Adrian, Crump and Moench (ACM)

#termstructure #economics #finance #atsm

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.

Main References

Adrian, T., Crump, R. K., and Moench, E. (2013). "Pricing the Term Structure with Linear Regressions". FRB of New York Staff Report No.340, Available at SSRN: https://ssrn.com/abstract=1362586
Jennison, F. (2017). "Estimation of the term premium within Australian Treasury Bonds". Australian Office of Financial Management Working Paper, 1.
Kim, Seung Hyun. (2024). "Asset Pricing and Term Structure Models". WORK IN PROGRESS.

Introduction

The Adrian, Crump and Moench (2013) (henceforth ACM) proposes an alternative approach to estimate the Gaussian ATSM Gaussian ATSM that only required the linear regressions.

Consider a general Gaussian ATSM: While Dai and Singleton Dai and Singleton and JSZ JSZ specify the short-run and risk-neutral dynamics and then drive the market price of risk, in ACM, we first specify the physical dynamics and then form the market price of risk. The derivation of linear relationships between bond excess returns will follow subsequently.

Using ACM, we can consistently recover the parameters related to the physical dynamics and the market price of risk via OLS. Then, the short rate dynamics and the bond price formula are calculated using simple estimation process.

Model and Solutions

Definition (ACM model).

The Adrian, Crump and Moench (ACM) model is stated as where is the orthogonal projection of on the space spanned by with respect to the norm.
Also, under the No-Arbitrage Condition No-Arbitrage Condition, there exists an SDF process with such that for the empirical SDF is given as where the market prices of risk are affine functions of the factors: following essentially affine essentially affine and extended affine extended affine models.

Excess Bond Returns

Remark (measurement error in ACM).

Since is the orthogonal projection of on the space spanned by , by the definition of orthogonal projection orthogonal projection, we have implying that the measurement error is uncorrelated with the factors .

Proposition (excess bond returns in ACM).

Given the assumptions of ^07381c Definition 1 (ACM model), and by letting that jointly follows the normal distribution given the information set up to , the bond excess returns can be decomposed as where

Proof.From the definition of one-period ahead expected excess rate one-period ahead expected excess rate, and log expression of the rate of return log expression of the rate of return, the one-period excess bond return for a maturity bond from time to time is given as and By the no-arbitrage condition of we have Now assume that and jointly follow the normal distribution given the information set up to , and . Then using Statistical Proof > ^5081bb Statistical Proof > Lemma 23 (MGF of multivariate normal distribution), we have Thus by taking logs on the both sides, which gives us By letting be the Consumption-CAPM > ^c57404 Consumption-CAPM > Definition 4 (beta-term), we therefore have

Now, we decompose the forecasting error of the excess bond returns into a component that is correlated with , and another component that is conditionally orthogonal: Since we have using least square estimator least square estimator, we can conclude that , and .

Note that the conditional mean of at time is zero. Then we have implying that and are uncorrelated conditional on time information.

Hence, we have which is the desired result. □

Corollary (excess bond returns under constant variance).

Let the model be ^07381c Definition 1 (ACM model) and let all other assumptions are remained from ^d34a76 Proposition 3 (excess bond returns in ACM). Now, suppose has constant variance . Then the excess bond returns are given as Furthermore, by letting be constant over time, we have

Proof.Since has constant variance , from the previous result of we have where the fourth equation holds since and are uncorrelated conditional on time information (see the proof of ^d34a76 Proposition 3 (excess bond returns in ACM)).

Then, the excess bond returns are given as Finally, by letting be the constant over time, we have which is the desired result. □

Affine Yields and Prices

Proposition (affine yields in ATSM).

In ^07381c Definition 1 (ACM model), the yields are given as affine functions of the factors as in the usual Gaussian ATSM: where

Proof.Remark that in ^07381c Definition 1 (ACM model), the dynamics are affine VAR(1) structure under affine market prices of risk, and short rate dynamics are also affine: Since dynamics are affine under affine market prices of risk and VAR(1) physical dynamics, and the short rate dynamics are also affine, with an added pricing error term , bond prices can be expressed as exponential-affine function:

From the definition of excess returns definition of excess returns, we have By matching the result with the obtained equation from ^90d4a3 Corollary 4 (excess bond returns under constant variance) we have implying that . Then, the equation can be re-arranged into Thus we have where and .

