having a crystal ball, but certainly for the saver it's a big step forward in terms of the awareness of the benefits of his investment.
However, in order to know the worth of my investment today, knowing the probability distribution is indeed necessary, but still not enough. There are in fact two problems to consider: (1) the distribution assigns many event probabilities at many possible values, but I need just the one value; (2) the distribution describes the coupons obtainable in 6 months, but I'm interested in a valuation today. The operators solve problem (2) by discounting the possible values at maturity by the time value of money, and problem (1) by taking a simple average of all possible values of the investment, once discounted (see Figure 1.3).
Figure 1.3 Calculation of the fair price of a 6-month floating-rate bond issued by Bank A
The number obtained following this procedure is the fair price at which the market, i.e. the whole set of financial operators, values the bond of Bank A. This price is unique because all the operators use the same procedure to calculate it, and objective because the estimate of the probability distribution of the final values of the bond is based on market data which all operators can access.
Of course, this does not mean that I cannot sell my bond for a lower price, for example 97; if I have an immediate need for money I will probably be willing to accept lower figures with the understanding that the “right price” is 100 and that the difference should be considered as a real loss. This understanding is taken for granted among the professionals, but unfortunately it is not part of the wealth of knowledge of the average saver; an unfair bank could well sell a bond which has a fair price of 93 to Mr Smith, for example asking him to pay 100, counting on the fact that Mr Smith doesn't have the tools to “understand” the benefits of the investment. If our saver was able to read the information of the probability distribution and the fair price in an understandable manner, the unfair bank would have little chance of placing the bond to the investor.
In the above example, to understand the relationship between probability distributions, risks and fair price, we have analysed a very simple bond, but the procedure stands as valid for any kind of financial product available on the market. In fact, it is precisely through observation and the proper reworking of the probability distribution that financial products are engineered.
In Figure 1.4, the probability distribution is constructed and the fair price of a bond is calculated, with a maturity of two years and paying four semi-annual coupons, based on the dynamics of our interest rate. As we can see, with the exception of the numbers of coupons considered, nothing changes in the valuation procedure previously described. In fact, in correspondence to a certain number of possibilities of the interest rate (first panel), we have different probability distributions for the four coupons every 6 months (second panel); adding these coupons and the principal returned at maturity, we obtain the probability distribution of the bond. Once this probability distribution is obtained, the possible values of the bond are discounted in order to take into account the time value of money, and finally the average of these discounted values is calculated (third panel); the only value that emerges from this procedure is the fair price of the financial product at stake.
Figure 1.4 Probability distribution of the values at maturity of a 2-year floating-rate bond issued by Bank A and calculation of the fair price
1.1.2 Swap Rate of a Floating Rate Bond
A floating rate bond like the one described in Figure 1.4 has uncertain results by definition, given that it is not possible to know beforehand the actual return that the investor will get; conversely, a bond at a fixed rate, such as a government bond, pays the same coupon regardless of changing market conditions. At a first reading, the two investments are therefore not comparable. However, the professional financial operators still have the need to compare the fixed rate with the floating rate transactions, and they do so by calculating a fixed rate that is representative of the operation at a variable rate: the swap rate. Let's try to understand this further.
Let's reconsider the 2-year floating rate bond issued by Bank A in Figure 1.4. The fair price of this bond is now 100. Now let's try to answer this question: given a fixed coupon bond with the same number of bond coupons from Bank A (four), which fixed interest rate should I pay to have a fair price equal to 100, that is, the same as our floating-rate bond?
Imagine being able to calculate this fixed rate and obtain a value equal to 1.4 %. Through this indicator we are saying that the holder of the floating rate bond will get on average the same return as the holder of a bond with an annual fixed coupon rate of 1.4 %; the bonds are different and will yield differently, but for professionals the two bonds are considered equivalent (always on average) for the purposes of comparison, so much so that they have the same value. In the first panel in Figure 1.5, the horizontal line represents the (fixed) swap rate in comparison to the possible developments of the floating rate. In the second panel, the fixed coupon corresponding to the swap rate is represented by the horizontal line.
Figure 1.5 Calculation of the swap rate of a 2-year floating-rate bond issued by Bank A
Let's take a last example and consider government bonds: if I have a CCT (the standard Italian floating-rate bond) and a BTP (the standard fixed rate bond) sharing the same maturity, and the bonds have the same fair price, the interest rate paid by the BTP will be the swap rate of the CCT.
Swap rates are very helpful for professionals because they can condense into one number for each contractual maturity (the so-called interest rate curve) all information relating also to floating interest rates. For now we have considered just one issuer at time (Bank A or the Italian Republic). Let's try to complicate things for a minute: consider a set of other banks belonging to the same banking system, like the European one (Eurosystem); by averaging the swap rates of every issuer, it is possible to get an image of the state of the banking system as a whole through the publication of a single interest rate curve.
We will return to this argument when it is time to analyse the functioning of the European banking system.
1.1.3 The Credit Risk
From the arguments made in the previous section it is clear that every bank can calculate its own swap rate according to variable interest rates that it pays, and these swap rates can be different according to the issuer.
How do we explain these differences?
As usual, we start with a very simple example. We have a very solid issuer, which basically cannot fail (e.g. in this historical period, the market considers that of Germany). In this case, applying the methodology previously described, it is fairly straightforward to calculate the swap rate for this issuer (1.1 % per annum). Since the swap rate is a fixed rate, we can construct the probability distribution of the bond of our solid issuer (Bond D) which pays the swap rate of 1.1 % per annum. Let's have a look.
The probability distribution calculated in the first panel of Figure 1.6 is rotated and shown in an enlarged form in the second panel. On the horizontal axis of the blue figure, the possible values of the bond at maturity are plotted, while on the vertical axis it is shown how often these possible values are going to be achieved. From the analysis of the distribution it is clear that investors at maturity will clearly get back the capital invested (€100), inclusive of the accrued coupons that are based on the swap rate of 1.1 % per annum (represented by the spread of the distribution around the value