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The risk-free rate of return, usually shortened to the risk-free rate, is the
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ...
of a hypothetical
investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing is ...
with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free rate can be obtained with no risk, any other investment having some risk will have to have a higher rate of return in order to induce any investors to hold it. In practice, to infer the risk-free interest rate in a particular currency, market participants often choose the yield to maturity on a
risk-free bond A risk-free bond is a theoretical bond that repays interest and principal with absolute certainty. The rate of return would be the risk-free interest rate. It is primary security, which pays off 1 unit no matter state of economy is realized at ti ...
issued by a government of the same currency whose risks of default are so low as to be negligible. For example, the rate of return on T-bills is sometimes seen as the risk-free rate of return in US dollars.


Theoretical measurement

As stated by Malcolm Kemp in chapter five of his book ''Market Consistency: Model Calibration in Imperfect Markets'', the risk-free rate means different things to different people and there is no consensus on how to go about a direct measurement of it. One interpretation of the theoretical risk-free rate is aligned to
Irving Fisher Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt d ...
's concept of inflationary expectations, described in his treatise ''The Theory of Interest'' (1930), which is based on the theoretical costs and benefits of holding currency. In Fisher's model, these are described by two potentially offsetting movements: # Expected increases in the money supply should result in investors preferring current consumption to future income. # Expected increases in productivity should result in investors preferring future income to current consumption. The correct interpretation is that the risk-free rate could be either positive or negative and in practice the sign of the expected risk-free rate is an institutional convention – this is analogous to the argument that Tobin makes on page 17 of his book ''Money, Credit and Capital''. In a system with
endogenous money Endogenous money is an economy’s supply of money that is determined endogenously—that is, as a result of the interactions of other economic variables, rather than exogenously (autonomously) by an external authority such as a central bank. ...
creation and where production decisions and outcomes are decentralized and potentially intractable to forecasting, this analysis provides support to the concept that the risk-free rate may not be directly observable. However, it is commonly observed that for people applying this interpretation, the value of supplying currency is normally perceived as being positive. It is not clear what is the true basis for this perception, but it may be related to the practical necessity of some form of (credit?) currency to support the specialization of labour, the perceived benefits of which were detailed by
Adam Smith Adam Smith (baptized 1723 – 17 July 1790) was a Scottish economist and philosopher who was a pioneer in the thinking of political economy and key figure during the Scottish Enlightenment. Seen by some as "The Father of Economics"— ...
in ''
The Wealth of Nations ''An Inquiry into the Nature and Causes of the Wealth of Nations'', generally referred to by its shortened title ''The Wealth of Nations'', is the '' magnum opus'' of the Scottish economist and moral philosopher Adam Smith. First published in ...
''. However, Smith did not provide an 'upper limit' to the desirable level of the specialization of labour and did not fully address issues of how this should be organised at the national or international level. An alternative (less well developed) interpretation is that the risk-free rate represents the time preference of a representative worker for a representative basket of consumption. Again, there are reasons to believe that in this situation the risk-free rate may not be directly observable. A third (also less well developed) interpretation is that instead of maintaining pace with purchasing power, a representative investor may require a risk free investment to keep pace with wages. Given the theoretical 'fog' around this issue, in practice most industry practitioners rely on some form of proxy for the risk-free rate, or use other forms of benchmark rate which are presupposed to incorporate the risk-free rate plus some risk of default. However, there are also issues with this approach, which are discussed in the next section. Further discussions on the concept of a 'stochastic discount rate' are available in ''The Econometrics of Financial Markets'' by Campbell, Lo and MacKinley.


