Interest Rate and Bond Definitions

Interest Rates and Bonds Featured

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

Definition (Zero Coupon Bond Price)
A Zero Coupon Bond (ZCB) is a trade-able contract that pays 1 unit at maturity and has no other payments before maturity (i.e. no coupons). We use the notation P(t, T) for the price at time t of a ZCB that matures at time T (a ZCB with time to maturity \tau = T-t years).

In the above we will often be a bit lax with the notation and use P_t = P(t, T) where T is fixed and known. Some important properties of ZCBs are:

  • P(t, T) is always deterministic since the value is always known at time T;
  • P(T, T) = 1;
  • 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 t) and some future date T 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.

Definition (Simply Compounded Spot Rate)
The simply compounded \tau year spot rate prevailing at time t is the constant simple interest rate implied by P(t, T). We use the notation L(t, T).

As with P_t we will sometimes use L_t where T is known.  With simple interest if we have an amount C_t that earns simple interest at rate L_t for a period of \tau = T-t years and where we get paid interest at the end of each year then we have

    \begin{displaymath} C_T = C_t + \underbrace{L_t C_t + L_t C_t + \cdots + L_t C_t}_{\tau} = C_t \left( 1 + L_t \tau \right) \end{displaymath}

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

    \begin{displaymath} P(t,T) = \left( \frac{1}{1 + L(t, T) \tau} \right) \Rightarrow L(t, T) := \frac{1 - P(t,T)}{P(t,T) \tau} \end{displaymath}

Definition (Annually Compounded Spot Rate)
The annually compounded \tau year spot rate prevailing at time t is the constant annually compounded interest rate implied by P(t, T). We use the notation y(t, T).

As with P_t we will sometimes use y_t.  A simple No-Arbitrage argument allows us to arrive at a formula for P_t in terms of y_t.  To do this our two portfolios are:

  • Portfolio A: This has value at t of V_t^A = P_t i.e. we purchase a ZCB at t;
  • Portfolio B: This has value at t of V_t^B = \left( \frac{1}{1 + y_t} \right)^\tau, where \tau = T - t.  For this we hold cash which earns an annually compounded interest at rate y_t;

Both portfolios have a value of 1 at time T.  By the NAP they must have the same value at time t otherwise we could construct an arbitrage portfolio.  For example if V_t^A < V_t^B we construct a portfolio with zero value at time t that has a non-zero value at time T as

    \begin{displaymath} V_t^C = \underbrace{V_t^A - V_t^B}_{\text{assets}} + \underbrace{\varepsilon_t}_{\text{cash}} = 0 \end{displaymath}

Where \varepsilon_t > 0.  Then

    \begin{displaymath} V_T^C = \underbrace{V_T^A - V_T^B}_{=0} + \varepsilon_T \geq 0 \end{displaymath}

Where \varepsilon_T \geq 0 is the value of cash \varepsilon_t which has earned interest between t and T.   We can do something similar for V_t^A > V_t^B and so come to the conclusion

    \begin{displaymath} P(t,T) = \left( \frac{1}{1 + y(t, T)} \right)^\tau \Rightarrow y(t, T) := \frac{1}{P(t,T)^{1/\tau}} - 1 = \frac{1 - P(t,T)^{1/\tau}}{P(t,T)^{1/\tau}} \end{displaymath}

Definition (m-thly Compounded Spot Rate)
The m-thly compounded \tau year spot rate prevailing at time t is the constant interest rate, with compounding m times per year, implied by P(t, T). We use the notation y^{(m)}(t, T).

In the same way as with y(t, T) we can use a No-Arbitrage argument to arrive at the formula

    \begin{displaymath} P(t,T) = \left( \frac{1}{1 + \frac{y^{(m)}(t, T)}{m}} \right)^{m \tau} \Rightarrow y^{(m)}(t, T) := \frac{m}{P(t,T)^{1/(m \tau)}} - m = m \left( \frac{1 - P(t,T)^{1/m \tau}}{P(t,T)^{1/m \tau}}\right) \end{displaymath}

Definition (Continuously Compounded Spot Rate)
The continuously compounded \tau year spot rate prevailing at time t is the constant and continuously compounded interest rate implied by P(t, T). We use the notation r(t, T).

The continuously compounded spot rate can be arrived at from the m-thly spot rate by allowing m to tend to infinity. In fact one definition of the exponent function for x \in \mathbb{R} is

    \begin{displaymath} \e^x := \lim_{p \to \infty} \left( 1 + \frac{x}{p}\right)^p \end{displaymath}

Using this definition and taking the limit as m \to \infty we get

    \begin{displaymath} P(t, T) = \e^{-r(t, T)\tau} \Rightarrow r(t,T) := - \frac{\ln P(t, T)}{\tau} \end{displaymath}

To see this we can write P(t,T) as

    \begin{displaymath} P(t,T) = \left[ \left( 1 + \frac{y^{(m)}(t, T)}{m}} \right)^{m} \right]^{-\tau} \end{displaymath}

and then let m \to \infty and use the definition of \e^x above and replace y^{(m)}(t, T) with r(t, T).  We will sometimes also use the notation \delta_t for r(t, T).

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).

Definition (Simply Compounded Forward Rate)
The simply compounded forward rate prevailing at time t, for borrowing between T_1 > t and T_2 > T_1, is the constant simple interest rate between T1 and T_2 implied by the pattern of ZCB prices at time t.  We use the notation F(t; T_1, T_2).

