Fixed Income Analytics
10 chapters
1. Money Market and Bond Equivalent Yields
[Verse 1]
When you buy a Treasury bill or short-term note
The yield that's quoted might make you take note
There's different ways to calculate what you'll earn
Money market basis is the first to learn
[Chorus]
Three sixty basis, that's the money way
Three sixty five for bond equivalent today
Convert between them with a simple trick
Multiply or divide to make the numbers click
Money market low, bond equivalent high
Same investment, different ways to quantify
[Verse 2]
Money market yield uses three sixty days
Actual dollars earned, that's how it pays
Take your discount, divide by face amount
Then by days to maturity, that's the count
[Chorus]
Three sixty basis, that's the money way
Three sixty five for bond equivalent today
Convert between them with a simple trick
Multiply or divide to make the numbers click
Money market low, bond equivalent high
Same investment, different ways to quantify
[Verse 3]
Bond equivalent yield wants three sixty five
More days in the year keeps the rate alive
It also compounds, assumes you reinvest
Higher than money market, that's the test
[Bridge]
From money market to BEY we go
Multiply by three sixty five over three sixty, now you know
The other direction, flip the fraction around
Three sixty over three sixty five brings it down
[Chorus]
Three sixty basis, that's the money way
Three sixty five for bond equivalent today
Convert between them with a simple trick
Multiply or divide to make the numbers click
Money market low, bond equivalent high
Same investment, different ways to quantify
[Outro]
Short-term instruments, two ways to see
Choose your convention, money market or BEY
Remember the numbers, three sixty and five
Keep your yield calculations accurate and alive
2. Spot Rates and the Spot Rate Curve
[Verse 1]
When you buy a zero coupon bond today
And hold it till it comes due
The yield you earn along the way
That's the spot rate, pure and true
No coupons paid, no reinvestment fear
Just one payment at the end
The spot rate makes the pricing clear
On this you can depend
[Chorus]
Spot rates are pure, spot rates are clean
Zero coupon yields, know what I mean
Plot them on a curve from short to long
Term structure singing its interest song
S-P-O-T, one maturity
Not like YTM with complexity
[Verse 2]
YTM assumes you reinvest
Each coupon at the same rate
But spot rates put this myth to rest
They isolate each payment date
Six month spot, one year spot too
Each maturity stands alone
Building blocks for me and you
In the interest rate zone
[Chorus]
Spot rates are pure, spot rates are clean
Zero coupon yields, know what I mean
Plot them on a curve from short to long
Term structure singing its interest song
S-P-O-T, one maturity
Not like YTM with complexity
[Bridge]
From the curve we can derive
Forward rates for future years
Bootstrap method keeps alive
The term structure that appears
Normal, inverted, or flat
The curve tells us where rates are at
[Verse 3]
Each point upon the spot rate curve
Shows the yield for that one term
Risk-free rates that help us observe
How time makes interest rates squirm
Discount factors we can find
Price any bond with ease in mind
[Chorus]
Spot rates are pure, spot rates are clean
Zero coupon yields, know what I mean
Plot them on a curve from short to long
Term structure singing its interest song
S-P-O-T, one maturity
Building blocks for you and me
[Outro]
From short term to long term rates
The spot curve never lies
Term structure it demonstrates
Pure yields before your eyes
3. Forward Rates and Forward Rate Curves
[Verse 1]
When you look at bonds today with prices that you see
There's hidden information about future rates to be
The market tells a story of what interest might become
Forward rates are waiting there for those who do the sum
[Chorus]
Forward rates show future's face
Implied by bonds in the marketplace
From spot to forward we can trace
The path that interest rates will chase
Lock it in, project ahead
Forward curves show where we're led
[Verse 2]
Take two bonds with different terms, compare their yields today
The longer bond must compensate for risk along the way
If one year pays three percent and two years pays four
The second year forward rate is higher than before
[Chorus]
Forward rates show future's face
Implied by bonds in the marketplace
From spot to forward we can trace
The path that interest rates will chase
Lock it in, project ahead
Forward curves show where we're led
[Bridge]
One plus spot rate to the power of time
Equals products of forwards that perfectly rhyme
If rates go up the curve is steep
If rates go down the slope falls deep
[Verse 3]
Plot these forwards on a graph from short term to long
You'll see the forward rate curve telling market's song
Upward sloping means rates rise, downward means they fall
Flat curve says they'll stay the same through it all
[Chorus]
Forward rates show future's face
Implied by bonds in the marketplace
From spot to forward we can trace
The path that interest rates will chase
Lock it in, project ahead
Forward curves show where we're led
[Outro]
Future rates implied today
Forward curves light up the way
