The basic technique of valuation under certainty is the so-called discounted cash flow models. The model simply expresses the value of a riskless asset as the present value of all its contractual and future flows calculated at the riskless rates. The process uses the set of discount rates that are applicable for each date of the future cash flows. The set of risk-free rates by maturity forms the term structure of risk-free rates. In practices such risk-free rates are derived from the prices of such risk-free assets such as Treasury bills and Government bonds. This section is a remainder of the valuation of future risk-free payoffs.

In a first stage, we use the zero coupon rates that apply to single cash flows. Then, we extend the discounting formulas to the yield-to-maturity (Ytm), and we extend the same conclusions to a multiple period framework with simple examples. The familiar discounted cash flow (DCF) model is first introduced used in a discrete time framework. The extension to continuous time is straightforward and reminded below.

Valuation in Discrete Time and the Rationale of Discounting

The value of an asset is the discounted value of the stream of future flows that it generates. The discount rates are market risk-free rates corresponding to the dates of the cash flows. The array of risk-free discount rates over all maturities forms the term structure of interest rates. Since they apply to single cash flows, they are also called zero-coupon rates because such zero-bonds have only one cash flow at maturity, all other intermediate payments, such as coupons for coupon bonds, being zero. Using the term structure of risk-free zero rates, the present value B(t, 7), as of t, of a stream of future flows Fk where k varies from t to the maturity T of the asset:

Discounting and borrowing or lending at market rates

FIGURE 11. 1 Discounting and borrowing or lending at market rates

The DCF rationale is directly related to financial transactions that would allow converting future cash flows into present cash flows and vice versa. The relevant discount rates are those that allow transferring flows across time through borrowing and lending. This is why market rates are relevant. For instance, consider a single flow at date 1. We can transfer it to today by borrowing against this flow, or borrowing an amount such that repayment exactly equals the future flow In the graph (Figure 11.1), up arrows are inflows and down arrows are outflows. The present value of X is obtained by borrowing today and repaying exactly X tomorrow. Capitalization transforms a present flow into a future flow. By lending today the present flow, the net flow today becomes zero and we get a future inflow at the future date equal to the proceeds of lending. The present value of 1000 at date 1 is 1000/(1 +yb), where yh is the borrowing rate. When lending, we use the lending rate. Hence, market rates are relevant for the DCF model.

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