Discounting, risk-free rate

A short-rate model can be used to derive a complete term structure. We can illustrate this by showing how the model can be used to price discount bonds of any maturity. The derivation is not shown here. Let P t, T) be the price of a risk-free zero-coupon bond at time t maturing at time T that has a maturity value of 1. This price is a random process, although we know that the price at time T will be 1. Assume that an investor holds this bond, which has been financed by borrowing funds of value C,. Therefore, at any time t the value of the short cash position must be C,= —P(t, T) otherwise, there would be an arbitrage position. The value of the short cash position is growing at a rate dictated by the short-term risk-free rate r, and this rate is given by... 

Under risk-neutrality assumption, the most appropriate discount rate is the risk-free rate. The model is more sensitive to the change of recovery rates, while less sensitive to the change in interest rates. If we consider a zero-coupon bond rated R with maturity at time T, the price is given by Equation (8.28) ... 

Therefore, the discount rate is a risk-free rate rf or risky rate depending on the hedge ratio ... 

In the first case the right discount rate to apply is the risk-free rate equal to 1.04%, while in the second case is the risky rate equal to 8.04%. Figure 9.30 shows the hedge ratio at each node. 

Conversely, if the option is held the value is discounted at each node at the risk-free rate of the binomial tree. It is given by (11.8) ... 

Here each premium P is calculated on the notional amount and multiplied by its appropriate daycount fraction. The resulting cash flow is then discounted at the risk-free rate and finally multiplied by its survival probability, that is, the probability that the relevant preminm payment will actually take place, since in the event of a default all future premium payments will be cancelled. 

Certainty equivalent approach (Keown etal., 2002) In this approach a certainty equivalent is defined. This equivalent is the amount of cash required with certainty to make the decision maker indifferent between this sum and a particular uncertain or risky sum. This allows a new definition of net present value by replacing the uncertain cash flows by their certain equivalent and discounting them using a risk-free interest rate. 

Risk-adjusted NPV (RPV) (Keown etal, 2002) This is defined as the net present value calculated using a risk-adjusted rate of return instead of the normal return rate required to approve a project. However, Shimko (2001) suggests a slightly different definition where the value of a project is made up of two parts, one forms the not at risk part, discounted using the risk-free return rate, and the other forms the part at risk discounted at the fully loaded cash plus risk cost. 

Thus OAS is a general stochastic model, with discount rates derived from the standard benchmark term structure of interest rates. This is an advantage over more traditional methods in which a single discount rate is used. The calculated spread is a spread over risk-free forward rates, accounting for both interest-rate uncertainty and the price of default risk. As with any methodol-ogy, OAS has both strengths and weaknesses however, it provides more realistic analysis than the traditional yield-to-maturity approach. Hence, it has been widely adopted by investots since its introduction in the late 1980s. 

Using the spot rate structure at Table 12.1, the price of this bond is calculated to be 98.21. This would be the bonds fair value if it were liquid and default free. Assume, however, that the bond is a corporate bond and carries an element of default risk, and is priced at 97.00. What spread over the risk-free price does this indicate We require the spread over the implied forward rate that would result in a discounted price of 97.00. Using iteration, this is found to be 67.6 basis points. The calculation is... 

To determine the correct rate to use, consider the corresponding price of the conventional bond when the share price is 63.47 at period fg. The price of the bond is calculated on the basis that on maturity the bond will be redeemed irrespective of what happens to the share price. Therefore, the appropriate interest rate to use when discounting a conventional bond is the credit-adjusted rate, as this is a corporate bond carrying credit risk—it is not default-risk free. However, this does not apply at a different share... 

All bond instruments are characterized by the promise to pay a stream of future cash flows. The term structure of interest rates and associated discount function is crucial to the valuation of any debt security and underpins any valuation framework. Armed with the term structure, we can value any bond, assuming it is liquid and default-free, by breaking it down into a set of cash flows and valuing each cash flow with the appropriate discount factor. Further characteristics of any bond, such as an element of default risk or embedded option, are valued incrementally over its discounted cash flow valuation. 

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