Risk Measures and the Impact of Asset Price Bubbles Robert A. Jarrow*and Felipe Bastos G. Silva May 7, 2014
Abstract This paper analyzes the impact of asset price bubbles on a firm’s standard risk measures, including value-at-risk (VaR) and conditional value-at-risk (CVaR). Comparing a bubble and non-bubble economy, it is shown that asset price bubbles may cause a firm’s traditional risk measures such as VaR and CVaR to decline. This decline is due to a reduced standard deviation and an increased right skew of the firm value’s distribution due to bubble expansion. This effect on a firm value’s moments due to the presence of price bubbles documents that traditional risk measures do not adequately capture the impacts of bubble bursting. We propose a new risk measure to account for losses associated with bubble bursting, a phenomenon that must be taken into consideration for the proper determination of equity capital. This additional risk measure is the expected holding period loss.
JEL Classification: G120, G170. Keywords: risk management; bubbles; capital determination; risk measures; Value-at-Risk; Conditional Value-at-Risk
* Samuel
Curtis Johnson School of Management, Cornell University, Ithaca, NY 14853 and Kamakura Corporation, Honolulu, Hawaii 96815. e-mail:
[email protected]. Samuel Curtis Johnson Graduate School of Management, Cornell University, Ithaca, New York 14853, email:
[email protected]. 1 The authors greatly acknowledge the support from the Global Association of Risk Professionals (GARP) Research Fellowship
1 Electronic copy available at: http://ssrn.com/abstract=2341641
1
Introduction
A refined understanding of the strengths and weaknesses of various risk measures, including valueat-risk (VaR) and its offshoots (see Jorion (1997)), has taken on an increased importance since the 2007 financial crisis and the passing of the Dodd-Frank Wall Street Reform and Consumer Protection Act (2010). With hindsight, we know that the computation and monitoring of various risk measures prior to the 2007 credit crisis did not adequately incorporate the housing price bubble. This omission was due, in part, to a lack of understanding of the economics of asset price bubbles, including their characterization and measurement. Since the 2007 crisis, the mathematical finance literature has made significant advances in the modeling and testing of asset price bubbles, see Jarrow and Protter (2010), Hong, Scheinkman, and Xiong (2006) and Hong, Scheinkman, and Xiong (2006) for a literature review. The purpose of this paper is to apply these new insights to determine the impact, if any, that asset price bubbles have on the common risk measures that are used in practice for the determination of equity capital. To analyze the impact of asset price bubbles on a firm’s risk measures and capital determination, we construct various hypothetical economies, both with and without bubbles. We simulate a firm’s value process in each of these economies, computing the firm’s standard risk measures. The results show that the existence of asset price bubbles causes a firm value’s distribution to have a possibly lower standard deviation and a higher right skew. These are the result of bubble expansion and bursting. The increased right skew combined with a reduced variance of a firm´s returns due to bubble expansion causes the firm’s VaR, and its CVaR to decline. Based on these measures alone, their declining values imply that in the presence of asset price bubbles, less equity capital is required. However, as shown by the additional risk measure proposed in the present paper - the expected 5-day loss - this conclusion is incorrect. This loss measure increases in bubble economies and is due to bubble bursting, which cause significant firm value losses on the bubble bursting paths. Since bubbles eventually burst, causing significant firm value loss, more equity capital should be held for these bubble bursting scenarios. Unfortunately, the severity of these bubble bursting scenarios are not adequately captured by the standard risk measures, whose computation is based 2 Electronic copy available at: http://ssrn.com/abstract=2341641
on the standard moments and quantiles of a firm value’s distribution over short time horizons in which the bubble bursting is unlikely. These bubble bursting scenarios are captured, however, in some correctly constructed risk measure such as the expected holding period loss. An outline for this paper is as follows. Section 2 presents the model, and section 3 describes the simulation experiment. Section 4 presents the results, while section 5 summarizes the implications of our analysis for risk management.
