Читать книгу The Incomplete Currency - Marcello Minenna - Страница 14

Imagine we are holders of a bond of Bank A, at a variable rate, with a duration of only 6 months. In this experiment, the bank cannot fail. At maturity, therefore, we have the assurance that the bank will return the invested capital (€100) plus a coupon that pays a variable interest rate. The value of the coupon will be uncertain, and will depend on the level reached by the interest rate in 6 months. With many rates possible, many coupon values are possible. For example, in Figure 1.1, nine possible values are considered for the coupon paid: only once does it reach a very low value of around €0.20, once it has a value of €1, three times the coupon pays €1.50, twice there is a coupon of €2 and on two other occasions the coupon exceeds €2.

Figure 1.1 Possible realisations of the random coupon depending on the possible values of the interest rate

What is happening is that not all levels of the rate can be reached with the same probability. This is fairly intuitive: if we observe a rate of 1.6 % today, it is more probable that in 6 months the rate will be 1.7 % as opposed to 5 % and therefore that you will get a coupon of just €1.70 instead of €5. Now imagine studying the market data today and being able to assign each possible future interest rate a precise probability: the value of the coupon in 6 months is still uncertain, but we have developed an accurate estimation of the probability of gain, which is graphically represented by a bell-shaped curve defined in technical jargon of distribution probability (see Figure 1.2).

Figure 1.2 Probability distribution of the values at the maturity of a floating rate bond issued by Bank A

The bell-shaped curves which represent the probability distribution contain a wealth of information on the bond that we purchased from Bank A: studying this, we can now say that it is very probable (90 % represented by the central and lower areas) that the coupon that will be cashed will not be greater than €2 (total investment of €102); at the same time there is reasonable certainty, more than 90 % (central and upper areas) that my coupon will not be less than €1 (total investment of €101). It's not the same as 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