Spot and forward hand in hand
Help us understand
4. Par Curves and Yield Curve Relationships
[Verse 1]
Picture bonds all trading at their face value today
That's what par curves show us in their special way
When coupon rate equals the yield that's required
Par value and price are perfectly aligned
[Chorus]
Par above spot when the curve slopes up high
Par below spot when it's flat or declines
Forward rates dancing between the two lines
Par curves, spot curves, forward curves combine
Remember the pattern, it's there by design
Par curves, spot curves, forward curves combine
[Verse 2]
Spot rates are pure, single payment discount rates
Par rates need coupons to balance their weights
When longer rates rise above the shorter ones
Par curve sits higher, that's how math runs
[Chorus]
Par above spot when the curve slopes up high
Par below spot when it's flat or declines
Forward rates dancing between the two lines
Par curves, spot curves, forward curves combine
Remember the pattern, it's there by design
Par curves, spot curves, forward curves combine
[Bridge]
Forward rates implied between any two points
Connect all the curves at their pricing joints
When forwards rise steep, par stays more stable
Mathematical magic keeps markets able
[Verse 3]
Bootstrap the spot rates from par bond prices
Build up the curve through financial devices
Each rate depends on the others before
That's the foundation of fixed income's core
[Chorus]
Par above spot when the curve slopes up high
Par below spot when it's flat or declines
Forward rates dancing between the two lines
Par curves, spot curves, forward curves combine
Remember the pattern, it's there by design
Par curves, spot curves, forward curves combine
[Outro]
Three curves together tell the market's story
Par curves show us bond pricing glory
When yields are calling, remember this song
The math will guide you, you can't go wrong
5. Bootstrapping the Spot Rate Curve
[Verse 1]
Government bonds are trading on the street
With different maturities, the puzzle's incomplete
We know their prices and their coupon rates
But the spot rate curve is what we need to create
Start with the shortest bond, it's plain to see
Six months to maturity, that's our key
No coupons paid, just pure discount
The spot rate's easy to account
[Chorus]
Bootstrap up, step by step we climb
One spot rate at a time
Use the known to find unknown
Build the curve from ground up home
Bootstrap up, the method's clear
Strip away what interferes
From bond prices we derive
The spot rates that keep markets alive
[Verse 2]
Now we take the one-year government note
With coupon payments, here's what we wrote
The six-month spot rate that we found before
Helps us discount the coupon floor
Present value of each payment stream
Using known spots in our scheme
The final payment gets the rate
That makes the bond price calculate
[Chorus]
Bootstrap up, step by step we climb
One spot rate at a time
Use the known to find unknown
Build the curve from ground up home
Bootstrap up, the method's clear
Strip away what interferes
From bond prices we derive
The spot rates that keep markets alive
[Bridge]
Eighteen months, two years, and more
Each new bond opens up the door
Discount every coupon flow
With the spot rates that we know
The last payment tells the tale
Of the spot rate without fail
Sequential solving is the way
Building curves day by day
[Verse 3]
Zero-coupon equivalents we create
From coupon bonds at every date
The risk-free curve emerges clean
From this bootstrapping machine
Each maturity gets its place
In the term structure we embrace
From short to long, the rates align
A perfect risk-free rate design
[Chorus]
Bootstrap up, step by step we climb
One spot rate at a time
Use the known to find unknown
Build the curve from ground up home
Bootstrap up, the method's clear
Strip away what interferes
From bond prices we derive
The spot rates that keep markets alive
[Outro]
Government bonds show us the way
To build the curve we use each day
Bootstrap method, tried and true
The spot rate curve is waiting for you
6. Matrix Pricing for Illiquid Bonds
[Verse 1]
When bonds don't trade and prices hide
The market's thin, no bids in sight
Illiquid bonds need special care
We'll find their worth with methods fair
Look around for bonds alike
Same rating, term, and credit type
[Chorus]
Matrix pricing, find the way
Comparable bonds will light the day
Interpolate between the lines
Credit spread and yield combines
Matrix pricing, don't you see
Valued bonds through similarity
[Verse 2]
Take a bond that's five years long
But trading data's weak or wrong
Find a three-year, find a seven
Both are trading, that's our heaven
Draw a line between their yields
The five-year price this method yields
[Chorus]
Matrix pricing, find the way
Comparable bonds will light the day
Interpolate between the lines
Credit spread and yield combines
Matrix pricing, don't you see
Valued bonds through similarity
[Bridge]
Credit rating must align
Industry and structure fine
Government curve as our base
Add the spread to find the place
When the market's dark and quiet
Matrix keeps our books compliant
[Verse 3]
Say our bond yields six percent
But is that number really meant?