2
The Model
To model the evolution of asset prices in the presence of price bubbles, we use the approach contained in Jarrow, Protter, and Shimbo (2007), in a continuous trading and finite horizon economy [0, τ ]. The randomness in the economy is modeled as a filtered probability space (Ω, F, F, P) where Ω is the state space, F is a σ-algebra, F = (Ft )t∈[0,τ ] is the information filtration, and P represents the statistical probability (as distinct from the risk-neutral probability introduced subsequently). Traded in this economy are a collection of Ne risky assets whose market prices are denoted by S = (Sj (t))t∈[0,τ ] = (S1 (t), S2 (t), S3 (t), ..., SNe (t)). These price processes are assume to be adapted to the information filtration F. Without loss of generality we assume that these assets have no cash flows. Also traded is a money market account (the zeroth asset), whose value grows at the Rt default-free spot rate of interest r(t), i.e. S0 (t) = e 0 r(s)ds . It is assumed that the money-market account’s value is also adapted to the filtration F.
2.1
The Firm
To investigate the impact of asset price bubbles on various risk measures, we first need to construct a hypothetical firm to which these risk measures can be applied. In the most abstract setting, a firm can be represented by a portfolio of traded assets (the left side of its balance sheet). Using this perspective, we consider a firm consisting of nj shares of risky assets j = 1, 2, ..., Ne . To simplify the analysis, these share holdings are fixed for our time horizon [0, T ] where T ≤ τ .
3
The firm’s time t ≥ 0 value is denoted:
V (t) =
PNe
j=1
nj Sj (t).
(1)
We normalize the initial firm value so that it equals one dollar at time 0, i.e. V (0) = 1. This is for convenience and it does not affect any of the results. The shares in each asset are initially chosen so that each asset has an equal percentage weighting in the firm’s asset portfolio, i.e.
wj =
1 Ne
or, equivalently nj =
2.2
1 Sj (0)Ne
f or all j.
(2)
The Asset Price Processes
To capture an economy with and without price bubbles, we assume that the risky asset’s prices follow a constant elasticity of variance (CEV) process, i.e.
dSj (t) = θj Sj (t)dt + βj Sj (t)α dZj (t)
(3)
where Zj (t) are Brownian motions on (Ω, F, F, P) with correlations dZi (t)dZj (t) = ρi,j dt for i, j = 1, 2, ...Ne . Note that in this evolution, the parameters θj , βj , ρij are asset specific (they differ across j), while α is identical across all firms. The α parameter is used to characterize whether or not the risky asset price process exhibits a price bubble. For the subsequent analysis it is important to emphasize that these risky asset price processes are correlated due to the correlations between the Brownian motions processes in expression (3). This correlation captures the systematic risk present in the asset price processes’ evolutions.
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2.3
Asset Price Bubbles
Following Jarrow, Protter, and Shimbo (2007, 2010), an asset price bubble is defined to occur when the market price for an asset exceeds its fundamental value. An asset’s fundamental value is that price an investor would pay to buy and hold the asset forever, without retrading. To determine an asset’s fundamental value, we need to impose some additional structure on our economy. The minimal structure required consists of two additional assumptions. First, one needs to assume that the economy has no arbitrage opportunities. This guarantees the existence of a risk-neutral S (t) is a Q local-martingale for all j. The riskprobability Q, equivalent to P, such that S0j (t) t∈[0,τ ]
neutral probability provides a method for computing present values. Second, one needs to assume that additional derivatives on the risky assets trade, and that these prices determine a unique risk-neutral probability Q consistent with the derivatives’ prices. For the remainder of the paper, we assume that both of these conditions hold. Under this additional structure, an asset’s fundamental value can be defined as: F Vj (t) = EQ
Sj (τ ) |Ft S0 (τ )
S0 (t)
(4)
where EQ (· |Ft ) is the time t conditional expectation under Q. As shown, the asset’s fundamental value is its expected discounted future payoff from liquidation at time τ . The asset’s price bubble is then the difference between the market price and its fundamental value, i.e. βj (t) = Sj (t) − F Vj (t) ≥ 0.
