the bond to sell at
its par value forces the bond to sell at
$100 here look at the formula over there
on the on the top right there it says
100 equals so here's the question let me
go back here whoops I'm going the wrong
way let me go back here so look suppose
that we're forcing that one forcing that
100. n57 suppose we forc that to be 100
the question is what are the par rates
that you would use in that denominator
to get a price of $100 and that's pretty
much what a par rate is of course you
have to start with the spot rates in
getting the uh to get the par rate as
well so notice the formula over there
looks exactly like what we did before
and uh all we're going to do is use some
algebra so let's go ahead and work
through an example here so here are spot
rates for 5 49 and 525 right calculate
the par rates
for
um uh this Government Bond so we're
going to do one year we're going to do
two year we're going to do three years
all right so let's go ahead and get out
our calculator look up at the uh look up
in the red box so we're going to on the
left hand side of the equal sign that's
always going to be 100 right we're
forcing it to be par value and we're
going to solve for that PMT in the
numerator because that PMT is the coupon
rate and the only way that the price can
sell for its par value of 100 is if the
coupon rate and the yield to maturity in
this case we're calling it the par rate
and the yield to maturity are the same
number so that PMT in the numerator is
going to be our uh variable to be solved
so you need a little bit of algebra boy
at the risk of offending you maybe you
guys are all over this and you've
already calculated that par rate right
right now but just for kicks and Giggles
as my cousin says on the right hand side
of the equal sign let's let's separate
let's do PMT ided
1.04 plus 100 divided
1.04 and then you can do the math and do
some algebra go to this side and
multiply product of the extremes equals
the product of the extremes that's the
way I learned it in high school maybe
you guys call it cross multiplication
but if you do all that you get 4 . 5%
now the one-year par rate is always
going to be equal to the oneyear spot
rate we didn't have to go through that
math but but nevertheless uh it's
probably a good idea to for us to have
gone through that math so that we can
get the two-year rate oh boy all right
so what are we doing now two-year rate
we're going to say 100 is equal to the
payment divided by the oneyear spot rate
right 1.04 five then we're going to add
the payment plus the 100 divided by well
just go back here right there's the 4.9%
1.04 N squared so on the right hand side
of the plus sign we need to chop that
into two so do PMT / 1.04 n^2 plus 100
divided by 1.04 n^ squared and then do a
little bit more algebra and a little bit
more adding adding and subtracting and
cross multiplying and you'll get it's
about 4. uh
89% and then if you do the same thing
you get 522 52% for that
three-year uh for that three-year par
rate all right so this is how you get
one two and threeyear Par rates and look
up at the losos yeah absolutely the
losos says calculate par rates but at
the end of this learning module there
are nine questions and none of them ask
you to compute this par rate but one of
them says something like hey what's the
relationship between the par rate and
the uh spot rate and what you're going
to say is this all right I want you to
look 45 489 522 whoops I went the wrong
way what is that 45 49 525 what do you
want to say the par rates are going to
be pretty close just a little bit less
than that spot rate and that's probably
sufficient so my advice is to just get
out a piece of scratch paper and work
through the algebra here so that you can
do it you've done if you do it once then
you can do it on the exam if the
Institute asked you to calculate which
of course it can ask any Los question on
the exam but I think there are better
questions and The Institute probably
agrees with me better questions rather
than um asking you to compute the par
rate all right so we did spot we did par
let's go ahead and do forward rates so
let me give you the math uh that I give
my students all the time suppose you
have $100 today and you have a two-year
window and you're yield is 10% so what's
going to happen that 100 grows to 110 at
the end of year 1 and it grows to 121 at
the end of year two okay the question
then becomes what happens if you don't
want to buy a 2-year security for your
2-year window but two
consecutive one-year Securities so you
buy a one-year security now let's
suppose that that's 10% so that the end
of that first year you have 110 well
what do you expect that rate to be one
year from today well in order for there
to be no Arbitrage then you expect that
rate one year from today to be 10%
because you're going to turn that 110
into the 121 that means you're
indifferent between buying a 2-year
security or two consecutive oneyear
Securities so that rate that rate that
is you expect to get at the end of year
one that's called a forward rate some
people called an implied forward rate
that makes perfect sense some people
call it a forward
yield now in my example which was super
simple by the way um I had 10% and 10%
and 10% so that worked out but what if
it's not 10% and 10% what if it's 10%
and 11% and 12% and 19% ah so we're
going to use that equation there up at
the top right you see the formula where
look on the on the right hand side of
the equal sign we're going to have 1 + z
right there's our zero rate or our spot
rate and what we're going to do is we're
just going to say that has to be equal
to some other spot rate out there times
the implied forward rate now you know
look you have an A and A B and A B minus
a if that's confusing you uh let's go
ahead and work through an example so uh
so you can see how easy this is are you
ready threeyear spot rate is 3 and a
