Pricing bonds

• Main concepts and types of bonds;
• Comments on basic and advanced methods for pricing bonds.

This post discusses the main characteristics and the more common valuation models of bonds. Bonds are financial instruments issued by companies and governments (from municipalities to countries) as a form of borrowing by these organizations.

In principle, bonds are simple instruments for which an investor pays a determined amount to the issuer (of the bond). At the maturity of the contract, a principal value is paid by the issuer to the investor. The value of the bond must then be lower than or at maximum equal to the principal. The specification of a bond must contain at least the value and frequency of possible coupons (intermediate values paid by the issuer), and the maturity. Other characteristics may also be described such as callability and puttability, convertibility, compensations, and special periods (in order to be callable, for instance).

Bond genealogy

The simplest type is the one that does not pay any intermediary coupons, but only the principal at maturity, the so called zero-coupon bond. A coupon-bearing bond pays intermediary values which may be previously specified or based on some floating index. When based on a floating index, one must specifies a fixing and a payment date, which not necessarily should be equal.

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A callable bond gives the issuer the right but not the obligation to call the bond before maturity, paying the discounted value of the principal as well as some compensation to the holder for the previous contract cancelling. Usually, callable bonds are cheaper than regular ones, since the issuer has more right in this contract. Conversely, a puttable bond gives the holder the right but not the obligation to put the bond against the issuer earlier than the maturity. This type of bond is usually more expensive because it gives more right to the holder. An issuer may choose to call a bond when the interest rates in the market drop down. In this case, the issuer would cancel an existing debt and issue a cheaper one. A holder would choose to put a bond when the interest rates rise, in this case the holder could buy a more profitable bond afterwards.
Another common type of bond is the convertible one. These bonds contain special clauses that allows its conversion to stock shares of the company issuer. 

First steps

Similar to other instruments, like swaps, one can valuate a bond $B(t,T)$ at time $t$ and maturity $T$ by considering the present value of the corresponding cash flows. Assuming a constant (continuous compounding) interest rate $r$, the price (value) of the bond could be given by:

$B(t,T) = e^{-r(T-t)}N$

where $N$ is the principal value. A non constant interest rate could be handled with an integral over time. For a coupon-bearing bond, the value would be given by:

$B(t,T) = \sum_i e^{-r_i(T_i-t)}C_i + e^{-r_N(T-t)}N$

where $C_i$ denotes the i-th coupon. The last term in the right side denotes the payment of the principal value at maturity. In the second expression above, it is introduced the possibility of different zero-rates $r_i$ depending on the time to maturity of each coupon. This is a prelude for more complex analysis later on.
A very useful metric that can be "evaluated in a glimpse" is the bond yield $y$. This value is obtained considering a constant value of interest rate for the entire bond life time. It is worth noticing that there is a certain relaxation between yield and the actual interest rate. However, evaluate the problem using a single parameter is much more interesting from the practical point of view. Bond yield $y$ could be calculated by means of:

$B(t,T) = \sum_i e^{-y(T_i-t)}C_i + e^{-y(T_N-t)}N$

where the value of $y$ can be obtained by some iterative method. Notice that the market bond value $B(t,T)$ is inversely proportional to the bond yield, the same relation holds to the level of the interest rate. Therefore, if the level of the interest rate drops down, the current value of a bond increases. Conversely, if interests rise, bond values go down.
An interesting risk measurement for investment in bonds is obtained with a simple Taylor expansion of the relative bond value:

$\Delta B=\frac{\partial B}{\partial y}\Delta y + \frac{1}{2}\frac{\partial^2 B}{\partial y^2}\Delta y^2$

Actually, it is more common to see expression above written as:

