1. 1
An estimation of stochastic
relative risk aversion from
interest rates
Joon Yoong Loh
Supervised by: Dr Matteo de Tina, Mr Ian Corrick
University of Bath
Department of Economics
April 2016
Abstract
I calibrate the risk aversion parameter for time separable utility functions
within a dynamic stochastic general equilibrium context, using a time
inhomogeneous single factor short interest rate model to imply a stochastic
process for the market price of risk using daily US LIBOR data. I show that while
elevating risk aversion alone smoothens consumption too much, simultaneously
adding random shocks to relative risk aversion can increase consumption variance
dramatically. I modify a Real Business Cycle (RBC) model, producing standard
deviations closer to the empirical data as compared to a baseline model with
constant relative risk aversion.
Word count: 9894

2. 2
Table of contents
1 Introduction 3
2 From interest rates to relative risk aversion
2.1 Model description
2.1.1 The market price of interest rate risk and risk neutrality 5
2.1.2 The ex-ante Sharpe ratio 8
2.1.3 Preferences and risk aversion 10
2.2 Estimation methodology
2.2.1 Data 12
2.2.2 The short rate process 12
2.2.3 The market price of risk and the bond Sharpe ratio 14
2.2.4 The volatility of log consumption 15
2.3 Empirical results and analysis 15
3 The economic model
3.1 The model 20
3.2 Model specifications 23
3.3 Empirical results and analysis 24
4 Discussion of the models and methodologies
4.1 Limitations of the time inhomogeneous single factor model
4.1.1 Information sets and conditional expectations 29
4.1.2 Time inhomogeneity and imprecise volatility estimates 33
4.2 Robustness of the estimation methodology
4.2.1 Cross model robustness 33
4.2.2 Robustness to choice of proxy and sample 35
4.3 Limitations of the time separable utility function in asset pricing 36
5 Conclusion 39
Appendix 40
References 41

3. 3
1. Introduction
The literature surrounding the calibration of relative risk aversion (RRA) has evolved
over time. A number of early pieces have advocated relatively low values. Arrow (1971)
suggested a value of approximately one based on theoretical grounds, while Friend and
Blume (1975) estimated values above 2 from the analysis of asset holdings of households.
Empirical exercises with dynamic models of the economy have also used similar ranges to
match aggregate quantities, including Kydland and Prescott (1982) and Long and Plosser
(1983). Ljungqvist and Sargent (2012) summarized that “reasonable” values of relative risk
aversion below 3 have mostly been the result of the observation of individual behavior in the
context of experiments and well understood gambles.
A turning point in the literature was Mehra and Prescott (1985) who highlighted a
significant weakness in a general class of frictionless general equilibrium models, the failure
to generate equity premiums anything close to the 6% observed in U.S. equity markets, a
problem since called the “equity premium puzzle”. Another key result was Hansen and
Jagannathan (1991), who deduced volatility bounds for the intertemporal marginal rates of
substation for a very general class of economic models extending beyond the simple power
utility function considered by Mehra and Prescott (1985). Since then, increased interest in
producing asset-market consistent models have resulted in work trying to either reproduce the
equity premium or satisfy the Hansen-Jagannathan volatility bounds.
These include Campbell and Cochrane (1999) who incorporated habit formation into
a power utility function, and Jermann (1998) who considered both habit formation and capital
adjustment costs. These authors successfully produced the equity premium with relatively
low power utility function exponents.
However, work with non-habit formation utility functions have suggested values of
relative risk aversion far beyond the values traditionally accepted as “reasonable”. Kandel
and Stambaugh (1991), using Epstein-Zin preferences, found that relative risk aversion of 29
was required to match the first moments of asset prices. Cochrane and Hansen (1992) and
Campbell and Cochrane (1999) find that relative risk aversion in excess of 40 was required
for consistency with the Hansen-Jagannathan bounds, while Tallarini (2000) used values up
to 100.
In addition to findings surrounding the level of relative risk aversion the notion of
time varying preferences have also received attention, although somewhat less. Sargent and

4. 4
Cogley (2005) introduce a model where agents, after some adverse financial event, begin
with a pessimistic probability law and then gradually update their preferences back to
unbiased rational expectations. Brandt and Wang (2003) specified a mean reversion process
for relative risk aversion dependent on surprises in inflation and consumption, using mean
values for relative risk aversion below 4.
Other research supporting this notion have come from areas outside of dynamic
economic modelling and have been empirically compelling. Guiso, Sapienza and Zingales
(2013) carried out a series of surveys on customers of an Italian bank, showing an increase in
risk aversion since the 2008 crisis even among individuals who did not experience large
financial losses. Ahmad and Wilmott (2007) obtained an empirical time series estimate for
the market price of interest rate risk, noting cyclical fluctuations in the time series. They
proposed a dual factor interest derivative valuation methodology featuring a stochastic
market price of interest rate risk.
While much of the work regarding the calibration of relative risk aversion in a
dynamic economic modelling context has been to choose values for relative risk aversion
such that the models become consistent with asset market or aggregate quantity data, I
approach the problem by trying to estimate some “true” process for risk aversion directly,
using the methodology proposed by Wilmott and Ahmad (2007). By calibrating a time
inhomogeneous single factor short rate model to USD LIBOR data between 1987 and 2006
(avoiding the financial crisis and the period thereafter), I estimate a daily time series for the
market price of interest rate risk. Then extending their methodology, I use the results to
derive another time series for the ex-ante bond Sharpe ratio. Using the Sharpe ratio as a proxy
for the market price of risk, I deduce a stochastic process for relative risk aversion in the
context of the simple power utility function by assuming random-walk consumption. I
estimate mean relative risk aversion at 49.89, somewhat concurring with Cochrane and
Hansen (1992) and Campbell and Cochrane (1999).
I then test my process by modifying a baseline RBC model calibrated by Fernandez-
Villaverde in 2005, allowing relative risk aversion to follow a stochastic process, and
compare the performance of the modified model with the baseline on empirical aggregate
quantity moments observed from the U.S. economy in the period 1987 to 2006. I find that
increased levels of fixed relative risk aversion greatly reduces consumption volatility,
confirming Rouwenhorsts’ (1995) finding that increased relative risk aversion leads to higher
degrees of consumption smoothing. However, adding a stochastic mean reverting evolution

5. 5
process to relative risk aversion causes consumption volatility to rise significantly, resulting
in a model that matches empirical consumption volatility better than the baseline model.
This paper is organised as follows: Section 2.1 introduces the model used to derive
relative risk aversion from interest rates which is largely based on financial mathematics.
Section 2.2 details the estimation methods employed, while Section 2.3 reports and analyses
the empirical results of the estimation exercise. Section 3 introduces and reports the results of
calibrating an economic model to the results in Section 2, comparing the modified RBC
model with the baseline. Section 4 provides critical discussion on the validity and limitations
of the methodologies, models and assumptions used in the exercise. Section 5 concludes the
paper.
2. From interest rates to risk aversion
2.1 The model
2.1.1 The market price of interest rate risk and risk neutrality
Interest rate modelling in finance is based on a concept of the instantaneous or spot
interest rate, representing the yield on a theoretical zero coupon bond of infinitesimal
maturity. At time t, investors observe a short rate 𝑟𝑡 and can invest a sum of money at time 𝑡
and receive a return of 𝑟 ∗ 𝑑𝑡 at time 𝑡 + 𝑑𝑡 where 𝑑𝑡 → 0. The evolution of the short interest
rate is modelled as a stochastic differential equation of the form:
𝑑𝑟𝑡 = 𝑢(𝑟𝑡, 𝑡)𝑑𝑡 + 𝑤(𝑟𝑡, 𝑡)𝑑𝑊 (5)
Where 𝑑𝑊 is an increment of Brownian motion, also called the Weiner process, and satisfies
the condition, 𝔼(𝑑𝑊) = 0. 𝑤(𝑟, 𝑡) represents the volatility of the interest rate 𝑟 while 𝑢(𝑟𝑡, 𝑡)
is the drift, representing the deterministic component of the process.
I assume that interest rates follow the process in (5) and that market participants have rational
expectations. Therefore, their expectations correspond exactly to the predictions of the short
rate model. The expected change in the interest rate in the time interval [𝑡, 𝑡 + 𝑑𝑡] is then
𝔼(𝑑𝑟𝑡) = 𝑢(𝑟𝑡, 𝑡)𝑑𝑡. Assuming the process above is the true process and that agents are
rational, we have that 𝔼 𝑡(𝑟𝑡+1|𝑟𝑡) = 𝑟𝑡 + 𝑢(𝑟𝑡, 𝑡)𝑑𝑡.
In this framework, a bond which begins in time 𝑡 and matures in time 𝑇 (𝑇 > 𝑡)is
considered a risky asset because its value is based on the short rate, which moves

6. 6
stochastically in the time interval [𝑡, 𝑇]. Although the investor is sure of his cash return at
time 𝑇, his bond experiences mark-to-market price fluctuations before then. On the other
hand, the investor who invests at the risk free rate in the interval [𝑡, 𝑡 + 𝑑𝑡] is certain of his
return.
A key result which for this section is a result of Ito’s formula, which allows us to find
the function of a stochastic process, which is itself a stochastic process1
:
If 𝑑𝑋𝑡 = 𝜇(𝑡, 𝑋𝑡)𝑑𝑡 + 𝜎(𝑡, 𝑋𝑡)𝑑𝑊𝑡
and there exists some function 𝑓 = 𝑓(𝑡, 𝑋𝑡) where 𝑡 is deterministic,
Then the process for 𝑓 is given by:
𝑑𝑓𝑡 =
𝜕𝑓
𝜕𝑡
𝑑𝑡 +
𝜕𝑓
𝜕𝑋
𝑑𝑋𝑡 +
1
2
𝜕2
𝑓
𝜕𝑋2
𝑑𝑡
Which upon substitution of process 𝑑𝑋𝑡 can be represented:
𝑑𝑓𝑡 = [
𝜕𝑓
𝑑𝑡
+
1
2
𝜎2
𝜕2
𝑓
𝜕𝑋2
+ 𝜇
𝜕𝑓
𝜕𝑋
] 𝑑𝑡 + 𝜎
𝜕𝑓
𝜕𝑋
𝑑𝑊𝑡
(6)
Let us now consider a derivative which value is dependent on the interest rate and time,
𝑉(𝑟, 𝑡; 𝑇), where 𝑟 evolves according to the process in (5), 𝑡 is the time of observation and 𝑇
is the time of maturity, a fixed parameter. Using (6) on the derivative give the stochastic
differential equation:
𝑑𝑉 = [
𝜕𝑉
𝑑𝑡
+
1
2
𝑤2
𝜕2
𝑉
𝜕𝑟2
+
𝜕𝑉
𝜕𝑟
𝑢] 𝑑𝑡 + 𝑤
𝜕𝑉
𝜕𝑟
𝑑𝑊
(7)
Another key equation is a variation of the Black-Scholes equation—but for bond pricing—
and is sometimes known as the Bond Pricing Equation. It is given by2
:
𝜕𝑉
𝜕𝑡
+
1
2
𝑤(𝑟, 𝑡)2
𝜕2
𝑉
𝜕𝑟2
+ (𝑢(𝑟, 𝑡) − 𝜆(𝑟, 𝑡)𝑤(𝑟, 𝑡))
𝜕𝑉
𝜕𝑟
− 𝑟𝑉 = 0
(8)
Substituting (8) into equation (7) gives:
𝑑𝑉 = [𝜆(𝑟, 𝑡)𝑤(𝑟, 𝑡)
𝜕𝑉
𝜕𝑟
+ 𝑟𝑉] 𝑑𝑡 + 𝑤
𝜕𝑉
𝜕𝑟
𝑑𝑊
(9)
Which can be rearranged into the following expression:
1
Detailed proofs and explanations found in Shreve (2000), Chapter 4. In particular, this result of the Ito
formula is found on page 141 under Chapter 4.4.1.
2
See Bollen (1997) pp 842

