Intra-Horizon Value at Risk (VaR-I)

Peter Urbani
Jan 24, 2024
6 min read

During the recent financial crises, many financial institutions suffered losses much larger than their Value-at-Risk (VaR) models predicted and found that traditional end-of horizon risk measures are inadequate for managing longer-term risks.

There has been a considerable amount of literature on the measurement of financial risk; however, almost all the existing risk measures, either the popular Value-at-Risk (hereafter VaR) or expected shortfall (hereafter ES), mainly focus on quantifying the possible large losses at the end of the predetermined time horizon. This focus may be appropriate when dealing with short-term risks such as those mandated by the Basel Banking Committee.

But, this risk measurement methodology is inadequate for measuring longer-term risks, since it does not take into account the “intra-horizon risk,” i.e. the possibility that the losses incurred before the end of the specified time horizon might trigger other problems such as position rebalancing, early liquidation, or margin calls.

So far there have been only three main papers studying the measurement of intra-horizon risk: Kritzman and Rich (2002), Boudoukh et al.(2004), and Bakshi and Panayotov (2010).

The standard VaR approach considers only terminal risk, completely ignoring the sample path of portfolio values. In reality interim risk may be critical in a mark-to-market environment. Sharp declines in value may generate margin calls and affect trading strategies. Boudoukh introduced the notion of MaxVaR, analogous to VaR in every way except it quantifies the probability of seeing a given loss on or before the terminal date rather than at the terminal date.

The formula for MaxVaR a.k.a Intra-Horizon VaR, VaR-I and Continuous VaR* is given as:

Where:

And:

= Cumulative Standard Distribution

Except in the special case of the normal distribution and also the Geometric Brownian Motion there are very few closed forms for Z. In the special case of the Normal distribution and Brownian motion we can appeal to the underlying symmetry of the normal distribution to invoke the “reflection principle” to find it.

The result follows from the properties of the first passage time of a Brownian motion with drift used in pricing Barrier Options. To get some intuition for this result consider the simpler case where m=0, i.e., when there is no expected drift in the (log) price process

The MaxVaR probability of hitting Z or lower on or before T is, interestingly, exactly twice the probability associated with hitting Z at the horizon, T. i.e., the VaR. In this simpler case the formula for the MaxVaR is based on the well-known “reflection principle” -- for every process with a sample path that touches the value Z and declines further there is one that rises from that z point on. These could be thought of as mirror image processes.

To summarize, the z percent VaR is the 2Z percent MaxVaR. Of course, this does not imply that the Z% MaxVaR is twice the Z% VaR.

In the next table we provide a comparison between the standard VaR and the MaxVaR in order to get some intuition for the magnitude of the adjustment which is required in order to account for interim risk. We set m=0, i.e., the expected return on the portfolio is assumed to be 0. In this case, while the probability of seeing a return of -1.645 standard deviations or worse at the end of period is 5%, there is a 10% probability of seeing this size move along the sample path prior to the terminal date. To put it differently, there is a 5% probability of seeing an end-of period return of less than –1.645 standard deviations (VaR), and a 5% probability of seeing a return of less than –1.960 standard deviations on or before the end of the period

(MaxVaR). At the 5% level, the ratio of the VaR to the MaxVaR is, therefore, 1.960/1.645=1.192. That is, the standard VaR calculation can be easily adjusted to account for the fact that value is observed continuously in the interim simply by inflating VaR by 19%. This “inflation” factor, the ratio of MaxVaR to VaR,

does not depend on volatility or the horizon, but it declines as the tail probability declines. Thus, the adjustment grows smaller in percentage terms as the tail event becomes less likely.

By using the concept from statistics of first-passage time (also known as first-hitting time) we can estimate the probability that our return will hit a particular hurdle e.g. -10% during the investment period, or estimate the maximum loss –$X during the period at some given confidence level e.g. 99%

In the absence of drift, things are symmetric about the barrier.The really fun bit is that the path-dependent joint

probability of breaching the barrier and arriving at –A is equal to the path-independent probability of just ending at

–A because to get to –A you have to breach the barrier somewhere. So by invoking the principle of reflection we

have turned a path-dependent joint probability into a path independent one. We can calculate Intra-Horizon Expected Shortfall similarly using an iterative process.

There are some very important implications that spring to mind immediately when examining that formula. First of all, the second term can never be zero or negative.

That means intra-horizon risk is always greater than end-of horizon risk.

For example, even keeping the Normal distribution, prohibiting jumps, and setting the drift to zero: at 1% VaR-I = 1.1072 * VaR. At 5% it’s 19.16% bigger.

Secondly, intra-horizon risk (probability) increases with time, whereas end-of-horizon risk (probability) decreases with time.

This adds fuel to the argument against time diversification. Various authors have argued that as the investment horizon increases (e.g. more years until retirement), the probability of loss declines and therefore an investor can afford to be more aggressive. In fact this is still taught in business schools, and forms a part of every investment planning questionnaire that I have seen; “how old are you?”.

Samuelson (1963) argued against the benefits of time diversification, saying that although the probability of loss declines, the magnitude of the potential loss increases. When we consider risk from an intra-horizon perspective, we see that not only does the magnitude of the potential loss increase, but so too does the probability of loss

Intra-Horizon VaR ( VaR-I ) is by no means a panacea to all the problems of Value at Risk. The simple closed form formulas shown here rely on the Normal distribution which we well know to be inadequate in capturing the fat tails typically associated with financial assets. More complex versions require Monte Carlo simulations to derive the first passage probabilities. In addition, the method shown here assumes that the distribution of returns remains stationary throughout the time horizon which in the real world it would most certainly not do. Nonetheless it is a helpful and useful addition the risk management armoury particularly for those for whom falling below a specified level at any stage may have adverse consequences.

References:

Boudoukh, Jacob (Kobi), Stanton, Richard H., Richardson, Matthew P. and Whitelaw, Robert F., MaxVaR: Long Horizon Value at Risk in a Mark-to-Market Environment (March 2004)

Kritzman, Mark and Rich, Don R., The Mismeasurement of Risk. Financial Analysts Journal, Vol. 58, No. 3, May/June 2002.

Yen, Simon H., Lin, Shao-chieh, Re-examining the Contributions of Intra-Horizon Risk Measures

*Due to possible confusion of abbreviations of Continuous VaR with Conditional VaR and I-VaR with Incremental-VaR I strongly suggest using either MaxVaR, VaR-I or Within Horizon VaR.

The small VBA Code snippet below implements the MaxVaR or VaR-I of Boudoukh, Richardson and Stanton. Expected Shortfall measures can be similarly coded with an extra loop iteratively incrementing the CL and then taking the average of the values beyond the original Confidence level.

'Intra Horizon, MaxVaR or Continuous VaR ( Normal ) 'without drift

'Method as per Boudoukh, Richardson, Stanton and Whitelaw 2004

'http://www.faculty.idc.ac.il/kobi/

'coded by Peter Urbani

Function VaRINORMAL(Mean, sigma, Time, CL)

M = 0 'Mean - Sigma ^ 2 / 2 'Drift term for geometric brownian motion

VaRN = Application.WorksheetFunction.NormSInv(1 - CL) sigma Sqr(Time) + M * Time

Inc = VaRN

Initial = Application.WorksheetFunction.NormSDist((Inc - M Time) / (sigma Sqr(Time))) + (Exp(2 M Inc / sigma ^ 2) Application.WorksheetFunction.NormSDist((Inc + M Time) / (sigma * Sqr(Time)))) - (1 - CL)

Target = Initial

While Target > 0#

Inc = Inc - 0.000001 'for faster calc reduce this by 10x