Where it is clear what the time interval for F(t; T_1, T_2) is we will often use F_t.  If we use No-Arbitrage arguments to express this in terms of ZCB prices we get

    \begin{displaymath} P\left(t, T_1\right) \left( \frac{1}{1 +F(t; T_1, T_2) \left(T_2 - T_1 \right)}\right) = P\left( t, T_2 \right) \Rightarrow F(t; T_1, T_2) = \left[ \frac{P(t, T_1)}{P(t, T_2)} - 1\right] \left( \frac{1}{T_2 - T_1}\right) \end{displaymath}

Definition (Annually Compounded Forward Rate)
The annually compounded forward rate prevailing at time t, for borrowing between T_1 > t and T_2 > T_1, is the constant annually compounded interest rate between T1 and T_2 implied by the pattern of ZCB prices at time t.  We use the notation g(t; T_1, T_2).

In a similar way to simple forward rates we can express this in terms of ZCB prices as

    \begin{displaymath} P\left(t, T_1\right) \left( \frac{1}{1 +g(t; T_1, T_2) }\right) ^{\left(T_2 - T_1 \right)} = P\left( t, T_2 \right) \Rightarrow g(t; T_1, T_2) = \left[ \frac{P(t, T_1)}{P(t, T_2)}\right]^{\left( \frac{1}{T_2 - T_1}\right)} - 1 \end{displaymath}

In most cases we will be dealing with T_2 - T_1 = 1 and so we can simplify the above relations for simple and annually compounded forward rates.  One particularly useful relationship will be

    \begin{displaymath} P_t \left( \frac{1}{1 + g_t}\right) = P_{t+1} \Rightarrow g_t P_{t+1} = P_t - P_{t+1} \end{displaymath}

The above can be used to collapse floating rate note prices down into simpler formulas and derive swap rate formulas.

Definition (Continuously Compounded Forward Rate)
The continuously compounded forward rate prevailing at time t, for borrowing between T_1 > t and T_2 > T_1, is the is the constant and continuously compounded implied by the pattern of ZCB prices at time t.  We use the notation f(t; T_1, T_2)

Again we use a No-Arbitrage argument to write this in terms of ZCB prices

    \begin{displaymath} P\left(t, T_1\right) \e ^{-f(t; T_1, T_2) (T_2 - T_1)} = P\left( t, T_2 \right) \Rightarrow f(t; T_1, T_2) = \left( \frac{1}{T_2 - T_1}\right) \ln \left[ \frac{P(t, T_1)}{P(t, T_2)}\right] } \end{displaymath}

If both P(t, T_1) = \e^{-y_1 T_1} and P(t, T_2) = \e^{-y_2 T_2} then by substituting into the above and rearranging we can write

    \begin{displaymath} f(t; T_1, T_2) = \frac{y_2 T_2 - y_1 T_1}{T_2 - T_1}. \end{displaymath}

We rearrange this again to get

    \begin{displaymath} f(t; T_1, T_2) = y_2 + T_1 \frac{y_2 - y_1}{T_2 - T_1} \end{displaymath}

if we then take the limit as T_2 \to T_1 and use y and T as our common values we arrive at

    \begin{displaymath} f(t; T_1, T_2) = y + T\frac{\d y}{\d T}. \end{displaymath}

Another formula for f(t; T_1, T_2) can be derived by using P_T = \e^{-yT} which gives y = - \frac{1}{T}\ln P_T.  We then write

    \begin{displaymath} f(t; T_1, T_2) = y + T \frac{\d y}{\d T} = - \left[ \frac{1}{T} \ln P_T + T \frac{\d}{\d y} \left( \frac{\ln P_T}{T}\right)\right] \end{displaymath}

using the product rule to evaluate this and then simplifying gives us the expression

    \begin{displaymath} f(t; T_1, T_2) = - \frac{\d}{\d T} \ln P_T. \end{displaymath}

Bonds

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.

Definition (Price of a Coupon Paying Bond)
The price of a coupon paying bond, with face value F, constant coupon amounts of C, redemption amount R and a remaining term of n years is denoted B and is given by
B := C \left\{ P_1 + P_2 + \cdots + P_n \right\} + R P_n.

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.

Definition (Par Yield of a Bond)
The par yield of a bond is the constant coupon rate (not amount) which causes the bond to be priced at par value when using the prevailing set of ZCB prices.

When considering this definition there are a couple of ways we can write this.  The par yield can be the value of c such that

    \begin{displaymath} 1 := c \left( \sum_{k=1}^n P_k \right) + P_n. \end{displaymath}

In this case the value of c is a percentage i.e. 0 \leq c \leq 1.  Another way I have seen this written/defined is as the value of c^* such that

    \begin{displaymath} 100 := c^* \left( \sum_{k=1}^n P_k \right) + 100 P_n. \end{displaymath}

This is really just the same thing but where now 0 \leq c^* \leq 100 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

    \begin{displaymath} P_n = \frac{1 - c_n \sum_{k=1}^{n-1} P_k}{1 + c_n} \end{displaymath}

where c_n is the coupon rate on a bond of remaining term n.

Definition (Gross Redemption Yield)
The gross redemption yield (GRY) of a bond is the constant interest rate y (in this case annually compounded) such that

    \begin{displaymath} B = C a_{\angle{n}}^{@ y} + RP_n^{@y}. \end{displaymath}

The GRY is essentially an average yield up to n.

 

Footnotes   [ + ]

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