7. 7
𝑑𝑉 − 𝑟𝑉𝑑𝑡 = 𝑤
𝜕𝑉
𝜕𝑟
𝜆𝑑𝑡 + 𝑤
𝜕𝑉
𝜕𝑟
𝑑𝑊
(10)
The left hand side represents a risky cashflow in a period [𝑡, 𝑡 + 𝑑𝑡], 𝑑𝑡 → 0, where
the investor has bought 1 unit of the derivative, funding the purchase by borrowing at the risk
free rate 𝑟. 𝑑𝑉 will be the random cashflow he will obtain from the derivative while 𝑟𝑉𝑑𝑡 is
his cost of funding. The investor effectively takes on a certain liability hoping to gain an
uncertain return, and the payoff from this transaction comprises of a deterministic component
𝑤
𝜕𝑉
𝜕𝑟
𝜆𝑑𝑡 and a random component 𝑤
𝜕𝑉
𝜕𝑟
𝑑𝑊.
To induce a risk averse investor into making this transaction, the deterministic
component must be significantly positive to compensate him for the random component. If
derivative 𝑉(𝑟, 𝑡; 𝑇) is a bond, then 𝜕𝑉/𝜕𝑟 is negative, meaning 𝜆 needs to be negative to give
the transaction payoff a positive expectation.
Another important representation of the value of the bond 𝑉(𝑟, 𝑡; 𝑇) is sometimes
called the fundamental asset pricing formula (FAPM), which states that the price of a
continuously compounded zero coupon bond is given by the following expectation under the
risk neutral measure ℚ3
:
𝑉(𝑟, 𝑡; 𝑇) = 𝔼ℚ [exp {∫ 𝑟𝑡 𝑑𝑡
𝑇
𝑡
} |ℱ𝑡]
(11)
where 𝔼ℚ(. |ℱ𝑡) is the expectation under the risk neutral probability measure ℚ given the
information available at time t, ℱ𝑡. Under the risk neutral measure, the interest rate evolves
according to the law:
𝑑𝑟 = [𝑢(𝑟𝑡) − 𝜆(𝑡)𝑤(𝑟𝑡)]𝑑𝑡 + 𝑤(𝑟𝑡)𝑑𝑊ℚ (12)
[
where 𝑑𝑊ℚ
is geometric Brownian motion under the risk neutral process. Essentially, this
process has a more positive drift compared to the real interest rate process in (1). The
expectation in (11) will be evaluated as though 𝑟𝑡 followed the risk neutral process (12) as
opposed to the real world process4
(5). The higher drift in the risk neutral process will result
in a slightly lower bond price, compensating the investor for taking on the uncertain cashflow
in (10).
3
Cerny (2004) Chapter 11.5, page 252 gives a more general form. However, the payoff of a zero coupon bond
is conventionally normalised to 1 in this case. Also see Shreve (2000) Chapter 5.2.4 page 218.
4
See Shonkwiler (2013) Chapter 3.4, page 90 and Trigeorgis (1996), page 102-103.

8. 8
If the prices of the derivatives 𝑉(𝑟, 𝑡; 𝑇) and the estimated of the model parameters
𝑢(𝑟, 𝑡) and 𝑤(𝑟, 𝑡) are known, it is then possible to find an implied value of 𝜆, the market
price of interest rate risk.
2.1.2 The ex-ante bond Sharpe ratio
The Sharpe ratio is understood to be the market price of risk, i.e. it is the return in
excess of the risk free rate on an investment per unit of volatility. For equities often expressed
as:
𝑆𝑅 𝐸
=
𝜇 − 𝑟
𝜎
(13)
Where 𝜇 is the expected return, 𝜎 is the volatility of returns and 𝑟 is the risk free rate.
While it has common for these parameters to be obtained statistically to form an ex-
post Sharpe ratio, 𝜇 and 𝜎 also correspond to the stochastic differential equation most
commonly used to model the price evolution of stocks, also called geometric Brownian
motion.
𝑑𝑆 = 𝜇𝑆𝑑𝑡 + 𝜎𝑆𝑑𝑊 (14)
In this case, assuming the above process is true, the Sharpe ratio becomes ex-post.
However, this distinction generally not emphasised because a common assumption is that that
the best measure of future performance is past performance, in which case the Sharpe ratio is
both ex-ante and ex-post. The ex-ante Sharpe ratio is a positive value since risk averse
investors require expected excess returns for a risky investment. On the other hand, the ex-
post Sharpe ratio could be negative depending on the sample taken. Investors could have
expected a positive return, but the realised outcome within the sample period could have been
a loss.
However, the market price of interest rate risk identified in the previous section is a
negative value, representing the modification to the short rate’s drift process. As it does not
pertain directly to returns, an equivalent Sharpe ratio for fixed income assets, particularly zero
coupon bond needs to be derived

9. 9
Since bonds are priced under the risk neutral measure5
, the dynamics of a bond price
𝑉(𝑟, 𝑡; 𝑇) is obtained by applying the Ito formula (6) on the risk neutral process of 𝑟𝑡 defined
in equation (12) to obtain:
𝑑𝑉 = [
𝜕𝑉
𝜕𝑡
+
1
2
𝑤(𝑟𝑡)2
𝜕2
𝑉
𝑑𝑟2
+
𝜕𝑉
𝜕𝑟
(𝑢(𝑟𝑡, 𝑡) − 𝜆(𝑟𝑡, 𝑡)𝑤(𝑟𝑡, 𝑡))] 𝑑𝑡
+
𝜕𝑉
𝜕𝑟
𝑤(𝑟𝑡, 𝑡)𝑑𝑊
(15)
Further, equation (11) implies that6
:
𝜕𝑉
𝜕𝑡
= 𝑟𝑡 𝑉(𝑟, 𝑡; 𝑇)
𝜕𝑉
𝜕𝑟
= −(𝑇 − 𝑡) 𝑉(𝑟, 𝑡; 𝑇)
𝜕2
𝑉
𝑑𝑟2
= (𝑇 − 𝑡)2
𝑉(𝑟, 𝑡; 𝑇)
(16)
Substituting (16) into (15) yields the expression:
𝑑𝑉 = [𝑟𝑡 +
1
2
𝑤(𝑟𝑡)2(𝑇 − 𝑡)2
− (𝑇 − 𝑡)(𝑢(𝑟𝑡) − 𝜆(𝑡)𝑤(𝑟𝑡))] 𝑉𝑑𝑡
+ [−(𝑇 − 𝑡)𝑤(𝑟𝑡)]𝑉𝑑𝑊
(16)
and then making the following definitions:
𝑚(𝑟, 𝑡) = 𝑟𝑡 +
1
2
𝑤(𝑟𝑡)2(𝑇 − 𝑡)2
− (𝑇 − 𝑡)(𝑢(𝑟𝑡) − 𝜆(𝑡)𝑤(𝑟𝑡))
𝑠(𝑟, 𝑡) = −(𝑇 − 𝑡)𝑤(𝑟𝑡)
(17)
(16) can be abbreviated to:
𝑑𝑉𝑡 = 𝑚(𝑟𝑡, 𝑡)𝑉𝑡 𝑑𝑡 + 𝑠(𝑟𝑡, 𝑡)𝑉𝑡 𝑑𝑊 (18)
Comparing the above expression with the stock price process (14) allows the equivalent ex-
ante Sharpe ratio for the zero coupon bond 𝑉(𝑟, 𝑡; 𝑇) to be derived:
𝑆𝑅 𝐸
=
𝜇 − 𝑟
𝜎
⟹ 𝑆𝑅𝑡
𝐵
=
𝑚(𝑟𝑡, 𝑡) − 𝑟
𝑠(𝑟𝑡, 𝑡)
= −𝜆(𝑟𝑡, 𝑡) +
𝑢(𝑟𝑡)
𝑤(𝑟𝑡)
−
1
2
𝑤(𝑟𝑡)(𝑇 − 𝑡)
(19)
5
See end of section 2.1.2
6
Workings are shown in Appendix

10. 10
2.1.3 Preferences and risk aversion
I consider an economy comprised of infinitely lived homogenous agents who have a
time separable lifetime utility function as below:
𝑈𝑡 = 𝔼 𝑡 ∑ {𝛽 𝑘
𝑐𝑡+𝑘
1−𝛾𝑡
1 − 𝛾𝑡
}
∞
𝑘=0
(20)
The representative agent gains utility from consumption, 𝑐𝑡. 𝛽 is the subjective discount
factor. 𝛾𝑡 is the relative risk aversion parameter which evolves according to some stochastic
process.
Within the context of the utility function, the market price of risk is defined as the
ratio of the standard deviation and expectation of the stochastic discount factor7
:
𝑀𝑃𝑅 =
𝜎(𝑚)
𝔼[𝑚]
(21)
Where 𝑚, the stochastic discount factor, is the intertemporal rate of substitution (IMRS) given
by:
𝑚 𝑡,𝑡+1 =
𝛽𝑢′(𝑐𝑡+1)
𝑢′(𝑐𝑡)
(22)
The market price of risk is the excess expected return of the market portfolio in excess of the
risk free rate per unit of risk, in other words the Sharpe ratio of the optimal market portfolio
which I will denote as 𝑆𝑅, so that:8
𝑆𝑅 = 𝑀𝑃𝑅 =
𝜎(𝑚)
𝔼[𝑚]
(23)
I further assume that the ex-ante bond Sharpe ratio I derived is a good approximation the
market portfolio Sharpe ratio, in that:
𝑆𝑅𝑡
𝐵
≅ 𝑆𝑅𝑡 (24)
One possible reservation about the assumption made in (24) is that market-moving
market participants with the power to affect prices tend to be institutional, while private
7
Tallarini (2000) pp. 512
8
See footnote 1 in Friend and Blume (1975) for one example of this definition.

