Risk-free asset

Efficient frontiers also invariably place Treasury bills as the risk-free asset. T-bills may be risk-free from a creditworthiness point of view, bnt it is not tenable that a three-month nominal asset is a risk-free instrn-ment for someone with, say, a 30-year savings horizon. If you are investing for 30 years, over which time you are interested in your prospective real returns, then a 30-year linker (to be held to maturity) is your riskfree asset, almost by definition. 100% invested in that bond becomes the lowest risk portfolio on yonr frontier. Efficient frontier analysis starts to lose its impact once this premise is accepted, not least because you do not have a large data sample of consecutive, nonoverlapping 30-year periods (for any asset) to produce robust analysis. ... 

At point Q the capital is completely invested in the risk-free asset. Risk, measured by the standard deviation of expected returns, is 0%. The more risky assets are included in the portfolio, the higher its expected return and risk. This is described by the line QZ. 

If the portfolio is completely invested in the risk-free asset, its expected return is rf with an expected volatility of Cp = Cf= 0. This defines point Q. If more risk is accepted, the expected return rp increases along with the expected volatility Op > 0. 

We calculate the continuously compounded rate of return on the risk-free asset to be... 

Under these four assumptions, the price of an asset can be described in present value terms relative to the value of the risk-free cash deposit M, and, in fact, the price is described as a Q-martingale. A European-style contingent liability with maturity date t is therefore valued at time 0 under the risk-neutral probability as... 

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. 

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. 

When valuing an option written on say, an equity the price of the underlying asset is the current price of the equity. When pricing an interest-rate option the underlying is obtained via a random process that described the instantaneous risk-free zero-coupon rate, which is generally termed the short rate. 

The presence of a risk-free real asset is of great value to the study of finance, lending itself to theoretical and behavioural analysis. In this section, we examine the relationship between inflation-linked bonds and nominal bonds, then show how this framework can be applied to the relationship with equities. 

We have compared linkers with other investment assets already, and that is really at the root of the issue. It is a question of opportunity cost. Risk-free real yields are determined by the prospective returns that other, riskier, assets in an economy are perceived to offer. 

The historical volatility of the difference between the reference asset yield and the yield on a risk-free benchmark. 

This risk-free rate is known as the implied repo rate, because the rate is similar to a repurchase reement carried out in the futures market. Generally, high implied repo rates indicate high futures prices, low rates imply low prices. The rates can be used to compare contracts that have different terms to maturity and even underlying assets. The implied repo rate for the contract is more stable than its basis, but as maturity approaches it becomes very sensitive to changes in the futures price, spot price, and (by definition) time to maturity. 

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

A number of option-pricing models exist. Market participants often use variations on these models that they developed themselves or that were developed by their firms. The best-known of the pricing models is probably the Black-Scholes, whose fundamental principle is that a synthetic option can be created and valued by taking a position in the underlying asset and borrowing or lending funds in the market at the risk-free rate of interest. Although Black-Scholes is the basis for many other option models and is still used widely in the market, it is not necessarily appropriate for some interest rate instruments. Fabozzi (1997), for instance, states that the Black-Scholes model s assumptions make it unsuitable for certain bond options. As a result a number of alternatives have been developed to analyze callable bonds. 

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... 

To compute the expected return on equity, which is denoted by E ROE),Vx. (2.50) is applied. In this expression, E (ROE) is computed as the sum of a risk free rate and a risk premium (cpRe). The former term represents the rate of return of an investment free of default risk available in the market and is usually equal to the yield to maturity offered by a government security. The latter, represents the expected amount of return above the risk-free rate in exchange for a given amount of variance (Pratt 2002, Applequist et al. 2000). One of the most commonly employed methods to estimate the risk premium is the Capital Asset Pricing Model (CAPM). For more details regarding this topic the reader is referred to Sharpe (1999). 

In an unadulterated world, MV analysis would simply be applied to after-tax return, risk, and correlations to produce tax-efficient portfolios. Regrettably, tax law is complex and there are a variety of alternative tax structures for holding financial assets. In addition, there are tax-free versions of some instruments such as municipal bonds. Further complicating the analysis is the fact that one must know the split between ordinary income and long-term gains to forecast MV model inputs. This muddles what would otherwise be a fairly straightforward portfolio-allocation problem and requires that tax structure subtleties be built into the MV framework to perform tax-efficient portfolio optimization. 

When it comes to applying the MV approach to taxable investing, the methodology is the same as that used for tax-free portfolios, except that four different after-tax return streams, risk measures, and four-square correlations are required for each asset. Table 6 compares the pre-tax and after-tax returns for assets typically considered in taxable portfolio allocation studies. 

The assumption of complete capital markets states that, as a result of arbitrage-free pricing, there is a unique probability measure Q, which is identical to the historical probability P, under which the continuously discounted price of any asset is a Q-martingale. This probability level Q then becomes the risk-neutral probability. 

This chapter examines a number of issues relevant to participants in the fixed-income markets. The analysis presented is based on government-bond trading and is confined to generic bonds that are default-free, with no consideration given to factors that apply to corporate bonds, asset- and mortgage-backed bonds, convertibles, or other nonvanilla securities, or to issues such as credit risk and prepayment risk. Nevertheless, the principles adduced are pertinent to all relative-value fixed-income analysis. 

---