Risk-free interest rate

For risk-free investments, such as U.S. Treasury bills, the required return (as a percent of the capital invested) is determined by supply and demand in the money markets. If the going risk-free interest rate is 5 percent per year, for example, an investor who puts up 100 expects to get at least 105 back next year. From another point of view, 100 promised for delivery next year is worth only 95.23 today, because the investor could take that 95.23, invest it in a risk-free security, and have the 100 a year hence. Not having access to the 95.23 today essentially deprives the investor of the opportunity to invest at the going interest rate. 

The cost of capital varies widely across types of research projects and with successive investments as the project progresses toward the market. (See appendix C for an explanation.) It also changes from day to day as the risk-free interest rate changes. But detailed data on the actual riskiness of particular projects invested at specific times simply do not exist. Consequently, the fully capitalized cost of R D associated with the NCEs entering testing in DiMasi s study can be only crudely approximated. 

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. 

Now, demanding the absence of arbitrage opportunities requires that the drift term equals the risk-free interest rate, which implies... 

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

The authors find that conversely to the price of a fixed-rate coupon payment, which is a decreasing function of the maturity T, with floating-rate notes, the value depends on the level of interest rates. In fact, when the interest rate is below the long-run average value, the increase of T reduces the value of the floater and vice versa. In addition, the price of a floating-rate note increases with rising risk-free interest rates. 

In debt capital markets the yield on a domestic government T-bill is usually considered to represent the risk-free interest rate, since it is a shortterm instrument guaranteed by the government. This makes the T-bill rate, in theory at least, the most secure investment in the market. It is common to see the 3-month T-bill rate used in corporate finance analysis and option pricing analysis, which often refer to a risk-free money market rate. 

This version of the hypothesis is the only one that permits no arbitrage, because the expected rates of return on all bonds are equal to the risk-free interest rate. For this reason, the local expectations hypothesis is sometimes referred to as the risk-neutral expectations hypothesis. 

The futures price can be analyzed in terms of the forward-spot parity relationship and the risk-free interest rate. Say that the risk-free rate is r-. The forward-spot parity equation (repeated as (6.8a)) can be rewritten in terms of this rate as (6.8b), which must hold because of the noarbitrage assumption. 

The risk free interest rate during the option s life... 

Calculate the price of a call option written with strike price 21 and a maturity of three months written on a non-dividend-paying stock whose current share price is 25 and whose implied volatility is 23 percent, given a short-term risk-free interest rate of 5 percent. 

Calculate the price of a put option on the same stock, given the same risk-free interest rate. 

Three-month risk-free Interest rate Nine-month risk-free Interest rate One-year risk-free Interest rate... 

The principle of arbitrage-free pricing requires that the hedged portfolio s return equal the risk-free interest rate. This equivalence plus an expansion of dC(F,t) produces partial differential equation (8.33). 

The volatility of the underlying asset s price returns The risk-free interest rate applicable to the life of the option... 

A rise in interest rates increases the value of most call options. For stock options, this is because the equity markets view a rate increase as a sign that share price growth will accelerate. Generally, the relationship is the same for bond options. Not always, however, since in the bond market, rising rates tend to depress prices, because they lower the present value of future cash flows. A rise in interest rates has the opposite effect on put options, causing their value to drop. The risk-free interest rate applicable to a bond option with a term to expiry of, say, three months is a three-month government rate—commonly the government bond repo rate for bond options, usually the T-bill rate for other types. 

As noted in Chapter 8, the value of an option is a function of five factors The price of the underlying asset The options strike price The options time to expiry The volatility of the underlying assets price returns The risk-free interest rate applicable to the life of the option... 

The fair price of a convertible bond is the one that provides no opportunity for arbitrage profit that is, it precludes a trading strategy of running simultaneous but opposite positions in the convertible and the underlying equity in order to realize a profit. Under this approach we consider now an application of the binomial model to value a convertible security. Following the usual conditions of an option pricing model such as Black-Scholes (1973) or Cox-Ross-Rubinstein (1979), we assume no dividend payments, no transaction costs, a risk-free interest rate, and no bid-offer spreads. 

P, is the price of the convertible bond is the price of the underlying equity C is the bond coupon r is the risk-free interest rate N is the time to maturity a is the annualized share price volatility c is the call option feature rd is the dividend yield on the underlying share... 

The one time-period or 180-day equivalent rate is (e° ) -1 or 2.0201 percent. This is therefore the risk-free interest rate to use. The volatility level of 10 percent is an annualized figure, following market convention. This may be broken down per time period as well, and this is calculated by multiplying the annual figure by the square root of the time period required. This calculation follows. 

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