11. 11
individuals or consumers are perceived as “noise traders”. Consequently, preferences among
investors may not reflect preferences among consumers.
(24) is a reasonable assumption because institutional investors use funds from private
individuals in their investments. Pension funds, hedge funds, asset managers and other buy-
side firms use funds from their clients. Consumers who do not invest store their money with
banks, which either lend the money or use it to make investments. Since the economic model
in Section 2 does not distinguish between consumers and institutional investors, we assume
all agents engage in investment activities directly or indirectly. Consequently, preferences
among investors is not distinguished from preferences among consumers.
Finally, in order to obtain an explicit relationship between the market price of risk and
relative risk aversion in the context of the time separable utility function I follow the approach
of Tallarini (2000) in assuming a log-normal random walk process for consumption:
log(𝑐𝑡) = log(𝑐𝑡−1) + 𝜖 𝑡
𝑐
𝜖 𝑡
𝑐
~𝑁(0, 𝜎𝑐
2
)
(26)
Which has the implication that the market price of risk can be expressed as9
:
𝑀𝑃𝑅 =
𝜎(𝑚)
𝔼(𝑚)
= (exp{𝜎𝑐
2
𝛾2} − 1)
1
2
(27)
This allows relative risk aversion to be computed directly given the market price of risk and
the volatility of consumption according to the equation:
𝛾𝑡 =
√log(𝑀𝑃𝑅𝑡
2
+ 1)
𝜎𝑐
(28)
Here, an important point to make is that the following evolutionary process for 𝛾𝑡:
𝛾𝑡 = 𝛾̅ + 𝜉(𝑦𝑡 − 𝛾̅) + 𝜖 𝑡
𝛾
ϵt
γ
~𝒩(0, σγ)
(29)
would be incorrect because the it would result in a non-zero probability of 𝛾𝑡 being negative,
which would be a violation of the equation (28).
9
A derivation of this property is available in Ljungqvist and Sargent (2012), pages 529-530

12. 12
Consequently, the 𝛾𝑡 process is instead defined as a function of some AR(1) ex-ante Sharpe
ratio process, 𝑆𝑅𝑡:
𝑆𝑅𝑡 = 𝑆𝑅̅̅̅̅ + χ(SRt−1 − 𝑆𝑅̅̅̅̅) + 𝑆𝑅̅̅̅̅ 𝜍 𝜀𝑡
𝑆𝑅
𝜎(𝜀𝑡
SR
) = 1, 𝔼(𝜀𝑡
𝑆𝑅) = 0
𝑀𝑃𝑅𝑡 = |𝑆𝑅𝑡|
𝛾𝑡 =
√log(𝑀𝑃𝑅𝑡
2
+ 1)
𝜎𝑐
(30)
where 𝑆𝑅̅̅̅̅ is the equilibrium value of the Sharpe ratio, 𝜒 is the rate of mean reversion and 𝜀𝑡
𝑆𝑅
is some error term with a standard deviation of 1. Expressing the volatility of the process as
𝑆𝑅̅̅̅̅ 𝜍, where 𝜍 is some constant, allows the volatility to scaled to the equilibrium value 𝑆𝑅̅̅̅̅.
When 𝛾𝑡 = 𝛾̅ = 1, and the variance of 𝜀𝑡
𝑆𝑅
is zero, we collapse into the commonly
used log utility functions. The Sharpe ratio within the model, 𝑆𝑅𝑡 represents the Sharpe ratio
on all the assets in the model economy.
2.2 Estimation methodology
2.2.1 Data
I follow Ahmad and Willmott (2007) in assuming that the 1-month US LIBOR rate is
a suitable proxy for the short rate, and calculate the slope of the yield curve using the
differential between the 2-month rate and 1-month rate.
Daily data for USD LIBOR from between 1987 and 2006 is used, just before the
subprime debt crisis. Post-crisis, exceptional prominence of central bank intervention and
suppressed interest rates make the environment unsuitable for this analysis.
2.2.2 The short rate process
Ahmad and Wilmott (2007) used a time inhomogeneous parameterisation of the short
interest rate model of the form:
𝑑𝑟𝑡 = 𝑢(𝑟𝑡)𝑑𝑡 + 𝑣𝑟𝑡
𝜂
𝑑𝑊𝑡
𝑢(𝑟𝑡) = 𝑣2
𝑟𝑡
(2𝜂−1)
(𝛽 −
1
2
−
1
2𝑎2
log (
𝑟𝑡
𝑟̅
))
(31)

13. 13
To estimate the volatility parameters 𝑣 and 𝜂, the time series 𝑑𝑟𝑡 is calculated and divided
into buckets, for which averages are calculated. Squaring both sides of (1) and using the
result10 that 𝑑𝑡2
= 0 and 𝑑𝑋2
= 𝑑𝑡, we have that:
𝔼[(𝛿𝑟)2] = 𝑣2
𝑟2𝜂
𝛿𝑡 (32)
Taking logs of (32) gives:
log(𝔼[(𝛿𝑟)2]) = 2 log(𝑣) + log(𝛿𝑡) + 2ηlog(r) (33)
The parameters 𝛽 and 𝑣 are then estimated using OLS between log(𝔼[(𝛿𝑟)2]) and log(𝑟).
The drift structure 𝑢(𝑟) is derived by using the steady-state probability density
function of r. Unlike equity prices, interest rates can be considered to be bounded and have a
steady state. The Fokker-Planck equation, or transition density function for a process 𝑑𝑟 with
probability density function 𝑝(𝑟, 𝑡) is given by:
𝜕𝑝
𝜕𝑡
=
1
2
𝑣2
𝜕2
𝜕𝑟2
[𝑟2𝜂
𝑝(𝑟, 𝑡)] −
𝜕
𝜕𝑟
[𝑢(𝑟)𝑝(𝑟, 𝑡)]
(34)
If there exists a steady state, then taking limits 𝑡 → ∞ gives:
𝜕𝑝∞
𝜕𝑡
= 0
(35)
i.e. the steady state distribution, by definition will not change with time. The Fokker-Planck
equation simplifies from a partial differential equation into an ordinary differential equation
with respect to the interest rate alone:
1
2
𝑣2
𝑑2
𝑑𝑟2
[𝑟2𝜂
𝑝∞(𝑟)] −
𝑑
𝑑𝑟
[𝑢(𝑟)𝑝∞(𝑟)] = 0
(36)
This is easily integrated to give us and expression for the drift term:
𝑢(𝑟) = 𝑣2
𝜂𝑟2𝜂−1
+ 𝑣2
1
2
𝑟2𝜂
𝑑
𝑑𝑟
[log(𝑝∞(𝑟))]
(37)
The one-month USD LIBOR rates are assumed to be log-normally distributed so that:
𝑝∞(𝑟) =
1
𝑎𝑟√2𝜋
exp {−
1
2𝑎2
(𝑙𝑜𝑔( 𝑟/𝑟̅))2
}
(38)
10
See Shreve (2000) page 141.

14. 14
Substitution of the lognormal distribution into (37) yields an expression for the drift term:
𝑢(𝑟) = 𝑣2
𝑟(2𝜂−1)
(𝛽 −
1
2
−
1
2𝑎2
log(𝑟/𝑟̅))
(39)
2.2.3 The market price of interest rate risk and bond Sharpe ratio
We begin by seeking an approximate solution to 𝑉(𝑟, 𝑡; 𝑇) of the form:
𝑉~1 + 𝑎(𝑟)(𝑇 − 𝑡) + 𝑏(𝑟)(𝑇 − 𝑡)2
+ ⋯ (40)
which is a Taylor series expansion about 𝑡 = 𝑇 with unknown coefficients. This is a good
approximation for short term bonds, i.e when 𝑇 − 𝑡 → 0. Substitution into the bond pricing
equation (8) yields the solution:
𝑉(𝑟, 𝑡; 𝑇)~1 − 𝑟(𝑇 − 𝑡) +
1
2
(𝑇 − 𝑡)2(𝑟2
− 𝑢 + 𝜆𝑤) + ⋯
(41)
Unity is the value of the bond at maturity. Before maturity, the bond price is lowered by 𝑟(𝑡 −
𝑇), with convexity correction of
1
2
(𝑇 − 𝑡)2(𝑟2
− 𝑢 + 𝜆𝑤). Further, using the results that
log(1 + 𝑟) ~𝑟 and 𝑟2
~0 for small values of 𝑟 allows (41) to be simplified to:
−
𝑙𝑜𝑔𝑉
𝑇 − 𝑡
~𝑟 +
1
2
(𝑢 − 𝜆𝑤)(𝑇 − 𝑡) + ⋯
(42)
where −
𝑙𝑜𝑔𝑉
𝑇−𝑡
is by definition the yield to maturity of the bond, so that the slope of the yield
curve is approximated by:
𝑆𝑙𝑜𝑝𝑒𝑡 =
(−
𝑙𝑜𝑔𝑉
𝑇 − 𝑡
− 𝑟)
𝑇 − 𝑡
=
1
2
(𝑢 − 𝜆𝑤)
(43)
The slope at the short end is found empirically, by taking the differences in yield on 2-month
LIBOR and 1-month LIBOR and dividing by the time interval expressed in fractions of a
year. Rearranging expression (43) gives:
𝜆 𝑡 =
𝑢(𝑟𝑡)
𝑤(𝑟𝑡)
− 2 ∗
𝑆𝑙𝑜𝑝𝑒𝑡
𝑤(𝑟𝑡)
(44)
The bond Sharpe ratio can then be computed directly using (19).

