# Monthly Archives: August 2014

## LONG-TERM BONDS: A Two-Factor Model 6

A comparison of the model implications in rows 1 and 7 shows that 10-year nominal bonds have a risk premium over three-month nominal bills of 2.06% per year, while 10-year indexed bonds have a risk premium over three-month indexed bills of 1.62% per year. These numbers, together with the 49-basis-point risk premium on three-month nominal bills over three-month indexed bills, imply a 10-year inflation risk premium (the risk premium on 10-year nominal bonds over 10-year indexed bonds) of slightly less than 1%. This estimate is consistent with the rough calculations in Campbell and Shiller (1996).

Rows 2 and 8 show that nominal bonds are much more volatile than indexed bonds; the difference in volatility increases with maturity, so that 10-year nominal bonds have a standard deviation three times greater than 10-year indexed bonds. This difference in volatility makes the Sharpe ratio for indexed bonds in row 9 considerably higher than the Sharpe ratio for nominal bonds in row 3.

## LONG-TERM BONDS: A Two-Factor Model 5

The first two columns of Table 1 report parameters and asymptotic standard errors for the period 1952-96. All parameters are in natural units, so they are on a quarterly basis. We estimate a moderately persistent process for the real interest rate; the persistence coefficient фх is 0.87, implying a half-life for shocks to real interest rates of about 5 quarters. The expected inflation process is much more persistent, with a coefficient фх of 0.9985 that implies a half-life for expected inflation shocks of almost 115 years! Of course, the model also allows for transitory noise in realized inflation Click Here.

The bottom of Table 1 reports the implications of the estimated parameters for the means and standard deviations of real interest rates, nominal interest rates, and inflation, measured in percent per year. The implied mean log yield on an indexed three-month bill is 0.85% for the 1952-96 sample period. Taken together with the mean log yield on a nominal three-month bill of 5.31% and the mean log inflation rate of 3.99% (both restricted to equal the sample means over this period), and adjusting for Jensen’s Inequality using one-half the conditional variance of log inflation, the implied inflation risk premium in a three-month nominal Treasury bill is 49 basis points. This fairly substantial risk premium is explained by the significant positive coefficient and the significant negative coefficient /37rm in Table l.

## LONG-TERM BONDS: A Two-Factor Model 4

The term structure of interest rates in the US

We estimate the two-factor term structure model using data on US nominal interest rates, equities and inflation. We use nominal zero-coupon yields at maturities 3 months, 1 year, 3 years, and 10 years from McCulloch and Kwon (1993), updated by Gong and Remolona (1996a,b). We take data on equities from the Indices files on the CRSP tapes. We use the value-weighted return, including dividends, on the NYSE, AMEX and NASDAQ markets. We take data on CPI inflation from the SBBI files on the CRSP tapes. Although the raw data are available monthly, we construct a quarterly data set in order to reduce the influence of high-frequency noise in inflation and short-term movements in interest rates.

## LONG-TERM BONDS: A Two-Factor Model 3

The variance term on the left hand side of (9) is a Jensen’s Inequality correction that appears because we are working in logs. The conditional covariance of the excess bond return with the log SDF determines the risk premium. In our homoskedastic model the conditional covariance is constant through time but dependent on the bond maturity; thus the expectations hypothesis of the term structure holds for indexed bonds. It is important to realize that constant risk premia do not imply constant investment opportunities because real interest rates are stochastic in our model read more.

Since Bn-1 > 0, the Jensen 7s-inequality-corrected risk premium is negative if Pmx > 0> znd positive otherwise. With positive f3mx) long-term indexed bonds pay off when the marginal utility of consumption for a representative investor is high, that is, when wealth is most desirable. In equilibrium, these bonds must have a negative risk premium. With negative on the other hand, long-term indexed bonds pay off when the marginal utility of consumption for a representative investor is low, and so in equilibrium they have a positive risk premium.

Equation...