Check the sector, check the grade
See what similar bonds have paid
Benchmark plus the credit spread
That's how matrix bonds are read
[Chorus]
Matrix pricing, find the way
Comparable bonds will light the day
Interpolate between the lines
Credit spread and yield combines
Matrix pricing, don't you see
Valued bonds through similarity
[Outro]
When liquidity runs dry
Matrix pricing gets us by
Fair value from the bonds we know
That's how illiquid pricing flows
7. Arbitrage-Free Valuation Framework
[Verse 1]
In the world of bonds there's a golden rule
No free money floating in this market pool
When prices don't align across the yield curve line
Someone's making profit that should not be mine
Arbitrage means profits with no risk at all
But efficient markets won't let that ball fall
So we build a framework to keep prices fair
Making sure no free lunch is floating in the air
[Chorus]
No arbitrage, no arbitrage
Keep the prices in their cage
Same cash flows, same present value
That's the rule we must pursue
No arbitrage, no arbitrage
Turn the pricing theory page
If you find a gap today
Market forces smooth the way
[Verse 2]
Start with government bonds, the risk-free base
Build your yield curve step by step with grace
Each maturity point must fit just right
No gaps or jumps that don't look quite right
Bootstrap method pulls the rates apart
From the curve we see, we build from the start
Short rates first, then longer ones in line
Creating spot rates, pure and so refined
[Chorus]
No arbitrage, no arbitrage
Keep the prices in their cage
Same cash flows, same present value
That's the rule we must pursue
No arbitrage, no arbitrage
Turn the pricing theory page
If you find a gap today
Market forces smooth the way
[Bridge]
Forward rates are locked in tight
Today's curve shows tomorrow's light
If the future doesn't match
Traders jump to fill that gap
Replication is the key
Build the same cash flows you see
If two bonds pay the same
Then their prices are the same
[Verse 3]
Corporate bonds add credit spread on top
Risk premium makes the yield rate hop
But even then the framework still applies
No free money hiding in disguise
Swap curves, LIBOR, treasury bills
All must follow these pricing rules and skills
The framework works for every bond type
Keep arbitrage completely out of sight
[Chorus]
No arbitrage, no arbitrage
Keep the prices in their cage
Same cash flows, same present value
That's the rule we must pursue
No arbitrage, no arbitrage
Turn the pricing theory page
If you find a gap today
Market forces smooth the way
[Outro]
Consistent pricing across every trade
That's how efficient markets are made
No arbitrage framework shows the way
Fair bond prices every single day
8. Macaulay Duration Basics
[Verse 1]
When you buy a bond today, cash flows come your way
Coupons every period, principal at end of day
But when do you really get your money's worth back?
Macaulay shows the timing, keeps your knowledge on track
[Chorus]
Weighted average time, that's what we're looking for
Each cash flow has a weight, by present value's score
Macaulay Duration, measures timing right
Weighted average time, brings the concept to light
[Verse 2]
Take each coupon payment, discount to present day
Multiply by time periods, that's the weighting way
Add them all together, divide by bond's price
Weighted average timing, calculated precise
[Chorus]
Weighted average time, that's what we're looking for
Each cash flow has a weight, by present value's score
Macaulay Duration, measures timing right
Weighted average time, brings the concept to light
[Bridge]
Longer maturity means duration goes higher
Lower coupon rates make the number climb higher
Higher yields will pull the duration down
It's the balance point where cash flows can be found
[Verse 3]
Think of it like balance, on a see-saw beam
Each cash flow is a weight, at its time point it seems
Where's the fulcrum point, where it balances out?