15. 15
2.2.4 The volatility of consumption
The parameters from equation (26) are estimated directly from US aggregate
consumption from 1947 to 2015, detrended using the Hodrick-Prescott filter. 𝜎𝑐 is obtained
from empirical data rather than estimated as a model-implied function of the technology shock
in the DSGE to keep the estimates of the variable 𝛾𝑡 strictly exogenous. While it would have
been more consistent to estimate this using data from 1987 to 2015, this resulted in a very low
volatility of consumption, pushing 𝛾𝑡 to values too high for a solution in the DSGE model.
2.2.5 Risk aversion
Finally, the time series is computed using equations (24) and (28):
𝑆𝑅𝑡
𝐵
≅ 𝑆𝑅𝑡 = 𝑀𝑃𝑅𝑡
𝛾𝑡 =
√log(𝑀𝑃𝑅𝑡
2
+ 1)
𝜎𝑐
2.3 Empirical results and analysis
The final short interest rate model is:
𝑑𝑟𝑡 = 𝑣2
𝑟𝑡
(2𝜂−1)
(𝜂 −
1
2
−
1
2𝑎2
log(𝑟𝑡/𝑟̅)) 𝑑𝑡 + 𝑣𝑟𝑡
𝜂
𝑑𝑊
Where the volatility parameters are estimated using OLS:
log(𝔼[(𝛿𝑟)2]) = 2 log(𝑣) + log(𝛿𝑡) + 2ηlog(r)
The drift parameters are estimated using maximum likelihood on:
𝑝∞(𝑟) =
1
𝑎𝑟√2𝜋
exp {−
1
2𝑎2
(𝑙𝑜𝑔( 𝑟/𝑟̅))2
}
Table 1 shows the estimated OLS coefficients, which have high confidence levels in
excess of 99.9%. Figure 1 plots log(𝔼[(𝛿𝑟)2]) against log(r), showing that a linear model
does fit the data well. The volatility parameters are summarised in Table 2. In particular, 𝜂 is
close to 1, so that volatility approximately scales with the interest rate:
𝑤(𝑟𝑡) ≅ 0.15 𝑟𝑡
In the estimation of the drift parameters, the mean value of the interest rate is 4.32%,
shown in Table 3. Figure 1 shows the lognormal fit against the density histogram of the
interest rates, demonstrating a reasonably good fit.

16. 16
Table 1. Regression results and parameters for volatility
Coefficient Value Error t-statistic Confidence
2 log(𝑣) + log(𝛿𝑡) -9.3325 0.6828 -13.668 >99.9%
2η 1.9913 0.2195 9.071842 >99.9%
𝑅2 0.4861
Table 2. Volatility parameters
Parameters
𝜂 0.9957
𝑣 0.1493
Table 3. Drift parameters
Parameter Value 95% confidence interval
𝑟̅ 0.0432 [0.0425, 0.0438]
𝑎 0.5547 [0.5442, 0.5655]
Figure 1. Short rate model parameter fits. Left pane: log(𝔼[(δr)2]) against log(r). Right pane: lognormal fit for
steady state distribution of interest rates.
The estimated time series for the bond Sharpe ratio and relative risk aversion is shown
in Figure 2. The average market price of risk in the period is 0.89 on an annual basis and 0.45
when converted to a quarterly basis. Relative risk aversion has a sample mean of 50 and
standard deviation of 30 quarterly averages are used.
The daily time series for the bond Sharpe ratio shows a very strong level of mean
reversion which is consistent with what most economists would expect. Another important
observation is that the Sharpe ratio estimated is occasionally negative, which is a
y = 1.9913x - 9.3325
R² = 0.4861
-21
-19
-17
-15
-13
-11
-5 -4 -3 -2
0
5
10
15
20
25
30
1.45 3.25 5.05 6.85 8.65 10.45 11.45

17. 17
Figure 2: Bond Sharpe ratio against the economy. Primary axis: Bond Sharpe ratio, daily and quarterly averages.
Secondary axis: Economic variables, filtered using the Hodrick-Prescott filter
counterintuitive result considering the Sharpe ratio is estimated is ex ante. This will be
discussed in greater detail in Section 4.1.1.
Figure 2 shows the pro-cyclical behaviour of the bond Sharpe ratio, where peaks and
troughs in GDP, Consumption and Investment (highlighted with grey bands) have historically
corresponded to peaks and troughs in the Sharpe ratio. An intuition behind this is that when
the economy is expanding, there exist more investment opportunities providing good returns
and lower risk. These opportunities, even if they exist in markets outside the interest rate
market would have an impact. For example, an economic boom may results in a preference
for equities and credit relative to bonds, causing bonds to sell at a discount. This naturally
increase the return on bonds since the payout is fixed. During slumps, institutional investors
such as hedge funds still have a mandate to seek out investment opportunities for their clients
and consequently have a smaller selection of good investments to choose from, decreasing the
Sharpe ratio.
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
-15
-10
-5
0
5
10
Bond Sharpe ratio, daily Bond Sharpe ratio, quarterly averages
GDP Consumption
Investment

18. 18
Figure 3: Relative risk aversion against the economy. Primary axis: Relative risk aversion, daily and quarterly
averages. Secondary axis: Economic variables, filtered using the Hodrick-Prescott filter
It is noteworthy that between 2004 and the crisis, the bond Sharpe ratio steadily worsened
even while the economy was growing, possibly reflecting the unsustainable growth in
mortgage related investments that caused the sub-prime crisis.
Figure 3 shows the behaviour of relative risk aversion. It shows similar properties to
the bond ratio during periods where the bond Sharpe ratio is positive. However deep troughs
in the Sharpe ratios result in small peaks in relative risk aversion. One implication of this is
that risk aversion would increase during very adverse periods of the economy.
The estimation results for the AR(1) process for the quarterly Sharpe ratios:
𝑆𝑅𝑡 = 𝑘 + 𝜒𝑆𝑅𝑡−1 + 𝑆𝑅̅̅̅̅ 𝜍 𝜀𝑡
𝑆𝑅
are reported in Table 4. Considering that the mean reversion process in (30) can also be
expressed as:
𝑆𝑅𝑡 = 𝑆𝑅̅̅̅̅(1 − 𝜒) + 𝜒𝑆𝑅𝑡−1 + 𝑆𝑅̅̅̅̅ 𝜍 𝜀𝑡
𝑆𝑅
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
-250
-200
-150
-100
-50
0
50
100
150
200
250
Bond Sharpe ratio, daily Bond Sharpe ratio, quarterly averages
GDP Consumption
Investment

19. 19
Table 4
Quarterly Sharpe ratio process parameters
Parameter Value Error t-statistic Confidence
𝑘 0.307619 0.166677 1.8456 90%
𝜒 0.647749 0.095595 6.77594 >99.9%
(𝑆𝑅̅̅̅̅ 𝜍)2 0.771191 0.086431 8.92262 >99.9%
𝑙𝑜𝑔𝐿 -103.1223
[
and comparing 𝑆𝑅̅̅̅̅(1 − 𝜒) with 𝑐 equation X, I can deduce the long term value of the Sharpe
ratio:
𝑆𝑅̅̅̅̅ =
𝑘
1 − 𝜒
= 0.8733
Considering that the mean reversion process in (30) can also be expressed as:
𝑆𝑅𝑡 = 𝑆𝑅̅̅̅̅(1 − 𝜒) + 𝜒𝑆𝑅𝑡−1 + 𝑆𝑅̅̅̅̅ 𝜍 𝜀𝑡
𝑆𝑅
and comparing 𝑆𝑅̅̅̅̅(1 − 𝜒) with 𝑐 equation X, I can deduce the long term value of the Sharpe
ratio:
𝑆𝑅̅̅̅̅ =
𝑘
1 − 𝜒
= 0.8733
Note that this is the Sharpe ratio on the annual basis. This implies a quarterly Sharpe ratio of
0.44, which is higher than historical stock market Sharpe ratios of between 20-30%11
. Given
𝑆𝑅̅̅̅̅, the value of 𝜍 then becomes 1.006, indicating that volatility scales to the level of the
Sharpe ratio.
The volatility of the random walk consumption model is estimating by taking standard
deviations of the expression:
log(𝑐𝑡) − log(𝑐𝑡−1) = 𝜉 + 𝜖 𝑡
𝑐
𝜖 𝑡
𝑐
~𝑁(0, 𝜎𝑐)
and an estimate 𝜎𝑐 = 0.008374 is used.
11
Cogley and Sargent (2005)

20. 20
Then, the steady state value for the relative risk aversion parameter is:
𝛾̅ =
√log(|𝑆𝑅̅̅̅̅|2 + 1)
𝜎𝑐
= 49.8856
As highlighted in the introduction, although this value for 𝛾̅ exceeds the traditional
single-digit values, it concurs with the findings of Cochrane and Hansen (1992) and Campbell
and Cochrane (1999), who found that relative risk aversion in excess of 40 was required to
satisfy the Hansen-Jagannathan bounds. Kandel and Stambaugh (1991) found a value of 29,
while Tallarini (2000) used values up to 100.
3 The production economy
3.1 The model
Here I present the model I use to study the effects of my calibration in section 2 on
business cycles. The model is based on a baseline Real Business Cycle (RBC) model
calibrated by Fernandez-Villaverde in 2005. However, I modify the utility function with
respect to consumption.
Preferences of the economic agent are defined over consumption 𝑐𝑡 and leisure, (1 − 𝑙 𝑡).
𝑈𝑡 = 𝔼 𝑡 ∑ 𝛽 𝑘
{
𝑐𝑡+𝑘
1−𝛾𝑡
1 − 𝛾𝑡
+ Ψ log(1 − 1 𝑡)}
∞
𝑘=0
(46)
The agent gains utility from consumption and is endowed with one unit of time, which he can
apportion between work, 𝑙 𝑡, and leisure, (1 − 𝑙 𝑡). The agent derives utility from leisure,
which is the time spent not working. 𝛽 is the subjective discount factor while 𝛾𝑡 is the relative
risk aversion parameter which evolves according to some stochastic process.12
The economy is also populated by homogenous firms which are subject to the
neoclassical production function:
𝑦𝑡 = 𝑘 𝑡
𝛼(𝑒 𝑧 𝑡 𝑙 𝑡)1−𝛼 (47)
where production in period 𝑡, 𝑦𝑡 requires two inputs: capital stock, 𝑘 𝑡, and labour employed,
𝑙 𝑡 which is augmented by technology, 𝑒 𝑧 𝑡, which evolves according to the law of motion:
12
Fernandez-Villaverde used the utility function: 𝑈𝑡 = 𝔼 𝑡 ∑ 𝛽 𝑘{log(𝑐𝑡) + Ψ log(1 − 1 𝑡)}∞
𝑘=0 .