half% the foure year spot rate is 4% so
the question is what is the oneyear
forward rate expected to be three years
from today what do we know we know we
know that we can buy a fouryear secur
security and get 4% right four four four
four right over that four-year period or
we can buy a threeyear
security and get three and a half three
and a half and three and A2 and then
we'll buy a one-year security to match
the two time Horizons what is that one
year forward rate 3 years from today
well look at the math down at the bottom
this is so simple all we're going to do
is we're going to compound that
1.035 out three
times and then we're going to multiply
it by one plus that forward rate and
we're only going to have to compound it
out one time right because that's going
to be from year three to four and that
must equal I mean look that's an equal
sign there 1.04 raised to the 4th power
so if if you do the uh uh what are we
doing here just divide both sides by
1.035 raised to the thir power and then
well that's raised to the one so
whatever that is subtract one you get
5515 of course of course you mean think
about it if we buy a four-year Bond
we're getting 4% every year if we buy a
three-year Bond we're only getting three
and a half% every year right so in that
fourth year we're expecting interest
rates to go up we need to make up for
the fact that over those first three
years we're losing out right by a half
percent so it's not surprising that the
interest rates and that forward rate is
expected to
rise now what did we do earlier we
calculated the bond price using spot now
we can calculate the bond price using
forward rates all right so let's look in
the middle there there's a table so that
forward rate is 1 and A2 2.2 and 2 all
right so what we did is that we went
back and we computed all those from uh
the original uh the original 2.5% coupon
rate three-year Bond all right so there
are the one-year rates so how do we do
this well what we're going to do is
we're going to Simply go ahead and say
all right what we know is that forward
rate that First Rate 1 and a half%
that's the spot rate so look down on the
right hand side of the equal sign we're
going to take 2.5 and just ER 2.5% of
100 gives me
$2.50 divide that by the one-ear forward
rate but then we need to compound that
out every time we go forward so in the
second year we're getting $2.50 so we're
going to discount it at well the 1 and a
half% times the 2.2% of course we add
one to it to do the compounding and then
in year three you just multiply all
three of those together so you see the
sequence in the denominator so if the
Institute add asks you for calculate the
bond price you get the
10993 and all we're going to do is say
something like oh oh oh the in the
denominator we're going to make the
adjustment using forward rates so here's
seems to me I mean in my classes when I
teach this guaranteed question is
compute bond price using spot rates and
compute bond price using forward rate so
make sure you know both of
those now what did I say in that
introduction know the spot part and
forward rates the second part was know
the spot part and forward curves okay so
here we go we have colors here
coordinating but let's go ahead and just
look at the graph here you ready so time
to maturity on the horizontal axis
yields on the uh vertical axis so note
note that the spot that's the that's the
solid orange line and then the par rates
they're going to be what was that
Finance term I used a little bit ago
teeny we the par rates are going to be
just a little bit underneath
that because of the mathematics that we
went through in that previous slide so
that's a good thing to remember spot and
Par rates they're about the same uh par
rates will but the par Curve will be
just a little bit below and then the
forward rate that looks like in this
example that it's going to be above so
watch this so let's do the comparison of
the curves then we have the relationship
and then we're done all right so this is
super important important here all right
so spot rates show a positive trend
right so this is an upward sloping yield
curve the spot and the par they align
closely and then the forward rates are
higher than both the spot and the par
rates is that going to always be the
case well no no that's going to be the
case right on this leftand uh column
here the upward sloping spot curve but
let's skip over to the right the
downward sloping spot curve the par
curve is going to be just slightly above
it and the forward curve is going to be
uh below it when we have a flat curve
well we're going to go back to that
previous learning module where it's a
flat yield curve and then we're back to
the uh kindergarten model not par curve
is equal to spot curve forward curve
equals the spot curve forward curve
equals the par curve so remember in the
middle that's the kindergarten model
that we could have talked about in that
previous learning module but we didn't
really want to uh complexify things
until we got to this point here now we
start with that kindergarten model move
out to the side and then these things
should make perfect sense based on the
math that we did uh over the last you
know what has been 25 minutes or so of
uh recording so this was fun what did I
say to you earlier make sure you know
everything about spot par and forward
curve so I think we did a pretty good
job of summarizing all that um I want
you to go and spend 9 minutes now on
those problems at the end of this
learning module um there's one or two
questions in there that you're going to
have to think about some of the things
that um I talked about in this learning
module that will lead you to that
correct answer so hey thanks for
watching and good luck
studying
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[긴급진단]TMF, TLT 투자자분들은 꼭 보세요, '이 타이밍'을 확인해야 합니다
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