That's Macaulay Duration, what it's all about
[Chorus]
Weighted average time, that's what we're looking for
Each cash flow has a weight, by present value's score
Macaulay Duration, measures timing right
Weighted average time, brings the concept to light
[Outro]
When rates change your bond price, duration shows the way
Weighted average timing, helps you plan your day
Macaulay Duration, now you understand
Weighted average time, in the palm of your hand
9. Modified Duration and Price Sensitivity
[Verse 1]
When interest rates start to climb or fall
Your bond's price moves, it's natural law
But how much will that price change be?
There's a measure called duration, you see
It's not just time until it's due
Modified duration shows what prices do
[Chorus]
Modified duration, price sensitivity
Divide by one plus yield, that's the key
Higher duration means more price swing
When rates change, it's a powerful thing
Mod D times rate change, price moves that way
Duration helps you sleep at night, trade by day
[Verse 2]
Take Macaulay duration first
Then modify it, here's how it works
Divide by one plus the yield rate
This adjustment seals your bond's fate
If duration's five and rates drop one percent
Your bond price rises, five percent it's sent
[Chorus]
Modified duration, price sensitivity
Divide by one plus yield, that's the key
Higher duration means more price swing
When rates change, it's a powerful thing
Mod D times rate change, price moves that way
Duration helps you sleep at night, trade by day
[Bridge]
Long-term bonds have higher duration
Short-term bonds, less price variation
Lower coupons mean more sensitivity
Higher coupons, more stability
It's an inverse relationship dance
When rates go up, bond prices can't advance
[Verse 3]
Portfolio managers use this tool
For hedging risk, it's pretty cool
Match your assets with liabilities
Duration matching brings tranquilities
Risk assessment becomes more clear
With modified duration always near
[Chorus]
Modified duration, price sensitivity
Divide by one plus yield, that's the key
Higher duration means more price swing
When rates change, it's a powerful thing
Mod D times rate change, price moves that way
Duration helps you sleep at night, trade by day
[Outro]
So remember when you're trading bonds today
Modified duration shows the price display
When interest rates shift up or down
This measure keeps you safe and sound
10. Money Duration and PVBP
[Verse 1]
Duration tells us how much prices change
When interest rates go up or down their range
But percentages don't show the dollar cost
We need to know exactly what we've lost
Modified duration gives us the percent
But money duration shows where dollars went
Take your portfolio value times duration
That's your money duration calculation
[Chorus]
Money duration, dollar sensation
Shows the real impact on your situation
PVBP, one basis point's fee
Price value tells you what the change will be
Money times duration equals dollar risk
PVBP makes it quick and crisp
[Verse 2]
A basis point is just one hundredth percent
Point zero one when rates are being bent
PVBP shows the dollar change you'll see
When rates move up by that small degree
Take a bond worth one million today
Duration is five, what does that say?
Money duration is five million flat
That's your dollar risk, imagine that
[Chorus]
Money duration, dollar sensation
Shows the real impact on your situation
PVBP, one basis point's fee
Price value tells you what the change will be
Money times duration equals dollar risk
PVBP makes it quick and crisp
[Bridge]
When rates rise one percent, you'll lose five million
When they fall the same, you gain that vision
But PVBP breaks it smaller still
One basis point change shows your bill
Fifty thousand lost per basis point
That's PVBP, right on the joint
Divide money duration by one hundred
Now rate changes can be easily numbered
[Verse 3]
Portfolio managers use these tools each day
To measure interest rate risk in every way
Money duration for the bigger swings
PVBP for the smaller things
Both convert duration to dollar terms
So you can see how your portfolio squirms
When interest rates decide to move around
These metrics keep you safe and sound
[Chorus]
Money duration, dollar sensation
Shows the real impact on your situation
PVBP, one basis point's fee
Price value tells you what the change will be
Money times duration equals dollar risk
PVBP makes it quick and crisp
[Outro]
From percentages to dollars clear
Money duration makes risk appear
PVBP for precision fine
Now you've got these tools in line
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