21. 21
𝑧𝑡 = 𝜌𝑧𝑡−1 + 𝜎𝑡
𝑍
𝜀𝑡
z
𝜀𝑡
𝑧
~𝒩(0,1)
(48)
The costs of these inputs to the firm is in the form of interest payments, 𝑟𝑡 on capital stock,
which can be interpreted payments on loans used to buy machinery, and wage payments, 𝑤𝑡
on labour supply. In addition, capital stock depreciates at the rate 𝛿, so that the firm has an
intertemporal profit function:
Π 𝑡 = 𝔼 𝑡 ∑{𝑦𝑡+𝑘 − 𝑘 𝑡 𝑟𝑡 − 𝑙 𝑡 𝑤𝑡 − 𝛿𝑘 𝑡−1}
∞
𝑘=0
(49)
Maximising the intertemporal profit subject to the production function by choosing ( 𝑘 𝑡, 𝑙 𝑡)
yields:
𝑟𝑡 = 𝛼𝑘 𝑡
𝛼−1(𝑒 𝑧 𝑡 𝑙 𝑡)1−𝛼
− 𝛿 = 𝛼
𝑦𝑡
𝑘 𝑡
(50)
𝑤𝑡 = (1 − 𝛼)𝑘 𝑡
𝛼(𝑒 𝑧 𝑡 𝑙 𝑡)1−𝛼
𝑙 𝑡
−1
= (1 − 𝛼)
𝑦𝑡
𝑙 𝑡
(51)
Additionally, the economy, which is composed of the agents and firms is constrained
by the following:
𝑐𝑡 + 𝑘 𝑡+1 − (1 − 𝛿)𝑘 𝑡 = 𝑤𝑡 𝑙 𝑡 + 𝑟𝑡 𝑘 𝑡 (52)
𝑤𝑡 𝑙 𝑡 + 𝑟𝑡 𝑘 𝑡 = 𝑦𝑡 represents the amount payed to the inputs of production, while 𝑐𝑡 + 𝑘 𝑡+1 −
(1 − 𝛿)𝑘 𝑡 is the sum of consumption and investment in new capital. Expenditure on
consumption and investment needs to be balanced by production.
Incorporating the solutions of the Firms’ maximisation into the household problem, I obtain a
simplified problem:
max
𝑐 𝑡,𝑙 𝑡
𝑈𝑡 = 𝔼 𝑡 ∑ {𝛽 𝑘
𝑐𝑡+𝑘
1−𝛾𝑡
1 − 𝛾𝑡
+ log(1 − 𝑙 𝑡)}
∞
𝑘=0
𝑠. 𝑡
𝑐𝑡 + 𝑘 𝑡+1 = 𝑘 𝑡
𝛼(𝑒 𝑧 𝑡 𝑙 𝑡)1−𝛼
+ (1 − 𝛿)𝑘 𝑡
𝑆𝑅𝑡 = 𝑆𝑅̅̅̅̅ + χ(SRt−1 − 𝑆𝑅̅̅̅̅) + 𝑆𝑅̅̅̅̅ 𝜍 𝜀𝑡
𝑆𝑅
𝜀𝑡
𝑆𝑅
~𝒩(0,1)
𝑀𝑃𝑅𝑡 = |𝑆𝑅𝑡|
(53)

22. 22
𝛾𝑡 =
√log(𝑀𝑃𝑅𝑡
2
+ 1)
𝜎𝑐
𝑧𝑡 = 𝜌𝑧𝑡−1 + 𝜎𝑡
𝑍
𝜖 𝑡
𝑧
𝜀𝑡
𝑧
~𝒩(0,1)
𝜌 𝑍,𝑆𝑅 = 0 𝑜𝑟 1
I consider two possibilities regarding the correlations between 𝜀𝑡
𝑆𝑅
and 𝜀𝑡
𝑆𝑅
. The first to
assume both the shocks are orthogonal, allowing the risk aversion shock to be analysed in
isolation. The second is to assume that both shocks are correlated, as motivated by the
observations made in Section 2.3. As a simplification, I test a model with perfect correlation
between technology and Sharpe ratio shocks.
The first order conditions for the model solutions are:
𝑐𝑡
−𝛾𝑡
= 𝛽{𝑐𝑡+1
−𝛾𝑡+1
(1 + 𝛼𝑘 𝑡
𝛼−1(𝑒 𝑧 𝑡 𝑙 𝑡)1−𝛼
− 𝛿)}
Ψ
(1 − 𝑙 𝑡)−1
𝑐𝑡
−𝛾𝑡
= (1 − 𝛼)𝑘 𝑡
𝛼(𝑒 𝑧 𝑡 𝑙 𝑡)1−𝛼
𝑙 𝑡
−1
= wt
𝑐𝑡 + 𝑘 𝑡+1 = 𝑘 𝑡
𝛼(𝑒 𝑧 𝑡 𝑙 𝑡)1−𝛼
+ (1 − 𝛿)𝑘 𝑡
𝑆𝑅𝑡 = 𝑆𝑅̅̅̅̅ + χ(SRt−1 − 𝑆𝑅̅̅̅̅) + 𝑆𝑅̅̅̅̅ 𝜍 𝜀𝑡
𝑆𝑅
𝜀𝑡
𝑆𝑅
~𝒩(0,1)
𝑀𝑃𝑅𝑡 = |𝑆𝑅𝑡|
𝛾𝑡 =
√log(𝑀𝑃𝑅𝑡
2
+ 1)
𝜎𝑐
𝑧𝑡 = 𝜌𝑧𝑡−1 + 𝜎𝑡
𝑍
𝜖 𝑡
𝑧
𝜀𝑡
𝑧
~𝒩(0,1)
𝜌 𝑍,𝑆𝑅 = 0 𝑜𝑟 1
(54)
Second order Taylor approximations are performed through Dynare to produce the model
estimates.

23. 23
3.2 Model specifications
Six models are estimated to analyse the impact of both an increased levels of risk
aversion and stochastic risk aversion separately and together. Their performance in generating
moments for the aggregate variables, output (y), consumption (c), investment (i) and wage
(w) are compared relative to the original baseline RBC model, B0. Their performance is
measured by how well they are able to generate the empirical moments of the US economy in
Table 6.
Referring to Table 5, I use values calibrated by Fernandez-Villaverde in 2005 for the baseline
parameters which are kept constant across each model. To study the effect of increasing risk
aversion to my estimated mean of 49.89, I compare model (HB) (High risk aversion,
baseline) with model B0.
Table 5
Parameter values for the models B0,LCs,LOs, HB, HCs, HOs. Models beginning with “L” and “H” represent
low and high values of the steady state aversion, 𝛾̅ = 1 and 𝛾̅ = 49.89 respectively. B0 and and HB have
constant risk aversion. Models ending with “Cs” have perfectly correlated shocks. Models ending with “Os”
have an orthogonal shocks
Values
𝛾̅ = 49.89 𝛾̅ = 1
Shocks
Parameters
High
orthogonal
shocks
(HOs)
High
correlated
shock
(HCs)
No risk
aversion
shocks
(HB)
Low
orthogonal
shock
(LOs)
Low
correlated
shock
(HCs)
No risk
aversion
shocks
(B0)
Baseline parameters (Constant across all models)
𝛼 0.33 0.33 0.33 0.33 0.33 0.33
𝛽 0.99 0.99 0.99 0.99 0.99 0.99
𝛿 0.023 0.023 0.023 0.023 0.023 0.023
𝜓 1.75 1.75 1.75 1.75 1.75 1.75
𝜌 0.95 0.95 0.95 0.95 0.95 0.95
𝜎𝑧 0.0104 0.0104 0.0104 0.0104 0.0104 0.0104
Risk aversion parameters
𝛾̅ 49.89 49.89 49.89 49.89 49.89 1
𝜎𝑐 0.008374 0.008374 - 0.008374 0.008374 -
𝑆𝑅̅̅̅̅ 0.893 0.893 0.017 0.008 0.008 0.017
𝜒 0.649 0.649 - 0.649 0.649 -
𝜍 1.006 1.006 - 1.006 1.006 -
𝜌 𝑆𝑅,𝑧 0 1 - 0 1 -

24. 24
Table 6
Business cycle empirical data. Sample moments of HP-filtered quarterly data taken from the Federal Reserve
Bank of St. Louis, 1987:1 to 2006:4
Autocorrelations Cross-correlations
Variable Std. dev 1 2 3 y c i w
y 0.0101 0.885 0.720 0.517 1 0.877 0.919 0.109
c 0.00859 0.900 0.767 0.599 0.877 1 0.754 0.346
i 0.0449 0.877 0.701 0.494 0.919 0.754 1 0.025
w 0.00987 0.684 0.562 0.401 0.110 0.346 0.025 1
Model (LOs) (Low relative risk aversion, orthogonal shocks) evaluates the effect of
introducing a stochastic process to risk aversion without altering its expected value from the
baseline model, while (LCs) (Low relative risk aversion, correlated shocks) are used to
evaluate the effect of correlation on the model economy. The choice of parameterisation of
the stochastic Sharpe ratio in (30) was:
𝑆𝑅𝑡 = 𝑆𝑅̅̅̅̅ + χ(SRt−1 − 𝑆𝑅̅̅̅̅) + 𝑆𝑅̅̅̅̅ 𝜍 𝜀𝑡
𝑆𝑅
Thus implying that the volatility in (LOs) and (LCs) (where 𝑆𝑅̅̅̅̅ = 1) is 𝜍.
Models (HOs) (High relative risk aversion, orthogonal shocks) and (HCs) (High
relative risk aversion, correlated shocks) correspond to my 2 proposed models in (53) and
allow for the simultaneous effect of elevated risk aversion levels and stochastic behaviour.
3.3 Empirical results and analysis
Table 7 shows the moments generated by the models with 𝛾̅ = 1. The impact of the shocks
on the aggregate quantity variables, y, c, i, and w are measured by their correlations with the
shocks.
The baseline RBC model (B0) underestimates volatility in consumption and
investment at 0.0034 and 0.011 compared to the empirical moments of 0.0859 and 0.0449,
while standard deviations for output and wage at 0.014 and 0.007 are largely in line with the
data. Its performance with autocorrelations are in line with the data, however auto-
correlations on consumption and investment are underestimated. Because the effect of the
technology shock 𝜀𝑡
𝑧
is positive on each variable considered, cross-correlations are positive
and generally line with the data, although slightly biased upward, with the exception of cross-
correlations with wage.
An orthogonal risk aversion shock 𝜀𝑡
𝑆𝑅
(I refer to this as the Sharpe ratio shock and
relative risk aversion shock interchangeably) is introduced to the baseline model, resulting in