Proxies for the risk-free rate

The return on domestically held short-dated government
bond Bond or bonds may refer to: Common meanings * Bond (finance), a type of debt security * Bail bond, a commercial third-party guarantor of surety bonds in the United States * Chemical bond, the attraction of atoms, ions or molecules to form chemica ...
s is normally perceived as a good proxy for the risk-free rate. In
business valuation Business valuation is a process and a set of procedures used to estimate the economic value of an owner's interest in a business. Here various valuation techniques are used by financial market participants to determine the price they are willing ...
the long-term yield on the US Treasury coupon bonds is generally accepted as the risk-free rate of return. However, theoretically this is only correct if there is no perceived risk of default associated with the bond. Government bonds are conventionally considered to be relatively risk-free to a domestic holder of a government bond, because there is by definition no risk of default – the bond is a form of government obligation which is being discharged through the payment of another form of government obligation (i.e. the domestic currency). In fact, default on government debt does happen, so if in theory this is impossible, then this points out a deficiency of the theory. Another issue with this approach is that with coupon-bearing bonds, the investor does not know ex-ante what his return will be on the reinvested coupons (and hence the return cannot really be considered risk-free). Some academics support the use of swap rates as a measurement of the risk-free rate. state: "…the riskless rate is better proxied by the swap rate than the Treasury rate for all maturities". There is also the risk of the government 'printing more money' to meet the obligation, thus paying back in lesser valued currency. This may be perceived as a form of tax, rather than a form of default, a concept similar to that of
seigniorage Seigniorage , also spelled seignorage or seigneurage (from the Old French ''seigneuriage'', "right of the lord (''seigneur'') to mint money"), is the difference between the value of money and the cost to produce and distribute it. The term can be ...
. But the result to the investor is the same, loss of value according to his measurement, so focusing strictly on default does not include all risk. The same consideration does not necessarily apply to a foreign holder of a government bond, since a foreign holder also requires compensation for potential foreign exchange movements in addition to the compensation required by a domestic holder. Since the risk-free rate should theoretically exclude any risk, default or otherwise, this implies that the yields on foreign owned government debt cannot be used as the basis for calculating the risk-free rate. Since the required return on government bonds for domestic and foreign holders cannot be distinguished in an international market for government debt, this may mean that yields on government debt are not a good proxy for the risk-free rate. Another possibility used to estimate the risk-free rate is the inter-bank lending rate. This appears to be premised on the basis that these institutions benefit from an implicit guarantee, underpinned by the role of the monetary authorities as 'the lendor of last resort.' (In a system with an
endogenous money Endogenous money is an economy’s supply of money that is determined endogenously—that is, as a result of the interactions of other economic variables, rather than exogenously (autonomously) by an external authority such as a central bank. ...
supply the 'monetary authorities' may be private agents as well as the central bank – refer to Graziani 'The Theory of Monetary Production'.) Again, the same observation applies to banks as a proxy for the risk-free rate – if there is any perceived risk of default implicit in the interbank lending rate, it is not appropriate to use this rate as a proxy for the risk-free rate. Similar conclusions can be drawn from other potential benchmark rates, including AAA-rated corporate bonds of institutions deemed '
too big to fail "Too big to fail" (TBTF) and "too big to jail" is a theory in banking and finance that asserts that certain corporations, particularly financial institutions, are so large and so interconnected that their failure would be disastrous to the great ...
.' One solution that has been proposed for solving the issue of not having a good 'proxy' for the risk-free asset, to provide an 'observable' risk-free rate is to have some form of international guaranteed asset which would provide a guaranteed return over an indefinite time period (possibly even into perpetuity). There are some assets in existence which might replicate some of the hypothetical properties of this asset. For example, one potential candidate is the 'consol' bonds which were issued by the British government in the 18th century.


Application

The risk-free interest rate is highly significant in the context of the general application of
capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into acc ...
which is based on the
modern portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversificat ...
. There are numerous issues with this model, the most basic of which is the reduction of the description of utility of stock holding to the expected mean and variance of the returns of the portfolio. In reality, there may be other utility of stock holding, as described by Robert J. Shiller in his article 'Stock Prices and Social Dynamics'. The risk-free rate is also a required input in financial calculations, such as the Black–Scholes formula for pricing
stock option In finance, an option is a contract which conveys to its owner, the ''holder'', the right, but not the obligation, to buy or sell a specific quantity of an underlying asset or instrument at a specified strike price on or before a specified dat ...
s and the
Sharpe ratio In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the performance of an investment such as a security or portfolio compared to a risk-free asset, after adjusting for its ...
. Note that some finance and economic theories assume that market participants can borrow at the risk-free rate; in practice, very few (if any) borrowers have access to finance at the risk free rate. The risk-free rate of return is the key input into
cost of capital In economics and accounting, the cost of capital is the cost of a company's funds (both debt and equity), or from an investor's point of view is "the required rate of return on a portfolio company's existing securities". It is used to evaluate ne ...
calculations such as those performed using the
capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into acc ...
. The cost of capital at risk then is the sum of the risk-free rate of return and certain risk premia.


See also

* Short-rate model *
Capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into acc ...
*
Beta (finance) In finance, the beta (β or market beta or beta coefficient) is a measure of how an individual asset moves (on average) when the overall stock market increases or decreases. Thus, beta is a useful measure of the contribution of an individual a ...


References

{{financial risk, state=collapsed Interest rates Financial risk