Stress Testing Methodology: Top-Down vs. Bottom-Up

In this article we'll review the strengths and applications of top-down vs. bottom-up stress testing and the model implications for each approach.

Choosing the right methodology for stress testing can be challenging but with a few basic considerations one can determine the best fit-for-purpose modelling approach that will provide accuracy and stay true to the narrative of the stress test. To start, it is important to understand the basics of the two approaches.

Top-Down Stress Testing
The term Top-Down typically applies to modelling the movement of output results based on some estimated correlation with driving variables. Let's walk-through a simple example to demonstrate the approach. Assume a regression model is run against a time series of Real Estate loan amounts from Balance Sheet data and we have a correlation coefficient calculated against time series data for the US Housing Price Index over the same lookback period. In a top-down approach we would use a stress scenario which would contain levels or a stressed shock for the driving variable, in this case US HPI, and apply that to the actual or base level of the results to be stressed, in this case Real Estate loans.

Let's illustrate this numerically, assume .5 is our correlation coefficient, Real Estate loan amount is $1MM and the Shock to US HPI is a -20% downturn. To estimate the top-down stressed amount for our Real Estate loan portfolio we calculate: $900K = $1MM + ($1MM * .5 * -.2) or a 100K loss at 10%.

Bottom-Up Stress Testing

To illustrate this properly we should move to a different area particularly the Trading book where we have a very different risk profile for stressed loss calculations. Assume we have a trading portfolio with only one US Equity priced at $100 and we hold only a single stock and the scenario narrative notes that under the stressed condition US Equities move down -20%. Ok, easy enough we’re at a $20 loss with the portfolio value moving from $100 to $80. However, now let’s say we have three products Equity, Equity Option and a Bond. Ok, this is an issue because our portfolio now has sensitivity to Equity Price, Equity Volatility and Interest Rates which can all move in a scenario and we may actually have a hedging effect/gain from the Equity Option. Very quickly we have what I like to refer to as "Portfolio Mechanics".

Essentially we have a mix of different Risk Factors that the portfolio is sensitive to such as Interest Rates, Equity Prices, Equity Volatility as well as varying trades that impact the overall value of the portfolio in different ways such as Equity vs. Equity Option.

In the Bottom-Up approach we would stress the inputs, rather than reporting line outputs ultimately allowing us to account for the Portfolio Mechanics and arriving at a more accurate loss estimate.

In the example, equity price shocks would be applied, equity volatility shocks would be applied and interest rate yield shocks would be applied. To estimate the loss numbers the portfolio would be revalued, where each economic input would be shocked and then each trade would get recalculated to determine the new price under the stressed economic condition. Again, because we’re stressing the inputs and moving up to revalue the portfolio we call this a Bottom-Up approach.

There are cases where a simple top-down approach is used in the trading book. However, this is usually considered a back of the envelope approach, as it only applies a stress multiplier to the Market Value and these shocks are often severely punitive. In trading risk this is the least preferable approach due to the lack of accuracy and higher loss estimates and it’s only resorted to when there are significant challenges in revaluing the portfolio.

Common Uses for Top-Down
Top-Down is typically preferred when a scenario must be projected to estimate the portfolio evolution over some long running economic cycle.

For example, in the Dodd Frank Act Stress Test there’s a 9 Quarter projection which requires banks to run multiple supervisory scenarios to evolve portfolios over the projected horizon. This is imposed on balance sheet and income statement items as well as market and credit risk specific calculations that have underlying trading portfolios with high complexity. However, if we’re just evolving balance sheet line items there are cases where a top-down model can be very useful for projections such as the Real Estate loan example described earlier.

Now let’s expand on that example and assume the following, we have the originally calculated .5 correlation to US HPI, 2017 Q4 year-end Actual balance of $1MM and we hold the portfolio constant with no business growth over the projected horizon. This is what that would look like in practice:

The calculation for the Real Estate loan estimate is % Change from Actual (this is change of HPI from Q4 2017 to the current quarter) * Business Growth * Variable Correlation * Real Estate loan Q4 2017 Actual then add the original base Real Estate loan Q4 2017 Actual.

Now we have a pretty realistic use case for Top-Down and we can easily explain the movement of the portfolio under our scenario conditions.

The table below contains an example of projected real estate loan estimates against a Fed baseline scenario:

Common uses for Bottom-Up
Ok, we’re going to continue on with the projection theme and the Trading book which is going to get really exciting since we’ll touch on "Scenario Expansion" which solves a unique problem for Bottom-Up stress testing.

Let’s continue with the three trade portfolio which is sensitive to Equity Price, Equity Volatility and Interest Rate. Now go back to the top-down example and remember we had a macroeconomic variable US HPI.

This is lifted from an US Federal Reserve supervisory scenario for projections and there’s a point now where these two different worlds converge. In projections, we’ll have broad sweeping economic drivers like GDP, CPI, HPI some risk factor specific drivers like Treasury curves or DJI, these are commonly referred to as "the Fed 28" as there’s 16 domestic drivers and 12 international ones useful to estimate foreign market movements. These broad sweeping variables differ significantly from the Equity and interest rate shocks we need but we can bridge the gap using a handy trick called “Scenario Expansion” and coincidentally we get to reuse mathematical concepts familiar from top-down stress testing.

Ok, let’s get into the weeds – the Fed 28 specifies a US 3M Treasury level (related to yield curves), VIX levels (related to equity volatility) and DJI (related to equity price). We can now regress these variables against more specific yield curves, equity volatilities and equity prices and again estimate correlation coefficients. Now we have an estimate of how high-level drivers move granular market data and can use that for a basis to revalue a complex trading portfolio. This is Scenario Expansion modelling. There are many different approaches to expansion modelling but they all essentially start with the fundamentals of estimating the correlation between some granular market parameters vs macroeconomic drivers.

Key Considerations
There are many options when choosing how to model stress scenarios, considering top-down and bottom-up and various different methodology tweaks that come in to play. Ultimately, when designing a reasonable modeling approach it is important to consider how driving variables effect the portfolio, how this relates to the original scenario narrative and how explainable and accurate the results are. Keeping these point in mind will ensure you select the appropriate approach for stress testing and sound risk management.

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