25. 25
the (LOs) model. As can be seen from Table 8, this shock dominates the technology shock in
its effect on both consumption and investment, accounting for 99.9% and 98% of variance
respectively. This, combined with the risk aversion shock’s positive correlations on
consumption and output and negative correlations with investment and wage results in wrong
way correlation for consumption-investment, consumption-wage and output-investment when
compared to the baseline model and empirical moments.
When technology and risk aversion shocks are perfectly correlated (model (LCs)),
only investment is negatively correlated with the shock, resulting in wrong-way correlation
for pairs involving investment. In both (LOs) and (LCs), the overall standard deviations are
far beyond the empirical observations, suggesting these modifications worsen the model.
Table 7
Moments for models with 𝛾̅ = 1. Moments of shocks highlighted in bold. 𝜺𝒕
𝒛
is the shock to technology, 𝜺𝒕
𝑺𝑹
is
the shock to the Sharpe ratio while 𝜺𝒕 is the simultaneous shock to both technology and the Sharpe ratio.
Autocorrelations Cross-correlations
Variable Std. dev 1 2 3 y c i w
Correlated shocks (HCs)
y 0.0211 0.562 0.270 0.079 1 0.937 -0.898 0.076
c 0.0935 0.495 0.182 -0.005 0.937 1 -0.995 0.398
i 0.0741 0.496 0.183 -0.005 -0.898 -0.995 1 -0.481
w 0.0033 0.750 0.520 0.319 0.076 0.398 -0.481 1
𝜺𝒕 1 - - - 0.658 0.790 -0.805 0.583
Orthogonal shocks (LOs)
y 0.0173 0.655 0.394 0.198 1 0.572 -0.424 0.466
c 0.0923 0.501 0.189 0.001 0.472 1 -0.986 -0.224
i 0.0836 0.505 0.195 0.001 -0.424 -0.984 1 0.344
w 0.0115 0.837 0.638 0.436 0.466 -0.224 0.344 1
𝜺𝒕
𝒛
1 - - - 0.348 0.007 0.064 0.210
𝜺𝒕
𝑺𝑹
1 - - - 0.458 0.790 -0.778 -0.041
Baseline (B0)
y 0.014 0.718 0.478 0.279 1 0.819 0.991 0.947
c 0.0034 0.804 0.615 0.439 0.819 1 0.888 0.960
i 0.011 0.708 0.462 0.261 0.991 0.888 1 0.981
w 0.0073 0.743 0.518 0.326 0.947 0.960 0.981 1
𝜺𝒕
𝒛
1 - - - 0.420 0.193 0.469 0.336

26. 26
Table 8.
Variance decomposition in the LOs model
Shock
Variable Technology shock, 𝜖 𝑧
Sharpe ratio shock, 𝜖 𝑆𝑅
y 68.74% 31.26%
c 0.13% 99.87%
i 1.88% 98.12%
w 38.99% 61.01%
Increasing the steady state risk aversion in the baseline model from 1 to 49.89
corresponds to the (HB) model which moments are recorded in Table 9. In this model, the
technology shock has a higher correlation with consumption at 0.334 than the baseline, 0.193,
indicating a higher sensitivity to shocks. However, standard deviations on consumption are
greatly reduced, confirming Rouwenhorst (1995) who found that higher risk aversion leads to
ahigher degrees of consumption smoothing. The standard deviations on output, at 0.0104,
however are much closer to the empirical value of 0.0101 than that produced by the baseline
model at 0.014. Investment variance, auto-correlations and cross-correlations are not notably
different from model (B0).
From here, I add an orthogonal shock to risk aversion, 𝜀𝑡
𝑆𝑅
, producing model (HOs).
The standard deviations of consumption, now at 0.0073 become significantly closer to the
empirical moment of 0.0085, compared to the baseline model at 0.0034. However, it
produces wrong-way correlations for consumption-investment and consumption-wage.
Because positive risk aversion shocks moves consumption up while decreasing investment
and wage. While the risk aversion shock accounts for 99.9% of consumption variance as in
the (LOs) model, the impact on investment has decreased to 27.45% as compared to the
(LOs) model. Investment consumption still falls short of the empirical estimates, at only
0.012.
The (HCs) model imposes perfectly correlated shocks. While keeping the desirable
standard deviations attained by the (HOs) model, combining shocks has resulted in all
correlations being positive, overcoming the key weakness of the (HOs) model. However,
cross-correlations involving consumption, when compared to the baseline model are
underestimated, especially consumption-investment. Autocorrelations on consumption
become smaller. Because 𝜀𝑡 has a very large correlation with consumption at 0.765, these
independent shocks prevent consumption from moving back to equilibrium in a predictable
manner, reducing auto-correlation. Comparing the (HCs) with the baseline model, the (HCs)

27. 27
produces less accurate auto-correlations and cross-correlations, however the standard
deviations are significantly more accurate for output and consumption.
Table 9
Moments for models with 𝛾̅ = 49.89. Moments of shocks highlighted in bold. 𝜺𝒕
𝒛
is the shock to technology,
𝜺𝒕
𝑺𝑹
is the shock to the Sharpe ratio while 𝜺𝒕 is the simultaneous shock to both technology and the Sharpe ratio.
Autocorrelations Cross-correlations
Variable Std. dev 1 2 3 y c i w
Correlated shocks (HCs)
y 0.0114 0.692 0.442 0.242 1 0.840 0.782 0.978
c 0.0075 0.505 0.196 0.007 0.840 1 0.319 0.737
i 0.0065 0.919 0.743 0.530 0.782 0.319 1 0.861
w 0.0093 0.717 0.480 0.285 0.978 0.737 0.861 1
𝜺𝒕 1 - - - 0.479 0.765 -0.042 0.393
Orthogonal shocks (HOs)
y 0.0105 0.710 0.465 0.265 1 0.131 0.794 0.969
c 0.0073 0.506 0.196 0.008 0.131 1 -0.499 -0.004
i 0.012 0.654 0.393 0.195 0.794 -0.499 1 0.849
w 0.01 0.730 0.498 0.302 0.969 -0.004 0.849 1
𝜺𝒕
𝒛
1 - - - 0.446 0.010 0.384 0.373
𝜺𝒕
𝑺𝑹
1 - - - 0.075 0.770 -0.407 -0.009
Baseline (HB)
y 0.0104 0.711 0.468 0.267 1 0.960 1 0.981
c 0.0002 0.744 0.519 0.327 0.959 1 0.957 0.996
i 0.0102 0.711 0.468 0.270 1 0.957 1 0.980
w 0.01 0.730 0.497 0.300 0.981 0.996 0.980 1
𝜺𝒕
𝒛
1 - - - 0.449 0.334 0.451 0.374
Table 8.
Variance decomposition in the HOS model
Shock
Variable Technology shock, 𝜖 𝑧
Sharpe ratio shock, 𝜖 𝑆𝑅
y 98.93% 1.07%
c 0.09% 99.91%
i 72.55% 27.45%
w 99.45% 0.55%
[

28. 28
One way to describe the dynamics at play is that when the agent becomes more risk
averse, he prefers to consume rather than invest, resulting in a substitution effect away from
investment. Labour also increases, and this reduces wage due to diminishing marginal returns
to labour. However, when a shock to risk aversion is accompanied by improved technology,
the marginal return on capital increases, encouraging the agent to invest. Marginal return on
labour increases as well increasing wage. At low levels of risk aversion, (LCs), the
technology effect is insufficient to overcome the risk aversion effect, resulting in wrong-way
correlation. However, at high levels of correlation, the technology effect is strong enough to
cause a net increase in wage and investment, resulting in positive cross-correlations. This is
because the effect of technology shocks on investment and wage is enhanced by higher levels
of risk aversion. The correlation between the technology shock and investment is 0.064 in
model (LOs), but is dramatically higher, at 0.384 in model (HOs). Similarly, the correlation
between the technology shock and wage is 0.210 in (LOs), but as increases to 0.373 when
mean relative risk aversion is 49.89. The obvious outperformance of the (HOs) model by the
(HCs) model is consistent with the analysis in Section 2.3, suggesting that risk aversion
shocks should be correlated with technology shocks.
However, wage moments close to the empirical data were not generated by any of the
models. This is likely to be because the RBC models considered assumed a perfectly
competitive labour market and do not incorporate sticky wages or labour unionisation.
4. Discussion of the model and methodology
4.1 Limitations of the time inhomogeneous single factor model
In many respects, both the Willmott & Ahmad and CIR can be considered
oversimplifications. Guo and Hardle (2010) highlighted structural breaks and regime switches
impacting LIBOR rates and proposed using a local parametric approach to identify these
windows in order to estimate time inhomogeneous processes. Models such as Hull-White
(1990) and Black-Karasinski (1991) have allowed parameters such as the mean, to be change
with time to allow an stronger fits with the data. Furthermore, the exploration of multivariate
models that capture the dynamics of the yield curve such as the Heath Jarrow and Morton
(1992) model, and the incorporation of jumps into interest rate stochastic difference equations
have received considerable attention. This section will highlight two weaknesses of the model
I use.

29. 29
4.1.1 Information set limitations of a single factor model
In the single factor model, the drift term 𝑢(𝑟𝑡) represents the expectation of where the
interest rate will move in the next period. However, the drift term conditions on a remarkably
narrow and simple information set, depending only on the current interest rate:
𝑢(𝑟𝑡) = 𝔼(𝑟𝑡+𝑑𝑡|ℐ𝑡), ℐ𝑡 = {𝑟𝑡} (55)
This implies that market agents considered are oblivious to all other sources of information,
including macroeconomic fundamentals, commodity prices and monetary policy changes.
Consequently, the drift estimates have very little effect on the estimates for the market price
of interest rate risk and bond Sharpe ratio. Consider the formula for the market price of
interest rate risk derived earlier:
𝜆 =
𝑢(𝑟𝑡)
𝑤(𝑟𝑡)
− 2 ∗
𝑆𝑙𝑜𝑝𝑒
𝑤(𝑟)
While the time series 𝜆 𝑡 has a mean of -0.81 and standard deviation of 1.88, the time series for
𝑢 𝑡 has a mean of 0.075 and standard deviation of 0.137. Figure 13 shows that excluding the
term 𝑢(𝑟𝑡)/𝑤(𝑟𝑡) produces near-identical estimations of the market price of interest rate risk.
Consequently, the market price of risk can also be estimated using the approximation:
𝜆̂ ≅ −2 ∗
𝑆𝑙𝑜𝑝𝑒
𝑤(𝑟)
(56)
Figure 4 shows that very little information is lost when the term 𝑢(𝑟𝑡)/𝑤(𝑟𝑡) is
ignored. The resulting empirical simplification in (56) allows more flexibility in modelling
volatility, allowing methods such as GARCH or the local parametric approach.
Considering the the bond Sharpe ratio:
𝑆𝑅 𝐵 = −𝜆(𝑟, 𝑡) +
𝑢(𝑟𝑡)
𝑤(𝑟𝑡)
−
1
2
𝑤(𝑟𝑡)(𝑇 − 𝑡)
The term
𝑢(𝑟𝑡)
𝑤(𝑟𝑡)
−
1
2
𝑤(𝑟𝑡)(𝑇 − 𝑡) is insignificant, allowing the Sharpe ratio to be expressed a
reflection of the market price of interest rate about zero, i.e:
𝑆𝑅 𝐵
̂ ≅ −𝜆(𝑟, 𝑡) (57)
As shown in Figure 4, this empirical simplification results in negligible loss of information.

