3.3 Discount Factors

In the equity model the price of a security reflects the uncertainty of the cash flow that it promises the investor. We saw that the no-arbitrage condition was equivalent to the existence of a price of $1 contingent claim on state [Maple Math] , [Maple Math] , for each [Maple Math] so that for every security

[Maple Math] .

(3.11)

As we explained in the first Chapter, [Maple Math] discounts the dollar of state [Maple Math] in two ways: First, the dollar is not a certain dollar; it is received only if state [Maple Math] occurs. (The discounting accounts for the riskiness of the cash flow.) Second, if state [Maple Math] occurs, the dollar is received in the future. Thus [Maple Math] takes care of both the time value of money and the uncertainty. We can therefore expect that the no-arbitrage condition in the debt market is equivalent to the existence of [Maple Math] satisfying the same equation. After all, our interpretation of the states of nature as time periods does not conceptually affect the mathematical formulation of the no-arbitrage condition. Indeed, our formulation of the maximiziation problem equation (3.5) and Definition 5 confirm the above analysis.

Thus, in a similar manner to the equity model, the
[Maple Math] here can be interpreted as the price of a dollar to be received in time [Maple Math] . The discount factors here, however, take care of the time value of money alone since the amount is to be received for sure. The similarities extend further. In the same manner in which we interpreted the basic building blocks of the equity securities as a dollar contingent on state [Maple Math] (state-contingent cash flows), here we can identify them as a dollar to be received in time [Maple Math] . Indeed, an examination of the bonds will verify that they are composed of the same basic units: a dollar to be received at time [Maple Math] .

Consider a bond maturing at time one and paying $105. It is composed of 105 units of the time one basic building block: a dollar to be received in time one. Suppose the same bond were to mature at time two, i.e., it pays $5 at time one and $105 at time two. Such a bond is composed out of 5 units of the time one building block and 105 units of the time two building block -- a dollar received at time two. Hence, in a parallel manner to the equity market, we should expect that the market assigns a price to a dollar to be received at time [Maple Math] . Let us denote these prices by [Maple Math] than these [Maple Math] 's would satisfy an equation for every bond [Maple Math] such as the following:

[Maple Math] .

(3.12)

There are some characteristics of the current market that do not exist in the equity market. We alluded to these a few paragraphs back. In the debt market, the [Maple Math] are the prices of one dollar at time [Maple Math] . We expect that in spite of the dollar being a sure amount obtainable at some future time [Maple Math] , its value (price) should be less than a dollar today. Having a dollar now is worth more than having it later, simply because by having it now one can invest it in a risk-free asset to generate more than a dollar in the future. Hence the [Maple Math] reflects the discounting of one dollar due to the time value of money.

The distinct feature of the debt market is a consquence of the fact that the time periods occur sequentially and only one state of nature occurs. A dollar from time one can be carried forward to time two, and at that time must be equal to at least a dollar. (If the dollar were invested at a positive rate of interest, it would be worth more.) We should expect therefore that if [Maple Math] then [Maple Math] . This is one property which the [Maple Math] 's do not enjoy in the model of the equity market. This property is referred to as the monotonicity condition of the discount factors . We should also imagine that [Maple Math] . After all, a dollar at time one should have less value than a dollar now. This is simply a time value of money argument. Footnote 6 Indeed we see that the solution to our example satisfies this condition.

The interpretation for equation (3.12) is identical to that of equation (1.7) of the equity market. Furthermore, the use of the discount factors in the context of the debt market is analogous to the use of the stochastic discount factors in the equity market. The [Maple Math] s, in the debt market facilitate valuation of cash flows across time periods exactly in the same manner as the stochastic discount factors allow the valuation of cash flows across states of nature.

Consider again the example which is summarized in Table 3.1. The NarbitB procedure confirmed that the no-arbitrage condition was satisfied for that market. Let us reexecute NarbitB applying it to this example in order to explain another part of the output from the procedure.

> NarbitB([[105,0,0],[10,110,0],[8,8,108]],[945/10,97,89]);

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

Since the no-arbitrage conditon is satisfied, we expect to find a solution to the system of equations below.

[Maple Math]

(3.13)

[Maple Math]

(3.14)

[Maple Math]

(3.15)

Furthermore, since the [Maple Math] s are prices, and given the time value of money arguments, we also expect that the [Maple Math] s will satisfy

[Maple Math] , [Maple Math] , [Maple Math] ,..., [Maple Math] , and [Maple Math] .

(3.16)

Let us submit thus to MAPLE and request that it solve the system of equations (3.13) and (3.16).

> solve ({d1*105=945/10,d1*10+d2*110=97,d1*8+d2*8+d3*108=89,d1<1, d2<d1,d3<d2,d3>0},{d1,d2,d3});

[Maple Math]

MAPLE confirmed that there is a solution to this system of equalities and inequalities. The reader may now compare this solution to the discount factors reported from the NarbitB procedure. They are the same (of course) and have the same meaning as the stochastic discount factors of the equity market. The meaning of the [Maple Math] s is as in the equity market. It is the price of $1 obtainable at time [Maple Math] . Furthermore, as in the equity market, the no-arbitrage condition is satisfied if and only if the following system of equalities and inequalities is consistent (i.e., a solution to it exists):

[Maple Math] +...+ [Maple Math] , [Maple Math] ,..., [Maple Math]

(3.17)

[Maple Math] , [Maple Math] ,..., [Maple Math]

(3.18)

[Maple Math] ,

(3.19)

[Maple Math] .

(3.20)


In fact, the existence of a solution to (3.17) can be used as a dual definition of the no-arbitrage condition. The Appendix will explore this point further. In the same manner as the
[Maple Math] 's are used in the equity market they are used in the debt market. We utilize the [Maple Math] 's to value different cash flows. The function Vdis, which is an output defined by the procedure NarbitB, allows us to value a given cash flow. Thus in our example, given the cash flow ( [Maple Math] ), we may value it by issuing the following command:

> Vdis([c1,c2,c3]);

[Maple Math]

Indeed, the Vdis procedure defined here acts in the same way as its equity market counterpart, albeit here we are valuing certain cash flows which will be obtained at different times in the future. In the equity market, we valued state-contingent cash flows. For example, we can ask, what should be the value of [Maple Math] in ( [Maple Math] ) if the present value of ( [Maple Math] ) is zero? This will be the solution to Vdis ([ [Maple Math] ])=0.

The reader may now try to value different cash flows across time periods using the Vdis function. Alternatively, it is also possible to define a new market using the NarbitB procedure and thus a new Vdis function which will be defined in such a way to allow the reader to value cash flows in the new market.

Note that for now the Vdis function allows the valuation of cash flows as long as the time of payments coincides with the payments of existing bonds in the market. Of course, in real-world markets, there may be a need to value a cash flow whose time of payment does not coincide with the cash flow from an existing bond. Hence, there is a need to extract, somehow, the discount factors for times which do not correspond to existing bond payments, i.e., for some time period [Maple Math] where [Maple Math] belongs to a continuous interval [Maple Math] . We shall soon see how this is done.

Footnotes

Footnote 6

In an exercise at the end of this Chapter, the reader is asked to explain why this must be the case using arbitrage arguments.

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