## 1.4 Note on empirical estimation of the market risk premium

Empirical testing and application of this model requires the specification of three parameters: the instantaneous drift * v * and instantaneous variance * s * of the short- term interest rate process as well as the market price of risk * q * .Since * v * and * s * are parameters of the observable short-term interest rate process, they can be obtained by statistical analysis of market data. As previously mentioned, the market risk premium is not directly observable. Although it may be calculated from equation (1.12) as * q * ( * r * , * t * )=( ¼ ( * t * , * T * ) ˆ’ * r * ( * t * ))/ ƒ ( * t * , * T * ), a more direct method of estimation may be applied. Once * v * and * s * are determined, the market risk premium may be estimated from the slope of the yield curve at the origin. From equation (1.18) we have * P * as a function of * T * and * z * ( * T * ) and hence:

Therefore and . Also:

and hence

so

However, from equation (1.1) we have:

Equating to (1.19) we have:

and hence we have found the market price of risk in terms of the slope of the term structure at the origin.

## 1.5 Specific model

Vasicek specifies the required parameters for the short-term interest rate process and market price of risk, to derive an explicit term structure of interest rates. He assumes the market price of risk is constant:

and that the short-term interest rate follows an Ornstein-Uhlenbeck process:

where ± , ³ and * s * are constants with ± > 0. This is often referred to as an elastic random walk which is a Markovian process with normally distributed increments . The instantaneous drift ± ( ³ ˆ’ * r * ), displays mean reversion, with the short-term interest rate being pulled to its long-term mean, ³ , with magnitude proportional to its deviation from this long-term mean. This implies that the Ornstein-Uhlenbeck process is characterised by a stationary distribution, unlike a random walk (Wiener process) which is not stable and can diverge to infinite values ^{ [4] } .

Substituting (1.20) into the bond price PDE (1.13), the term structure equation becomes:

This is a linear, second-order partial differential equation, which can easily be solved by applying the boundary condition (1.14). Allowing the bond price to be of the form:

equation (1.21) becomes ^{ [5] } :

Since the right-hand side is a function of the short-term interest rate * r * , while the left-hand side is a function of * t * and * T * only, the following must hold:

Now, solving (1.23) with boundary condition * B * ( * T * , * T * ) = 0 gives:

Rearranging (1.24), we have:

and hence ln

Substituting (1.25) and (1.26) into (1.22), to obtain the bond price:

Equations (1.7), (1.8) and (1.9) give the dynamics of the bond price process and the mean and standard deviation of the instantaneous rate of return of a time * T * maturity bond. Substitute the above bond price formula to calculate the mean and standard deviation in terms of the short-term interest rate parameters and market price of risk:

One can see that the standard deviation (and hence the variance) of the instantaneous rate of return increases with the bond's term to maturity and the return in excess of the short-term interest rate is proportional to this standard deviation, with the proportionality constant being the market price of risk.

Now define to be the long-term rate of interest. The bond price becomes:

and from equation (1.1) the form of the term structure is:

Since (1.28) reduces to the short-term interest rate * r * ( * t * ) for = 0 and lim _{ ’ˆ } * R * ( * t * , ) = * R * ( ˆ ) it is consistent with previous definitions. Clearly the attainable shapes of the term structure depend on the value of * r * ( * t * ). See Figure 1.1. Let us examine the how the value of * r * ( * t * ) affects the slope of the term structure. From (1.28):

Figure 1.1: Possible shapes of the term structure. ³ = 0.14, ± = 0.5, * s * = 0.25, * q * = 0.2

For a monotonically increasing term structure, we require , for all . From (1.29) we have:

which is positive if because

Hence, the term structure is monotonically increasing if:

Additionally, since ˆˆ [0, ˆ ), * e * ^{ ˆ’± } ˆˆ (0, 1] and 1 ˆ’ * e * ^{ ˆ’± } ˆˆ [0, 1), so:

and hence, for a monotonically increasing term structure, we require:

For a monotonically decreasing term structure, we require for all . Making a substitution similar to that used when determining a monotonically increasing term structure ^{ [6] } , we need to show:

Since 1 ˆ’ * e * ^{ ˆ’± } 0 this is true if and only if:

However, this inequality does not hold because ^{ [7] } :

for some interval ˆˆ [0, *] where * depends on the value of the speed of reversion, ± . Hence the above substitution is inconsistent with our argument. However, substituting and in equation (1.29) we require:

This inequality holds if and only if:

This inequality is examined in the appendix and can be shown to hold ^{ [8] } for all values of . Hence we can write the following:

if since . Hence the slope is monotonically decreasing if:

that is:

The yield curve takes on a humped shape for values of * r * ( * t * ) such that:

^{ [4] } Vasicek emphasises that he is not trying to provide the best characterisation of the short-term interest rate process, but is merely specifying an example in the absence of conclusive empirical results.

^{ [5] } Here * A * _{ t } and * B * _{ t } denote the derivatives with respect to * t * .

^{ [6] } Here, substitute for and in equation (1.29).

^{ [7] } This result is shown numerically in the appendix.

^{ [8] } This statement is true only if ± > 0. However this is an assumption in the formulation of the model in (1.20).