The Term Structure of Interest Rates Spot, Par, and Forward Curves (2024 CFA® Level I Exam – FI 9)

Summary

The 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

00:00

📚 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.

05:02

📈 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.

10:03

🔢 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.

15:06

🔄 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.

20:08

📉 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.

Keywords

💡Spot Rates

Spot rates, often referred to as zero rates in the video, are the interest rates applied to a single-payment zero coupon bond if it were purchased today and held to maturity. In the context of the video, spot rates are used to discount future cash flows from a bond to determine its present value. For instance, each cash flow of the bond (like annual coupon payments) is discounted at a unique spot rate corresponding to its specific time to maturity, emphasizing the differentiation in interest rates for different time frames.

💡Par Rates

Par rates are the interest rates that, when used to discount a bond's future cash flows, equate the bond’s price to its face (or par) value. The video explores how par rates ensure that a bond priced at its face value offers a return exactly equivalent to its coupon rate. Jim, the instructor, explains how to calculate these rates using algebraic manipulation of the spot rates, helping ensure that the bond’s discounted cash flows add up to exactly $100 (its par value).

💡Forward Rates

Forward rates are projected future interest rates derived from spot rates, representing the expected return on a bond if invested in the future rather than today. The video discusses forward rates in the context of no-arbitrage conditions, where they help illustrate the expected yields on future investments. Jim explains how to calculate forward rates by determining the rate needed in the future to equate investing in shorter-term bonds consecutively to investing in a longer-term bond directly.

💡Yield Curve

The yield curve graphically represents the relationship between interest rates (or yields) and different time maturities. In the video, Jim outlines different shapes of the yield curve (upward sloping, downward sloping, flat) and explains how these shapes imply different interest rate scenarios over time. For instance, an upward sloping curve indicates higher interest rates for longer-term investments compared to shorter-term ones.

💡Arbitrage Free

Arbitrage free refers to a pricing model that does not allow for risk-free profit through arbitrage. Jim uses this term to describe the theoretical underpinnings of bond pricing that ensure no opportunity to make a risk-free profit exists by exploiting price discrepancies between theoretically equivalent financial instruments. This concept reinforces the validity of using spot, par, and forward rates to model bond prices accurately.

💡Coupon Rate

The coupon rate is the annual interest rate paid by the bond issuer on the bond's face value. The video uses the coupon rate in examples to calculate the annual payments bondholders receive. For example, a bond with a 10% coupon rate and a $100 par value would pay $10 annually. This rate is crucial for calculating the present values of these cash flows using different interest rate models.

💡Par Value

Par value is the principal amount of a bond that is repaid to the bondholder at maturity. In the video, Jim frequently references this term when discussing the repayment of the bond’s face value at the end of its term, alongside the final coupon payment. Par value is central to bond pricing formulas where it is discounted back to the present using respective rates.

💡Zero Coupon Bond

A zero coupon bond is a bond that does not make periodic interest payments and is instead issued at a discount to its par value and pays its face value at maturity. The video uses this concept to explain how each future cash flow of a regular coupon-bearing bond can be treated as a separate zero coupon bond for the purposes of valuation.

💡Yield to Maturity

Yield to Maturity (YTM) is the total return anticipated on a bond if the bond is held until it matures. In the video, YTM is used as a critical measure for calculating the present value of all future cash flows from the bond, assuming the rate remains unchanged over the bond’s life, underpinning the importance of understanding different rate models.

💡Macroeconomic Conditions

Macroeconomic conditions refer to the overall state of the economy which can influence interest rates and bond prices. In the video, Jim mentions how different macroeconomic conditions can affect interest rates in each year of a bond's life. This reference helps to contextualize the importance of using spot, par, and forward rates to adapt the bond valuation to real-world economic fluctuations.

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