(5)
Since the fundamental value normalized by the value of the money market account is a martingale (being a conditional expectation, see expression (4)), a bubble exists if and only if the asset’s normalized price is a strict local martingale and not a martingale under Q. For the CEV process, using Jarrow, Kchia, and Protter (2011), one can show that the asset’s Sj (t) normalized price S0 (t) is a martingale (no bubbles) when α ≤ 1, and a strict-local martingale (bubbles) for α > 1. Note that the boundary case of α = 1 gives the geometric Brownian motion underlying the Black-Scholes-Merton (BSM) option pricing model, which we call the BSM economy.
5
The BSM economy has no price bubble.
3
The Experiment
To determine the impact of bubbles on a firm’s risk measures, we will perform a simulation experiment. Simulation is needed to determine the firm value’s probability distribution over a given time interval. In our experiment we fix the time period for the standard risk measures to be 5 days (one business week). However, since the proposed risk measure is designed to capture the effects of bubble bursting, the simulation is performed over a longer time horizon within which are embedded fifty consecutive 5-day subintervals. These 5-day consecutive subintervals are used for our risk measure computations. The longer time horizon simulated is one year (250 business days). We use simulation because an analytic characterization of the firm value’s probability distribution using the CEV process is unavailable (see Emanuel and MacBeth (1982) and Schroder (1989) for a deeper discussion on the CEV model). The simulation experiment we perform is as follows. We construct a collection of different economies, some with bubbles and some without, by varying the CEV parameter α from 0.3 to 1.7 in steps of size 0.1. In each of these different economies, we compute the standard risk measures to determine the impact that bubbles have on their values.
3.1
The Risk Measures
This section defines the various risk measures considered in the experiment, where tNd = 250 days and tHP = 5 days represent the simulated time horizon and the consecutive subperiods, respectively. The index k ∈ {1, 2, ..., Np } represents different 250 day simulated paths (as also discussed in Section 3.3) and Np is the total number of 250 day paths considered in the experiment.
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3.1.1
The Standard Moments
To help characterize the firm value’s distribution at the end of 5 days (tHP = 5 days), we compute its mean, standard deviation, and skewness, i.e. Np 1 X Vk (tHP ), EV (tHP ) = Np k=1
ST DEV (tHP ) =
1 Np − 1
Np X k=1
1 Np
Skewness(tHP ) =
1 Np
Np P
!2 Np X 1 Vk (tHP ) − Vk (tHP ) , and Np k=1
Vk (tHP ) −
k=1 Np P
(6)
Vk (tHP ) −
k=1
1 Np
1 Np
Np P
(7)
!3 Vk (tHP )
k=1
!2 23 N p P Vk (tHP )
.
(8)
k=1
The above moments characterize the firm’s value, including both positive and negative returns. For risk management purposes, we are often only interested in losses. To understand the left tail of the firm value’s distribution, we compute various loss (or risk) measures.
3.1.2
VaR
Although its limitations (unrelated to bubbles) have been widely recognized in the literature (see Jorion (1997) and Alexander (2001)), Value-at-Risk (VaR) is still a widely used risk measure due to the Bank for International Settlements Accords (see Engelmann and Rauhmeier (2006) and Cornford (2005) on the Basel II Accord). Value-at-risk, for a given confidence level c, is:
V aR(c) = −(Quantile1≤k≤Np (c) − V (0)) = −(Quantile1≤k≤Np (c) − 1) (9) (" # ) Np P 1 IVk (tHP )≤x ≤ c and IVk (tHP )≤x is an indicator function where Quantile1≤k≤Np (c) = sup Np x
k=1
that takes value 1 if Vk (tHP ) ≤ x and 0 otherwise. Note that we define the risk measures in the context of our simulation. In the limit (when Np goes to infinity), these definitions are equivalent 7
to the standard definitions.
3.1.3
CVaR
A related risk measure to VaR is conditional VaR (CVaR), which is an attempt to capture the magnitude of the mass in the left tail, defined as: Np P
CV aR(c) =
(1 − Vk (tHP )) IVk (tHP )