30. 30
Figure 4. Left pane: Plot between u(r_t )/w(r_t ) − 2 ∗ Slope/w(r) and −2 ∗ Slope/w(r). Right pane: Bond
Sharpe ratio vs market price of interest rate risk.
This can explain why the counter-intuitive result of negative ex-ante Sharpe ratios
were obtained during some periods. Equation (57) implies the sign of the market price of
interest rate risk is almost completely determined by the slope of the yield curve, so that
periods with upward sloping yield curves have positive Sharpe ratios, while periods with
downward sloping yield curves have negative Sharpe ratios.
To build a better picture of the limitations of the model, it is first useful to review the
main theories on what drives the yield curve slope. The Pure Expectations Theory states that
the slope of the yield curve depends on investors’ expectations for future interest rates. An
expected rise results in an upward slope, while an expected drop would reduce the slope of the
yield curve. The Liquidity Preference Theory states that the slope of the yield curve also
reflects a term or liquidity premium which compensates investors for the risk inherent in tying
up their money for longer periods, including price uncertainty13
. The drift estimates are very
small, implying very neutral expectations on the direction of the short interest rate.
Consequently, changes in the slope of the yield curve have been explained almost completely
by changes in the market price of interest rate risk.
Figure 5, shows how periods of persistently downward sloping yield curves in 1990,
2001 and 2008 (market with grey bands) have always been followed by decreases in the
interest rate. It is therefore reasonable to conclude that the real-world market agents had
reasons to expect interest rates to move down, pushing down the yield curve slope. In the
period 2001-2002, the Federal Reserve, led by Greenspan cut the Federal Funds rate
13
PIMCO Japan Ltd (2006)
-25
-20
-15
-10
-5
0
5
10
-30 -20 -10 0 10
𝑢(𝑟_𝑡)/𝑤(𝑟_𝑡)−2∗𝑆𝑙𝑜𝑝𝑒/𝑤(𝑟)
-2∗𝑆𝑙𝑜𝑝𝑒/𝑤(𝑟)
-10
-5
0
5
10
15
20
25
-30 -20 -10 0 10
BondSharperatio
Market price of interest rate risk

31. 31
continuously from 6% in January to 1.75% in December. While this would have affected
expectations, single factor mode, oblivious to this, incorrectly predicted an increase in interest
rates toward the estimated mean. A better model is likely to have much more significant
expectations, which should result in a strictly positive bond Sharpe ratios and a smaller
variability in the market price of interest rate risk.

32. 32
Figure 5. Interest rate and yield curve slope timeline. Lines show the 1-month LIBOR rates. Dotted lines show the mean interest rate. Area in dark blue shows the slope of the
yield curve, estimated as the difference between 2-month and 1-month rates. Area in light blue shows the 1-day drift term 𝑢(𝑟𝑡) multiplied by 20days, a rough proxy for the
expected increase in the next month. Purple area shows negative Sharpe ratios. Grey strips highlight periods with overall negative yield curve slopes.
-0.03
0.02
0.07
0.12
0.17
-8
-6
-4
-2
0
2
4
6
8
10
Date
Jul-87
Jan-88
Jul-88
Jan-89
Jul-89
Jan-90
Jul-90
Jan-91
Jul-91
Jan-92
Jul-92
Jan-93
Jul-93
Jan-94
Jul-94
Jan-95
Jul-95
Jan-96
Jul-96
Jan-97
Jul-97
Jan-98
Jul-98
Jan-99
Jul-99
Jan-00
Jul-00
Jan-01
Jul-01
Jan-02
Jul-02
Jan-03
Jul-03
Jan-04
Jul-04
Jan-05
Aug-05
Jan-06
Aug-06
Feb-07
Aug-07
Feb-08
Aug-08
Feb-09
Aug-09
Feb-10
Aug-10
Feb-11
Aug-11
Feb-12
Aug-12
Feb-13
Aug-13
Feb-14
Aug-14
Feb-15
Aug-15
Drift -ve bond Sharpe ratio Slope 1M Series6

33. 33
4.1.2 Time inhomogeneity and imprecise volatility estimates
Another weakness of the time inhomogeneous single factor model in this context is
time inhomogeneity. In the case of the WA model, volatility 𝑤𝑡 is a function of interest rate
levels, so that the parameter 𝜆 𝑡 is essentially a calibrated function of the interest rate level and
the slope of the yield curve:
𝜆̂ 𝑡 ≅ −2 ∗
𝑆𝑙𝑜𝑝𝑒𝑡
𝑣𝑟𝑡
𝛽
This parameterisation means that points in time with very different “true” volatilities but
similar interest rate levels will have the same volatility estimates. In this context, better ways
to estimate volatility could include GARCH or local change point (LCP) methods14
.
4.2 Robustness of the estimation methodology
The empirical time series obtained from the short rate model, namely the market price
of interest rate risk and the bond Sharpe ratio is difficult to validate because observable data
does not exist. Consequently, it is important to ascertain that the time series’ of the market
price of risk and the bond Sharpe ratio obtained is not a statistical artefact of the particular
model specification and the dataset used.
A few criteria that this time series, if reliable, should have are:
i. Economic meaning and interpretation
ii. Cross model robustness
iii. Robustness to choice of proxy and sample
The first criterion has already been satisfied in section 5.1, where the pro-cyclical
characteristics and possible interpretations of the Sharpe ratio time series have been discussed.
This applies by extension to the time series for the market price of interest rate risk since the
two quantities are closely related.
4.2.1. Cross model robustness
An alternative model I use is the Cox-Ingersoll-Ross (1985), or CIR model specified
as follows:
14
See Spokoiny (2009)

34. 34
𝑑𝑟𝑡 = 𝛼(𝜇 − 𝑟𝑡)𝑑𝑡 + √ 𝑟𝑡 𝜎𝑑𝑤𝑡
(58)
Where 𝛼 represents the speed of reversion to some constant mean 𝜇. And volatility is scaled
to the square root of the interest rate. These model parameters need to be estimated using
maximum likelihood estimation as it is nonlinear process.
Fortunately, the CIR process has closed form density functions:
𝑝(𝑟𝑡+Δ𝑡|𝑟𝑡; 𝜃, Δ𝑡) = 𝑐𝑒−𝑢−𝑣
(
𝑣
𝑢
)
𝑞
2
𝐼 𝑞(2√ 𝑢𝑣)
𝑤ℎ𝑒𝑟𝑒
𝑐 =
𝛼
𝜎2(1 − 𝑒−(𝛼Δ𝑡))
𝑢 = 𝑐𝑟𝑡 𝑒−𝛼Δ𝑡
𝑣 = 𝑐𝑟𝑡+Δ𝑡
𝑞 =
2αμ
σ2
− 1
(59)
where 𝐼 𝑞(2√ 𝑢𝑣) is the modified Bessel function of the first kind, of order 𝑞.
The resulting market price of interest rate risk and bond Sharpe ratios are nearly
identical, providing some confirmation of my results.
Figure 6. Bond Sharpe ratios generated by the Willmott & Ahmad parameterisation (area) compared to the Cox
Ingersoll and Ross parameterisation (line).
-5
0
5
10
15
20
Willmott & Ahmad CIR

35. 35
4.2.2 Robustness to choice of proxy and samples
In my estimation, I followed the Ahmad and Wilmott (2007) in assuming that the 1-
month LIBOR rate is a reasonable proxy for the short rate. In this section, I investigate the
effect of calibration using rates closer to the ideal— overnight LIBOR rates as short rate
proxy and the 1-week to overnight yield spread as a proxy for the yield curve slope, thus
restricting myself to a smaller sample between 2001 and 2006. I compare this modification
not only to my original calibration, but to a calibration using 1-month and 2-month rates on a
sample between 2001 and 2006 to isolate the effects of using proxies further from the ideal,
and using larger samples.
The estimates of the quarterly Sharpe ratio process:
𝑆𝑅𝑡 = 𝑆𝑅̅̅̅̅ + χ(SRt−1 − 𝑆𝑅̅̅̅̅) + 𝑆𝑅̅̅̅̅ 𝜍 𝜀𝑡
𝑆𝑅
Are summarised in table 9.
Table 9
Sharpe ratio process parameters
Value
Overnight-1 week 1 month-2months
Parameter 2001-2006 2001-2006 1987-2006
𝑘 0.280565 0.236827 0.307619
𝜒 0.772911 0.84581 0.647749
(𝑆𝑅̅̅̅̅ 𝜍)2 1.92013 1.11873 0.771191
𝑙𝑜𝑔𝐿 -41.8832 -35.4 -103.1223
𝑆𝑅̅̅̅̅ 1.235 1.536 0.8733
𝜍 1.122 0.689 1.006
𝛾̅ 67.89 81.31 49.89
The results in table 9 suggest that the process is moderately sensitive to the selection
of both the sample and the rate. Both the estimations taken between 2001-2006 have a higher
estimate of the mean Sharpe ratio than my benchmark estimation. Within this period, the
estimation with the overnight rate is lower than with the 1-month rate. However, in Figure 7,
the overnight Sharpe ratio shows a significantly higher degree of volatility when observed on
a daily basis.

36. 36
Figure 7. Comparison between calibrations overnight- 1 week for the time period 2001-2006 (primary axis), 1-
month-2 months for the time periods 2001-2006 and 1987-2006 (secondary axis).
4.3 Limitations of the time separable utility function in asset pricing
The analysis I performed using the economic model did not cover asset market
implications because the time separable utility function is for suited to this purpose15
. Hansen
and Jagannathan (1991) considered a general asset pricing model, where there exist a vector
of assets with payoffs at some time 𝜏 > 0, 𝒙, and prices of these payoffs at time 𝑡 = 0, a
vector 𝒒. By simply imposing a weak restriction on asset prices in that the expected prices are
equal to the expected payoffs discounted by the intertemporal marginal rate of substitution:
𝔼[𝒒] = 𝔼[𝒙𝑚] (60)
they derived a condition that must be fulfilled in order for that weak restriction to hold, also
called the Hansen-Jagannathan bounds:
|𝔼[𝜉]|
𝜎(𝜉)
≤
𝜎(𝑚)
𝔼[𝑚]
(61)
15
Ljungvist and Sargent (2012) Chapter 14.6 page 529.
-20
0
20
40
60
80
100
120
140
-100
-80
-60
-40
-20
0
20
40
60
02-Jan-01 02-Jan-02 02-Jan-03 02-Jan-04 02-Jan-05 02-Jan-06
1M2M/1987-2006 ON1W/2001-2006 1M2M/2001-2006

37. 37
Where 𝜉 is the asset returns above the risk free rate estimated from the model. This implies
that an asset pricing model (which would obviously need to the basic restriction) should, in
theory generate moments that satisfy the above bounds.
Figure 8 compares the volatilities and expectations of the IMRS produced by state-
separable and Epstein-Zin (a recursive utility function) in their ability to fulfil the Hansen-
Jagannathan bounds, investigated in Tallarini (2000). In the case of the time-separable utility
function, increasing risk aversion, while increasing the volatility of the IMRS, also reduces
the mean, preventing the CRRA utility function from satisfying the Hansen-Jagannathan
bounds. This is because the risk aversion parameter in the CRRA function, in addition to
expressing preferences with regards to atemporal gambles, simultaneously alters preferences
regarding intertemporal substitution.16
For this reason, Epstein-Zin utility functions, which
successfully distinguish between preferences regarding atemporal gambles and intertemporal
substitution are more suitable for the analysis of asset pricing implications.
However, choosing the state-separable utility function over the Epstein-Zin utility
function considered by Tallarini (2000) has not affected the derivation of risk aversion from
the market price of risk because equation (27):
𝜎(𝑚)
𝔼(𝑚)
= (exp{𝜎𝑐
2
𝛾2} − 1)
1
2
holds for both utility functions under the assumption of random walk consumption17
.
16
See Ljungvist and Sargent (2012) Chapter 14.6 page 531.
17
See Tallarini (2000), page 513-515

38. 38
Figure 8: Solid line: Hansen-Jagannathan volatility bounds. Circles: mean and standard deviations for IMRS
generated by Epstein-Zin preferences with random walk consumption. Pluses: mean and standard deviations for
IMRS generated by Epstein-Zin preferences with trend stationary consumption. Crosses: Mean and standard
deviation for IMRS for CRRA time separable preferences. The coefficient of relative risk aversion takes on
values of 1, 5, 10, 20, 30, 40, 50. Taken from Tallarini (2000).