Note that this result is the same form of standard linear difference equations for affine term structure models with homoskedastic shocks. The only difference is the appearance of term, which is due to the result of a maturity-specific return fitting error that is conditionally orthogonal to the state variable innovations (see the previous proof the previous proof). □

Note that we have which implies that even if the yield pricing errors are serially uncorrelated, the return pricing errors must be cross-sectionally and serially correlated. This is an undesirable implication, but ACM assumes that the return pricing errors are serially uncorrelated.

Estimation Process

Stacking Matrix for Excess Bond Return

Suppose the sample consists of maturities, denoted as . Then for any time , the stacking equation for ^90d4a3 Corollary 4 (excess bond returns under constant variance) gives us Since we have then we can simplify the expression into i.e.

Then, the collecting observations for into the matrix is which equivalent to

Three-Step Estimation Procedure

The excess bond returns formular from ^90d4a3 Corollary 4 (excess bond returns under constant variance) allows for the three-step estimation procedure. Remark that the parameters we need to estimate consists of three blocks:

Step 1: Estimate Physical Dynamics and Short Rate Parameters

Physical Dynamics Parameters

We first estimate the dynamics via ordinary least squares. Here, it is assumed that the observable factors are constructed either by principal components or the use of macroeconomic variables.

Re-expressing the dynamics into the stacking matrix form, we have i.e. Then, given these , the estimates of the parameters , are given as i.e. Next, the estimates for are obtained from where .

Short Rate Parameters

From the short rate dynamics, we have which can be represented in the stacked matrix term as i.e. Then the estimates for the short rate parameters are given as

Step 2: Estimate the Excess Bond Return Regression

From the previous step, we have obtained and . Then, using we can estimate the factor innovations through

Now, for the excess bond return regression where we can estimate the parameters , , and using via OLS: i.e. Also, from we can estimate the return pricing error variance

Step 3: Estimate the Market Price of Risk Parameters

From the definitions of we have Then, the ordinary least squared estimates give us

Using the obtained , , , and , we first estimate as Then, from the definition we can estimate by where is the th row of .

Finally, we can estimate the market prices of risk parameters through and the market prices of risk is derived as

Recursive Function

Note that, we can calculate the bond prices as an exponential-affine function using the recursive function given from ^71f8bb Proposition 5 (affine yields in ATSM), where Also, the excess bond return can be calculated from Using the estimated parameters in Three-Step Estimation Procedure Three-Step Estimation Procedure, the decomposition of yields into an expectation component (risk-neutral yields) and a term premium is as follows:

model-fitted bond yields: calculate the fitted prices and fitted yields using the estimated parameters for the recursive funciton.
risk-neutral yield: let the market prices of risk into zero, i.e. and , then recalculate the bond yields.
term premium is the difference between the model estimated yields and risk-neutral yields.

Inferences

Under some regularity assumptions, the joint asymptotic distribution of , , and can be derived. The detailed proof of consistency and asymptotic normality of the estimates will be discussed in Consistency of ACM Estimation Consistency of ACM Estimation, and Asymptotics of ACM Estimates Asymptotics of ACM Estimates, respectively.

Remark that, since the ACM model uses the relationship between the excess bond returns and the factors for the estimation, the consistency of its estimation required independence of return pricing errors. Thus for deciding the estimation methods, which errors to be restricted will be an important criteria.

Bootstrap Bias-Correction Technique

While the ACM method allows for decomposition of yields into an expectation component(risk-neutral yields) and a term premium via the simple linear regressions, there have been doubts regarding the unbiasedness of parameter estimation.

The potential for the bias is a function of the persistent nature of interest rates (slow-mean reversion) and the sample size of data available (number of interest rate cycles captured) which make the OLS estimation difficult.

Jennison (2017) introduced a bootstrap technique to correct the bias in the ACM, originally suggested by Bauer et al (2014).

The mean bias-correction procedure is applied as follows:

Estimate the VAR parameters using OLS as per ACM Step 1: Estimate Physical Dynamics and Short Rate Parameters
Specify a number of simulations, for instance, .
For each simulation, randomly select a sample of state variables where and .
For each simulation , estimate the VAR parameters using OLS, and store these parameters for each simulation.
Calculate the average across the simulation size , and denote it into .
Calculate the bias-corrected parameters and apply the Killian (1998) stationary correction technique.
Replace with and continue with model estimation as per ACM.