# Bond valuation

Bond valuation is the process by which one may determine the fairest pricing at which to buy a bond. Just as with all of the various types of securities and capital investments, what is theoretically a fair value for any kind of bond is based on how much value the stream of cash flow it generates produces, as well as the value it is expected to produce going forward. Because of this, a bond’s value level is garnered through the discounting process of the expected cash flow level compared to what is present right now by way of using appropriate rates of discounting. This is a process that is known as bond reckoning. As a practical measurement, this rate of discount is typically determined through referencing other kinds of instruments that are of a similar nature, so long as there actually are similar kinds of financial instruments in existence. For any given price, there are several different measures of the yield.

Since some bonds come with options embedded into them, this level of valuation is a more challenging factor, and as such it combines the discount with a level of option pricing. Judging by what kind of option is present, the price of the option as it is calculated can either be added to the price of the straight portion of the bond or subtracted from it. After these calculations have taken place, the bond is said to have been valued.

The Valuation of the Bond

As it was stated in the above section, a straight bond is one that does not have any options embedded into it. The most fair pricing of what is called a straight bond is most often found out through discounting the cash flow that is expected to come out of it at a discount rate that is seen as appropriate. The general formula that applies most often is initially talked about. While this relationship with the present value does reflect one potential approach to deciding how much a bond is reasonably valued at, as a practical matter the price of a bond is typically decided through referencing other, more liquid types of financial instruments. The most common types of approaches are known as arbitrage free pricing and relative pricing, and these will be discussed at a later point in this article. When it is necessary to recognize that there will never be total certainty in the interest rates of the future and that no discount rate can ever truly be represented by one number that remains static, the use of stochastic calculus is employed from time to time. Times when this recognition is important include the times when someone writes an option on a bond.

In times when the price of a bond on the market is lower than that of its par or face value, this is known as the bond selling at a discount. By contrast, if the price within the market for the bond is one higher than its original face value, this is known as the bond being sold at a premium. The relationships that connect the price and the yield are discussed in more detail below.

The Approach of Present Value

The formula for calculating the price of a bond uses a variable defined as PV, or the present value. For the purposes of this formula, one may assume that only recently a coupon was paid. At other times during the repayment process, there are adjustments that one can make.

The F is the value on the bond’s face, whereas iF is the contracted rate of interest. The periodic payment necessary for interest or the coupon payment is C. The total of how many payments must be made is N. The market rate of interest, the yield required of the bond or the appropriate level of yield until the bond matures is known as i. M is the level of value that the bond will hold at maturity, which is typically the same as the value on the bond’s face. Finally, the market’s price for the bond is known as P.

The Approach of Relative Pricing

This is an extension of the approach that was mentioned above, and it goes by a sort of benchmark that determines how much the bond is going to cost. As a typical rule, a form of government security is the benchmark determinant. This relates to relative valuation. In this sense, the amount of the bond’s yield upon its maturity can be found through the credit rating that underlies the bond itself, with relation given to some type of government security that has a similar time until it reaches its maturity or the total duration of the bond. This difference is known as the credit spread. For bonds that are of a higher qualitative level, a smaller amount of spread exists between such a bond’s minimal level of return and the yield to maturity of a benchmark type of security. The requirement for such a return is utilized for a discount on the bond’s cash flow, which replaces i within the formula given in order to find the price.

Approaching it Arbitrage Free Style

There is an option that is different from the ones that were mentioned above, which essentially thinks of a bond as a package of both the coupon and face value cash flows. In this way, an individual can consider a bond to be a no coupon debt instrument that has cash flow which will essentially mature on the date when the entity issuing the bond pays the value. As a result of this, instead of going by only one discounted rate, an individual can utilize several different rates to calculate a discount, with each type of cash flow being discounted at a unique rate of its own. Here, every type of cash flow is distinctly discounted but at the same rate of a no coupon type of bond that corresponds to the entire bond’s date of completion. This also carries the same credit spread and uses an equal level of credit worthiness when it is possible to do so. The same issuer’s other bonds can also be used for this comparison.

Through the use of such an approach, the price of any given bond would be best to reflect what is known as an arbitrage free type of pricing, since any deviance away from such a price would undoubtedly be exploited and bonds would rapidly be repriced to their correct levels of value. In such a case, one applies what is known logically as a rational price that relates to any kinds of assets where the cash flow levels are identical. For some detail, the coupon amounts and the coupon dates of a bond is known to some degree of certainty. As a result of this, either a fractional amount of or a multiple of the no coupon type of bond that will correspond to the bond’s date of final pay off, and this can be made specific in order to produce levels of cash flow that are exactly the same as the bond produces. As a result of this, today’s price for any given bond has no choice but to be the same as the total amount of every type of cash flow it produces at a discount equal to the discount rate that is hinted at through the valuation of a ZCB that corresponds to this value. If such a thing were not the case in the first place, the arbitrageur would be able to finance such a purchase in any kind of bond instance or ZCB based on which option was the least expensive through simply short selling the other one in the pair and then combining the zero figures or the coupons themselves to meet any cash flow requirements that are necessary. After this, the no risk or pro forma level of arbitrage profit would be however much difference existed between the values of these two types of assets.

The Approach of Stochastic Calculus

When an individual either models an option on a bond or any other kind of IRD or interest rate derivative, the individual is wise to note that the rates of interest that will come in the future can never be completely known. Because of this natural level of uncertainty, the discounted rate or rates that are mentioned in the sections above, regardless of whether they are for every coupon or for any individual coupons, can never be deterministically represented by a single fixed number. When one wants to know beyond the fixed numbers, it is wise to use stochastic calculus.

There is a partially differential equation that determines the calculus amount with any zero coupon bond. There are elements of probabilities that are neutral to risk, and there are random variables that represent the discount rate. This can also be related to Martingale pricing. The analyst needs to select a short rate model that he or he wishes to use in determining the bond price. Among some of the different approaches, there are the Chen model, the framework of HJM and that of Black-Derman-Toy.

Every model has the potential to have both a solution within a closed form. However, in some cases it may also be necessary to use either a simulation or a lattice of the model’s implementation in order to find the solution. This can be related to Jamshidian’s trick.

The Dirty Price and the Clean Price

In situations where a bond’s value is not precise on the date of its coupon, the price that is calculated when one utilizes any of the methods above is going to use the amount of interest accrued. This involves making use of any level of interest that may be due to the individual who owns the bond after the prior coupon date, which is a process that is known as the convention of day count. As a general rule, what is known as the clean price is generally a more stable one than the dirty price. The dirty price is called the dirty price, the cash price or even the all in price of the bond. By contrast to this, the clean price is what the bond is said to be worth without adding in any of the interest that has thus far been paid or added up to be paid later on. Naturally, a dirty price is going to drop at the moment the bond becomes ex interest and there is not a coupon payment coming. A lot of markets will only quote a bond’s price on its clean basis. After such a purchase is squared away, the level of interest that has been accrued becomes added to the clean price’s quote in order to get to the actual amount that must be paid.