Before we really get into the meat of any posts I think it’s a good idea to start with some basic definitions and notation around interest rates. I have chosen some conventions which I think are clear and avoid confusion and in most cases consistent with other literature I have come across (though not all).
Although I provide definitions here and formulae for many of the terms I am only going to provide a few brief derivations of the formulas and other posts will deal with more involved derivations and concepts individually. We start with the most basic instruments, zero coupon bonds, then move onto a few different types of interest rates before finishing on a few definitions related to coupon paying bonds.
Mostly, this post is just a mind dump of various related and important definitions and formulas that I might want to refer to and update later!
Zero Coupon Bonds
In the above we will often be a bit lax with the notation and use where is fixed and known. Some important properties of ZCBs are:
- is always deterministic since the value is always known at time ;
- a ZCB is a trade-able contract;
Zero Coupon Bonds are the most basic of financial quantities and are used to build up values of future payments and of other interest rates. We can express ZCB prices in terms of other interest rates and often recover them in this way since financial markets typically quote interest rates rather than ZCB prices.
Spot Rates of Interest
At a high level a spot rate of interest is the constant rate of growth in value of cash between now (time ) and some future date with some compounding type and interval that is implied by ZCB prices. We start with simple interest and annual compounding then move onto to compound interest and different compounding intervals.
As with we will sometimes use where is known. With simple interest if we have an amount that earns simple interest at rate for a period of years and where we get paid interest at the end of each year then we have
Using this and a simple No-Arbitrage argument1See the annually compounded spot rate for an example. we can express the simple spot rate in terms of ZCB prices as
As with we will sometimes use . A simple No-Arbitrage argument allows us to arrive at a formula for in terms of . To do this our two portfolios are:
- Portfolio A: This has value at of i.e. we purchase a ZCB at ;
- Portfolio B: This has value at of , where . For this we hold cash which earns an annually compounded interest at rate ;
Both portfolios have a value of at time . By the NAP they must have the same value at time otherwise we could construct an arbitrage portfolio. For example if we construct a portfolio with zero value at time that has a non-zero value at time as
Where . Then
Where is the value of cash which has earned interest between and . We can do something similar for and so come to the conclusion
In the same way as with we can use a No-Arbitrage argument to arrive at the formula
The continuously compounded spot rate can be arrived at from the -thly spot rate by allowing to tend to infinity. In fact one definition of the exponent function for is
Using this definition and taking the limit as we get
To see this we can write as
and then let and use the definition of above and replace with . We will sometimes also use the notation for .
Forward Rates of Interest
The key difference between a forward and spot rate of interest is that a spot rate is the constant rate between now and some future date where a forward rate is a constant rate between some future date and some later future date which is implied by the current pattern of zero coupon bonds. We can again express it in simple, annual and continuous compounding and we will mainly want to relate it to ZCB prices as this then allows us to relate it to any other rates or types of compounding we choose (e.g. we can mix simple forward rates with continuous compounded spot rates).
Where it is clear what the time interval for is we will often use . If we use No-Arbitrage arguments to express this in terms of ZCB prices we get
In a similar way to simple forward rates we can express this in terms of ZCB prices as
In most cases we will be dealing with and so we can simplify the above relations for simple and annually compounded forward rates. One particularly useful relationship will be
The above can be used to collapse floating rate note prices down into simpler formulas and derive swap rate formulas.
Again we use a No-Arbitrage argument to write this in terms of ZCB prices
If both and then by substituting into the above and rearranging we can write
We rearrange this again to get
if we then take the limit as and use and as our common values we arrive at
Another formula for can be derived by using which gives . We then write
using the product rule to evaluate this and then simplifying gives us the expression
For now we are only going to deal with the simpler case of annual coupon paying bonds and for cases where a coupon has just been paid.
Note this is the price of the bond and may differ from the value of the bond (which may come from a required return and a required gross redemption yield).
Often the redemption amount will be the face value and we will want to express the coupon as a percentage of the face value of the bond.
When considering this definition there are a couple of ways we can write this. The par yield can be the value of such that
In this case the value of is a percentage i.e. . Another way I have seen this written/defined is as the value of such that
This is really just the same thing but where now rather than being a percentage. A useful formula we can derive from this (which allows us to easily obtain a yield curve from swap rates – which are par yields) is
where is the coupon rate on a bond of remaining term .
The GRY is essentially an average yield up to .
Footnotes [ + ]
|1.||⇧||See the annually compounded spot rate for an example.|