$\Delta B=-BD\Delta y + \frac{1}{2}BC\Delta y^2$

where $D=\frac{1}{B}\frac{\partial B}{\partial y}$ is known as duration and $C=\frac{1}{B}\frac{\partial ^2B}{\partial y^2}$ is known as convexity. Duration and convexity are related to the sensibility of the bond price with the yield (or interest rate) value in a first and second order, respectively. Therefore, duration and convexity are measurements of risk of determined bond. One can estimate the change in the bond value for determined variations of the yield value. In a financial language, duration and convexity are related (but are not equal) to the greeks delta and gamma (briefly discussed here). Another useful interpretation is the following. Duration denotes the average time for a holder recovers the invested amount in the bond. Convexity indicates the variance of this estimate recovering time. In this sense, as higher the duration and convexity of a bond, higher is the associated risk. Coupon-bearing bonds, for instance, present lower duration and convexity compared to zero-coupon bonds and then with lower risk.

Stochastic interest rate

Previous section assumes a constant interest rate. In this section, this assumption is disregarded. In fact, the assumption hereafter is a stochastic interest rate. In principle, at the initial time one allegedly  has the information of the interest rate curve adopted by the market, therefore one can calculate the present value, that is, $B(0,T)$. However, what happens when one needs to calculate the bond price at a time $t<T$? In this case, a possible solution is the use of short-rate models. These models present a stochastic description of the instantaneous rate. Being stochastic, the present value of the bond must be obtained from an expectation operator, that is:

$B(t, T) = E[ exp(- \int_t^T r(u)du) | F(t)]$

where $E[... |F(t)]$ denotes the expected value given the information available at time $t$, which is indicated by the filtration $F(t)$.

There are several possible short rate models that could be applied in this situation. The choice is based on some market situation as well as the possibility of reuse for other interest rate instruments. A common (but not the only, of course) model is the CIR (Cox-Ingersoll-Ross) whose stochastic differential equation (SDE) can be written as:

$dr=\theta(\Gamma-r(t))dt + \sigma \sqrt{r(t)}dW$

Parameter $\Gamma$ indicates a long term average interest rate. $\theta$ denotes a mean rate of return towards $\Gamma$. The volatility $\sigma$ indicates the level of variability of the interest rate. Finally, $W$ denotes a Brownian motion in the risk-neutral measure (see this discussion using Black-Scholes equation). A similar stochastic process has already been discussed in a previous post, because of the mean reverting and stationary property.
Taking a close look in the expression of the CIR model above we can realize some interesting characteristics. The first term in the left hand side is called drift term of the SDE. In this case, one can see that when $r(t)$ is below $\Gamma$ the drift term is positive, making the process tend upwards, that is, the value of $r(t)$ increases towards the long term average. A similar process happens when $r(t)$ is larger than $\Gamma$. Parameter $\theta$ controls the "speed" of this mean return process.
The last term on the right hand side is known as diffusion term. The presence of this term characterizes a stochastic process. The Brownian motion term $dW$ describes the uncertainty of the process, mimicking the random behavior of the market. The volatility $\sigma$ indicates the magnitude of this uncertainty. Large values of $\sigma$ are directly associated with markets of large risk. At last, the term $\sqrt{r(t)}$ is the skew of the process. Particularly for this model, this term assures positive values for $r(t)$. When $r(t)$ is close to or even zero, the diffusion term becomes negligible with respect to the drift term. Without uncertainty, the process will tends to the long term average value, making certain non negative values.
Short-rate models as the one exemplified above actually describes instantaneous forward rate that are not quoted in the market. Moreover, short-term rates are calibrated using the time structure of the forward interest rate. Therefore, in this case, the "feeling" of the market is not so relevant in the model. At last, interest rates are quoted for determined accrual period, that is, forward interest rate (such as Libor, for instance) are discrete, not continuous. In this sense, one of the current top developments in financial engineering is the Libor market model. This complex model shall be topic for several posts in the future.

I hope you enjoyed the post!

Live long and prosper.

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Diogo de Moura Pedroso

LinkedIn: www.linkedin.com/in/diogomourapedroso