## LONG-TERM BONDS: A Two-Factor Model 2

Campbell, Lo and MacKinlay (1997) note that £m>t+i only affects the average level of the real term structure and not its average slope or time-series behavior. Accordingly, we can either drop it or identify its variance with an additional restriction. We follow the second approach and introduce equities in the model. We assume that the unexpected log excess return on equities is affected by shocks to both the expected and unexpected log SDF:

Th...

## LONG-TERM BONDS: A Two-Factor Model

Specification of the model

Our focus in this paper is the microeconomic problem of portfolio choice for an individual investor facing exogenous bond returns. In order to generate empirically reasonable and theoretically well-specified bond returns, however, we start by writing down a general equilibrium bond pricing model. We consider a discrete-time, two-factor homoskedastic model that allows for non-zero correlation between innovations in the short-term real interest rate and innovations in expected inflation.

The real part of the model is determined by the stochastic discount factor (SDF) Mt+1 that prices all assets in the economy. In a representative-agent framework the SDF can be related to the marginal utility of a representative investor, but here we simply use it as a device to generate a complete set of bond prices. We assume that Mt+i has the following lognormal structure, a discrete-time version of Vasicek (1977):

## LONG-TERM BONDS: Introduction 3

We assume that the investor’s preferences are of the form suggested by Epstein and Zin (1989, 1991); the investor has constant relative risk aversion and constant intertemporal elasticity of substitution in consumption, but these parameters need not be related to one another. Epstein-Zin preferences nest the traditional power-utility specification in which relative risk aversion is the reciprocal of the intertemporal elasticity of substitution there.

We show that the investor’s demand for long-term bonds can be decomposed into a “myopic” demand and a “hedging” demand. Myopic demand depends positively on the term premium, and inversely on the variance of long-term bond returns and the investor’s risk aversion. As risk aversion increases, myopic demand shrinks to zero. Hedging demand, on the other hand, is proportional to one minus the reciprocal of risk aversion. It is zero when risk aversion is one but accounts for all bond demand when risk aversion is infinitely large. We show that an infinitely risk-averse investor with zero intertemporal elasticity of substitution in consumption will choose an indexed bond portfolio that is equivalent to an indexed perpetuity, that is, a portfolio that delivers a riskless stream of real consumption. In this way we are able to support the commonsense view that long-term bonds are appropriate for long-lived investors who desire stability of income.

## LONG-TERM BONDS: Introduction 2

Merton (1969, 1971, 1973) studied the intertemporal portfolio choice problem with time-varying investment opportunities, introducing the important concept of intertemporal hedging demand for financial assets, but he did not obtain explicit solutions for portfolio weights. Recently a number of authors such as Balduzzi and Lynch (1997), Barberis (1998), Brandt (1998), and Brennan, Schwartz, and Lagnado (1996, 1997) have used numerical methods to solve particular long-run portfolio choice problems, while Kim and Omberg (1996) and Campbell and Viceira (1999) have derived some analytical results, but these papers generally concentrate on the choice between cash and equities rather than the demand for long-term bonds.

In this paper we study intertemporal portfolio choice in an environment with random real interest rates. We use an approximation technique developed in our earlier papers (Campbell 1993, Campbell and Viceira 1999) to replace the intractable portfolio choice problem with an approximate problem that can be solved using the method of undetermined coefficients. We use the approximate solution to understand the demand for long-term bonds.

## LONG-TERM BONDS: Introduction

Long-term bonds have been issued for centuries, and they remain extremely common financial instruments. It is natural to suppose that bonds have been popular because they meet the needs of an investor clientele. Investment advisers and financial journalists, for example, often say that bonds are appropriate for long-term investors who seek a stable income.

Curiously, modern financial economics has little to say about the demand for longterm bonds. In the early postwar period Hicks (1946), following Keynes (1930) and Lutz (1940), argued that investors would naturally prefer to hold short-term bonds and would only hold long-term bonds if compensated by a term premium. Modigliani and Sutch (1966) countered that some investors might have a preference for longterm bonds (a long-term “preferred habitat”), and such investors would require a premium to go short, not a premium to go long. However Modigliani and Sutch were vague about the characteristics of investors that would lead to a long-term preferred habitat. They took it as a given that some investors would desire stable wealth at a long rather than a short horizon there.