Risk-free rate

The opportunity cost of capital for pharmaceutical R D is higher than the interest rate on safe investments, such as insured bank deposits or government bonds, but just how high the cost of capital for pharmaceutical R D projects is depends on how investors evaluate the risks of these investments, (See appendix C for a detailed discussion of the cost of capital.) The risk and, therefore, the cost of capital varies across different projects and even within the same R D project at different stages of development. The cost of capital for any investment also varies from year to year with underlying changes in the risk-free rate of interest (e.g., on bank deposits). Thus, the full cost of R D varies widely over time and across projects. 

Myers and Shyam-Sunder observed that the appropriate risk-free rate is the short-term Treasury bill rate, but this must be adjusted for forecasts that will govern... 

The realized market risk premium (over the risk-free rate) is highly volatile over time, while expected risks are assumed to be stable over long periods. Therefore, the market risk premium is typically estimated over a long period of time (198). Myers and Shyam-Sunder found an arithmetic mean of 8.7 percent for excess market return over the Treasury bill rate for the period 1926-89 (285). The market risk premium declined in the post-war period, however, and the premium for the period 1947-88 was 8.3 percent (285). 

Specifically, OTA assumed the pretax cost of debt is 9 percent for all three samples, the risk-free rate is... 

Because these estimates of the cost of capital are based on high estimates of the risk-free rate and the market premium, they should not be viewed as accurate estimates of the actual cost of capital over the period. Moreover, the cost of capital is a moving target over time a single estimate provides only a rough approximation of its value. Yet, they do provide a reasonably accurate (indeed, a conservative) test of differences in the cost of capital among the samples of firms examined by Baber and Kang. 

The value of the risk-free rate of return, r, can be estimated as being equal to the interest rate paid on U.S. Treasury Bills or other guaranteed savings instruments, approximately 6% in mid-20(X). The interest rate equivalent to the general equity market, was found to be approximately 9% by Fisher and Lorie (1968). Over the past 30 years this figure has risen to 11%. The effective tax rate, t, can... 

Risk-free rate Risk-free rate... 

Using the lognormal property, we can describe the distribution of the risk-free rate as ... 

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... 

The general rule of corporate bonds is that they are priced at a spread to the government yield curve. In absolute terms, the yield spread is the difference between the yield to maturity of a corporate bond and the benchmark, generally a yield to maturity of a govermnent bond with the same maturity. Corporate bonds include a yield spread on a risk-free rate in order to compensate two main factors, liquidity premium and credit spread. The yield of a corporate bond can be assumed as the sum of parts of the elements as shown in Figure 8.1, in which the yield spread relative to a default-free bond is given by the sum of default premium (credit spread) and liquidity premium. 

This equation can also be defined in terms of yield spread that reflects the yield premium required by a bondholder above the risk-free rate. The credit spread is given by Equation (8.17) ... 

In order to solve the probability of default, reduced-form models adopt a different approach. They are mainly based on debt prices rather than equity prices. In fact, they do not take into account the fundamentals of the firm and the default event is determined as an exogenous process without considering the underlying asset movements. In addition, the models are mainly based oti X t), that is the default intensity as a function of time. In particular, these models use the decomposition of the risky rate (risk-free rate and risk premium) in order to determine the default probabilities, recovery rates and debt values. Although structural models have the advantage to foUow a reliable measure of credit risk, that is the firm value, reduced-form approach overcomes the Umitatimi in which the balance sheet is not the unique indicator of the default prediction. 

The credit spread is defined as the difference between the risky rate of a defaul-table bond and the risk-free rate of a default-free bond. In this case, with bonds priced at par, between coupon and risk-free rate, the pricing is performed like a valuation of a straight bond, including the default risk adjustment. The price is given by Equation (8.25) ... 

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) ... 

The risk-free rate affects both elements, option-free bond and embedded option. Conversely, the credit spread is applied to the risk-free rate in order to find the price of the option-free bond. If the credit spread is also included into the option pricing model, the option value rises. For instance, consider the scenario in which the risk-free rate is 1.04% and the option value is 0.46. If the risk-free rate is 7.04%, then the option value increases to 0.66. Figure 9.16 shows the effect of a different interest rate level. 

If beta = 0, the investment does not add new risk to market portfolio and the expected return should be equal to risk-free rate (Rj=R ... 

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

If the hedge ratio is 1, we have a risk-free rate. In this point, the position of the investor is long above the underlying share price and also receiving a coupon. At a hedge ratio of 1, the option will move identically to the underlying asset. 

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. 

In contrast, for putable bonds, the right to exercise the option is held by the bondholder. In fact, putable bonds allow the bondholder to sell the bond back before maturity. Conversely to callable bonds, this happens when interest rates go up (risk-free rate increases, or the issuer s credit quality decreases). In fact, the bondholders may have the advantage to sell the bond and buy another one with higher coupon payments. 

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) ... 

The discussion is easily expanded to include risky floaters (e.g., corporate floaters) without a call feature or other embedded options. A floater pays a spread above the reference rate (i.e., the quoted margin) to compensate the investor for the risks (e.g., default, liquidity, etc.) associated with this security. The quoted margin is established on the floater s issue date and is fixed to maturity. If the market s evaluation of the risk of holding the floater does not change, the risky floater will be repriced to par on each coupon reset date just as with the default-free floater. This result holds as long as the issuer s risk can be characterized by a constant markup over the risk-free rate. 

The Duffle-Singleton modeling approach considers the three components of risk for a credit risky product, namely the risk-free rate, the hazard rate, and the recovery rate. 

X = annualized hazard rate (p = recovery value r = risk-free rate s = credit spread... 

Expiry in Six Months Risk-free rate = 10% Strike = 70 bps Credit spread = 60 bps Volatility = 20% Mean Reversion Model Price Standard Black Scholes Price Difference Between Standard Black Scholes and Mean Reversion Model Price... 

Simplified, and in present valne terms with a risk-free rate r PV(C) = il-pj)Ce ... 

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. 

Expression (3.14) is the formula for pricing zero-coupon bonds when the spot rate is the nonconstant instantaneous risk-free rate r(s) described above. This is the rate used in formulas (3.12), for valuing a money market account, and (3.15), for pricing a risk-free zero-coupon... 

The forward strategy can be used to imply the forward price, provided that the current price of the underlying and the money market interest rate are known. FIGURE 6.2 illustrates how this works, using the one-year forward contract whose profit/loss profile is graphed in figure 6.1 and assuming an initial spot price, P, of 50, a risk-free rate, r, of 1.05 percent, and a payout yield, R, of 1 percent. 

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