39. 39
5 Conclusion
In Section 2, I calibrated a time-homogeneous single factor short interest rate and
used the Black-Scholes formula for interest rates to estimate a time series for the bond Sharpe
ratio. Assuming that log consumption follows a random walk, I obtained a steady state value
for relative risk aversion consistent with findings by Cochrane and Hansen (1992) and
Campbell and Cochrane (1999).
The simulations in Section 3.3 have shown that the inclusion of stochastic relative risk
aversion can increase the volatility of consumption produced by an RBC model to levels
more in line with empirical data. While increasing overall levels of relative risk aversion
alone within the baseline RBC model results in excessive consumption smoothing, the
introduction of time variance increases consumption variance once again. I conclude that
increasing relative risk aversion results in over-smoothing of consumption unless it is allowed
to vary. Further, using correlation to impose a pro-cyclical characteristic on relative risk
aversion, in addition in being appealing to economic intuition, is also essential in ensuring
that aggregate quantities retain their positive cross-correlations. The model I calibrated,
model (HCs) is an improvement from the baseline model, although not a complete
improvement, in that auto-correlations and cross-correlations were smaller than estimated
from empirical data.
There is further scope in introducing stochastic relative risk aversion to asset pricing
models using Epstein-Zin preferences, investigating the consistency of stochastic risk
aversion with the equity premium and Hansen-Jagannathan bounds in the context of recursive
utility.
In Section 4 I showed that the methodology used to calibrate the risk aversion process
in Section 2 firstly generates negative bond Sharpe ratios, which are unintuitive, and is
sensitive to the choice of the short rate proxy and the sample chosen. The usage of a more
accurate and robust model, either time varying or multivariate could capture more
information, potentially providing a more accurate estimate of the bond Sharpe ratio.

40. 40
Appendix
Derivation of equations (16) from equation (11)
𝑉(𝑟, 𝑡; 𝑇) = 𝔼ℚ [exp {∫ 𝑟𝜏 𝑑𝜏
𝑇
𝑡
} |ℱ𝑡]
𝜕𝑉
𝜕𝑡
= 𝔼ℚ [
𝜕
𝜕𝑡
exp {∫ 𝑟𝜏 𝑑𝜏
𝑇
𝑡
} |ℱ𝑡] by linearity of the derivative
= 𝔼ℚ [(
𝜕
𝜕𝑡
∫ 𝑟𝜏 𝑑𝜏
𝑇
𝑡
) exp {∫ 𝑟𝑡 𝑑𝑡
𝑇
𝑡
} |ℱ𝑡] by chain rule
= 𝔼ℚ [𝑟𝑡 exp {∫ 𝑟𝑡 𝑑𝑡
𝑇
𝑡
} |ℱ𝑡] = 𝑟𝑡 𝔼ℚ [exp {∫ 𝑟𝑡 𝑑𝑡
𝑇
𝑡
} |ℱ𝑡] since 𝑟𝑡 is known under ℱ𝑡
= 𝑟𝑡 𝑉(𝑟, 𝑡; 𝑇)
𝜕𝑉
𝜕𝑟
= 𝔼ℚ [
𝜕
𝜕𝑟
exp {∫ 𝑟𝜏 𝑑𝜏
𝑇
𝑡
} |ℱ𝑡]
= 𝔼ℚ [(
𝜕
𝜕𝑟
∫ 𝑟𝜏 𝑑𝜏
𝑇
𝑡
) exp {∫ 𝑟𝑡 𝑑𝑡
𝑇
𝑡
} |ℱ𝑡] = 𝔼ℚ [(∫
𝜕
𝜕𝑟
𝑟𝜏 𝑑𝜏
𝑇
𝑡
) exp {∫ 𝑟𝑡 𝑑𝑡
𝑇
𝑡
} |ℱ𝑡]
= 𝔼ℚ [(∫ 1 𝑑𝜏
𝑇
𝑡
) exp {∫ 𝑟𝑡 𝑑𝑡
𝑇
𝑡
} |ℱ𝑡] = 𝔼ℚ [(𝑇 − 𝑡) exp {∫ 𝑟𝑡 𝑑𝑡
𝑇
𝑡
} |ℱ𝑡]
= (𝑇 − 𝑡)𝔼ℚ [ exp {∫ 𝑟𝑡 𝑑𝑡
𝑇
𝑡
} |ℱ𝑡] = (𝑇 − 𝑡) 𝑉(𝑟, 𝑡; 𝑇)
𝜕2
𝑉
𝑑𝑟2
=
𝜕
𝜕𝑟
[
𝜕𝑉
𝜕𝑟
] =
𝜕
𝜕𝑟
[(𝑇 − 𝑡) 𝑉(𝑟, 𝑡; 𝑇)] = (𝑇 − 𝑡)
𝜕
𝜕𝑟
[ 𝑉] = (𝑇 − 𝑡)2
𝑉(𝑟, 𝑡; 𝑇)

41. 41
References
Ahmad, R. and Wilmott, P., 2007. The Market Price of Interest-rate Risk: Measuring and
Modelling Fear and Greed in the Fixed-income Markets. Wilmott magazine, January 2007,
pp 64-70.
Arrow, K.J., 1971. Essays in the theory of risk-bearing. Amsterdan: North-Holland
Publishing Company.
Brandt, M.W., and Wang, K.Q.,2003. Time-varying risk aversion and unexpected inflation,
Journal of Monetary Economics 50, 1457-1498
Bollen, N.P.B., 1997. Derivatives and the Price of Risk, The Journal of Futures Markets, Vol.
17, No. 7, 839-854.
Campbell, J.Y. and Cochrane, J.H., 1999. By Force of Habit: A Consumption-Based
Explanation of Aggregate Stock Market Behaviour, The Journal of Political Economy,
Volume 107, Issue 2, 205-251.
Capinski, M., Kopp, E.K., Traple, J., 2012. Stochastic Calculus for Finance. UK: Cambridge
University Press.
Cerny, A., 2004. Mathematical Techniques in Finance: Tools for Incomplete Markets. New
Jersey: Princeton University Press.
Cochrane, J.H. and Hansen, L.P., 1992. Asset Pricing Explorations for Macroeconomics,
Journal of Monetary Economics 7, 115-165.
Cogley, T. and Sargent, T.J., 2008. The Market Price of Risk and the Equity Premium: A
legacy of the Great Depression?, Journal of Monetary Economics 55, 454-476.
Fernande-Villaverde, J., n.d. Real Business Cycles, lecture notes. Available from:
http://www.dynare.org/documentation-and-support/examples/rbc.pdf [Accessed 12th March
2016]
Friend, I. and Blume, M.E., 1975. The Demand for Risky Assets, The American Economic
Review, Volume 65, Issue 5, 900-922.
Guiso, L., Sapienza, P. and Zingales, L., 2013. Time Varying Risk Aversion, NBER Working
Paper No. 19284.
Guo, M. and Hardle, W.K., 2010. Adaptive Interest Rate Modelling, SFB 649 Discussion
paper 029.
Hansen, L.P. and Jagannathan, R., 1991. Implications of Security Market Data for Models of
Dynamic Economics, The Journal of Political Economy, Vol. 99, No. 2, 225-262.
Heath, D., Jarrow, R., Morton, A., 1992. Bond Pricing and the Term Structure of Interest
Rates: A New Methodology for Contingent Claims Valuation, Econometrica, Volume 60,
Issue 1, 77-105.
Jermann, U.J.,1998. Asset pricing in production economies, Journal of Monetary Economics
41, 257-275.

42. 42
Kandel. S and R.F. Stambaugh, 1991. Asset Returns and Intertemporal and Intertemporal
Preferences, Journal of Monetary Economics 27, 39-71.
Kladviko, K. (2007). Maximum likelihood estimation of the Cox-Ingersoll-Ross process: the
matlab implementation, working paper. Available from
http://dsp.vscht.cz/konference_matlab/MATLAB07/prispevky/kladivko_k/kladivko_k.pdf
[Accessed on 5th December 2015].
Ljungqvist, L. and Sargent, T.J., 2012. Recursive Macroeconomic Theory, 3rd
edition.
Cambridge, Mass: The MIT Press, eBook Collection (EBSCOhost), EBSChost, viewed 6
April 2016.
Long, J.B. and Plosser, C.I., 1983. Real Business Cycles, Journal of Political Economy,
University of Chicago Press, vol. 91(1), pages 39-69
Mehra, R. and Prescott, E.C., The Equity Premium: A Puzzle, Journal of Monetary
Economics 15, 145-161.
PIMCO Japan Ltd, 2006. Yield Curve Basics. Japan: PIMCO. Available from:
https://japan.pimco.com/EN/Education/Pages/YieldCurveBasics.aspx [Accessed on 6th April
2016]
Rouwenhorst, K.G., 1995. Asset pricing implications of equilibrium business cycle models.
In: Thomas, F.C. (Ed.), Frontiers of Business Cycle Research. NJ, Princeton: Princeton
University Press, 294-330.
Shonkwiler, R.W., 2013. Finance with Monte Carlo. New York, Heidelberg, Dordrecht,
London: Springer.
Spokoiny, V., 2009. Multiscale Local Change Point Detection with Applications to Value-at-
Risk, The Annals of Statistics, Vol. 37, No.3, 1405-1436.
Tallarini, T.D., 2000. Risk-sensitive business cycles, Journal of Monetary Economics 45, 507-
552
Trigeorgis, L., 1996. Real Options: Managerial Flexibility and Strategy in Resource
Allocation. London: The MIT Press