A risk-free bond is a theoretical bond that repays

The calculation is related to an Arrow-Debreu security. Let's call the price of the risk-free bond at time $t$ as $P(t,\; t+1)$. The $t+1$ refers to the fact that the bond matures at time $t+1$. As mentioned before, the risk-free bond can be replicated by a portfolio of two Arrow-Debreu securities, one share of $A(1)$ and one share of $A(2)$.
Using formula for the price of an $n$ Arrow-Debreu securities
$A(k)=p\_k\; \backslash frac,\; ~~~~~\; k=1,\backslash dots,\; n$
which is a product of ratio of ''the intertemporal marginal rate of substitution'' (the ratio of marginal utilities across time, it is also referred to as the ''state price density'' and ''the pricing kernel'') and ''the probability'' of state occurring in which the Arrow-Debreu security pays off 1 unit. The price of the portfolio is simply
$P(t,t+1)=A(1)+A(2)=p\_1\backslash frac+p\_2\backslash frac$
$P(t,t+1)=\backslash mathbb\_t^\backslash Bigg;\; href="/html/ALL/l/frac\backslash Bigg.html"\; ;"title="frac\backslash Bigg">frac\backslash Bigg$

interest
In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is disti ...

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 time $t+1$. So its payoff is the same regardless of what state occurs. Thus, an investor experiences no risk by investing in such an asset.
In practice, government bonds of financially stable countries are treated as risk-free bonds, as governments can raise taxes or indeed print money to repay their domestic currency debt.
For instance, United States Treasury notes and United States Treasury bonds are often assumed to be risk-free bonds. Even though investors in United States Treasury securities do in fact face a small amount of credit risk
A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased co ...

, this risk is often considered to be negligible. An example of this credit risk was shown by Russia, which defaulted on its domestic debt during the 1998 Russian financial crisis.
Modelling the price by Black-Scholes model

In financial literature, it is not uncommon to derive the Black-Scholes formula by introducing a continuously rebalanced ''risk-free portfolio'' containing an option and underlying stocks. In the absence ofarbitrage
In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between th ...

, the return from such a portfolio needs to match returns on risk-free bonds. This property leads to the Black-Scholes partial differential equation satisfied by the arbitrage price of an option. It appears, however, that the risk-free portfolio does not satisfy the formal definition of a self-financing strategy, and thus this way of deriving the Black-Sholes formula is flawed.
We assume throughout that trading takes place continuously in time, and unrestricted borrowing and lending of funds is possible at the same constant interest rate. Furthermore, the market is frictionless, meaning that there are no transaction costs or taxes, and no discrimination against the short sales. In other words, we shall deal with the case of a ''perfect market''.
Let's assume that the ''short-term interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...

'' $r$ is constant (but not necessarily nonnegative) over the trading interval $;\; href="/html/ALL/l/,\_T^*.html"\; ;"title=",\; T^*">,\; T^*$maturity date
Maturity or immaturity may refer to:
* Adulthood or age of majority
* Maturity model
** Capability Maturity Model, in software engineering, a model representing the degree of formality and optimization of processes in an organization
* Developme ...

''. Let $B(t,\; T)$ stand for the price at time $t\; \backslash in;\; href="/html/ALL/l/,\_T.html"\; ;"title=",\; T">,\; T$ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

$dB(t,\; T)\; =\; rB(t,\; T)dt~~~,~~~\; B(0,\; T)\; =\; e^\; ~.$
We consider here a ''risk-free'' bond, meaning that its issuer will not default on his obligation to pat to the bondholder the face value at maturity date.
Risk-free bond vs. Arrow-Debreu security

The risk-free bond can be replicated by a portfolio of two Arrow-Debreu securities. This portfolio exactly matches the payoff of the risk-free bond since the portfolio too pays 1 unit regardless of which state occurs. This is because if its price were different from that of the risk-free bond, we would have an ''arbitrage
In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between th ...

opportunity'' present in the economy. When an arbitrage opportunity is present, it means that riskless profits can be made through some trading strategy. In this specific case, if portfolio of Arrow-Debreu securities differs in price from the price of the risk-free bond, then the arbitrage strategy would be to buy the lower priced one and sell short the higher priced one. Since each has exactly the same payoff profile, this trade would leave us with zero net risk (the risk of one cancels the other's risk because we have bought and sold in equal quantities the same payoff profile). However, we would make a profit because we are buying at a low price and selling at a high price. Since arbitrage conditions cannot exist in an economy, the price of the risk-free bond equals the price of the portfolio.
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

, taken with respect to the probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...

$\backslash mathbb\; P=\backslash $, of the intertemporal marginal rate of substitution. The interest rate $r$, is now defined using the reciprocal of the bond price.
$1+r\_t=\backslash frac$
Therefore, we have the fundamental relation
$\backslash frac=\backslash mathbb\_t^\backslash Bigg;\; href="/html/ALL/l/frac\backslash Bigg.html"\; ;"title="frac\backslash Bigg">frac\backslash Bigg$Example

Suppose that the ''probability of state 1'' occurring is 1/4, while ''probability of state 2'' occurring is 3/4. Also assume that ''the pricing kernel'' equals 0.95 for state 1 and 0.92 for state 2. Let the pricing kernel denotes as $U\_k$ . Then we have two Arrow-Debreu securities $A(1),~A(2)$ with parameters $p\_1=1/4\; ~~,~~U\_1=0.95~,$ $p\_2\; =\; 3/4\; ~~,~~\; U\_2=0.92~.$ Then using the previous formulas, we can calculate the bond price $P(t,t+1)=A(1)+A(2)=p\_1U\_1+p\_2U\_2=1/4\backslash cdot\; 0.95+3/4\backslash cdot\; 0.92=0.9275~.$ The interest rate is then given by $r=\backslash frac-1=\backslash frac-1=7.82\backslash \%~.$ Thus, we see that the pricing of a bond and the determination of interest rate is simple to do once the set of Arrow-Debreu prices, the prices of Arrow-Debreu securities, are known.See also

* Risk-free interest rate *List of economics topics
The following outline is provided as an overview of and topical guide to economics:
Economics – analyzes the production, distribution, and consumption of goods and services. It aims to explain how economies work and how economic agen ...

* Black-Scholes model
* Arrow-Debreu security
In financial economics, a state-price security, also called an Arrow–Debreu security (from its origins in the Arrow–Debreu model), a pure security, or a primitive security is a contract that agrees to pay one unit of a numeraire (a currency ...

References

{{Bond market Financial risk Government bonds