## Wednesday, September 19, 2012

### PV01 vs Historical VaR

The world of fixed income is very much impacted by PV01, yet Market Risk analysts hang onto historical Value at Risk as if it is the be all for measuring potential downside. In my opinion this is a bit busted and I will explain why in this short blog post.

We must scrap historical VaR
There are many ways to measure risk, however one of the most popular methods is Value at Risk and we have discussed this statistical technique at length in various places on this blog [ What is wrong with VaR ]. Nevertheless, today we question why Market Risk analysts are hanging onto historical VaR when price volatility for a fixed income portfolio is measured using PV01.

To understand this problem let's look at both historical VaR and PV01.

Historical VaR
The process around historical VaR is really quite straight forward and bases potential losses on the performance of a portfolio by looking at actual previous returns.  So much has been written on this subject that I am not going to delve into the semantics of historical VaR, except to say Investopedia describes the calculation quite clearly below.

The historical method simply re-organizes actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective.
Investopedia | Understanding Historical VaR

For a portfolio of stocks, higher volatility of returns will translate to a wider curve along the x-axis of the chart above and consequently an increase in the value at risk number. Now, although historical VaR is full of problems, this is fine for equity positions but what about applying this kind of thinking to the world of fixed income and a portfolio of bonds?

PV01
Let's take a look at PV01 or the price variance of a vanilla bond for a single basis point change in prevailing interest rates.

The bond example above has tenure of 5 years but an effective maturity of 4.4012 years. This is found by calculating the bonds duration as shown in step 1 of the diagram; where a straight forward mathematical procedure is carried out by taking the sum of the bonds contractual cash flows, multiplied by the time to each cash flow over the present value of bonds entire cash flow. In the second step, we can calculate the PV01 through perturbation of the bond's yield as a measure of its absolute price.

The PV01 can be thought of as the price variance of the bond for a single basis point change in prevailing interest rates and in the simple example above, the 5 year bond with a yield to maturity of 6% has a PV01 of 0.04327. This would equate to a price variance of 432.70 for a million euro position on this specific instrument and the PV01 estimate is only accurate at the single point of time when the calculation is carried out.

Generating VaR from PV01
The two key driving factors that directly alter a bond's price is either a change in prevailing interest rate volatility and/or the swing of issuer risk on the paper, where the latter is shown when a specific bond trades at discount from its expected value, yet the interest rates haven't shifted. If the paper was to trade at discount from its expected forward price, the bond has a perceived increase in credit risk, while the opposite is being shown in the diagram below.

Yield Curve Shift | An example

Both PV01 and the credit spreads are forward estimates of the true risk in a bond, not historical prices and in my opinion, historical VaR is not a good forward indicator of the market risk in the fixed income instrument class.

The 'big number one' reason why historical VaR is utterly erroneous for measuring risk in a bond is that the payment of a coupon will reduce the bonds sensitivity to interest rates through time or concisely as the bond ages through time, its PV01 will slowly reduce, it naturally erodes as time passes and will only increase if interest rate volatility does in an equivalent manner or the credit spreads widen.

The next point that is more concerning is stress testing. The measure of PV01 allows hypothetical changes of the prevailing interest rate horizon and can be used as a shock test for the bonds forward Mark To Market value. PV01 is actually the corridor to understanding this relationship between prevailing interest rate volatility and the price variance of the value of the bond. Historical Mark to Market can't be used to shock the bonds par curve and consequently if a Market Risk analyst is using Historical VaR as a measure of risk, they will not be able to carry out any reliable stress test of their fixed income book.

Now fixing this problem and moving away from historical VaR is going require a different model and we'll discuss that in a separate blog posting.  Most importantly, the point I am making here is that historical VaR for fixed income portfolios is dodgy and erroneous.

1. Martin
I am getting old, this really brings me back, PVBP was state of the art risk management before VAR in the 1980s.

I believe PVBP is still useful used as a real-time risk measure for vanilla bonds, with convexity added of course.

But unfortunately it does not answer the question, how likely is a 1 BP change or a 10 or 20 BP change, one still has to know that and that comes from historical prices.
PVBP gives the P&L for a 1 BP change it does not help with possible losses across the portfolio over a holding period? But there again VAR is not great at that either, different problems.

From Pat McConnell

2. I like your post, the fact that your site is a little bit different makes it so interesting, I get fed up of seeing the same old boring recycled stuff all of the time.

3. Pat, I agree PVBP is not new however the argument I am presenting here is that historical VaR on a fixed income portfolio is a load of rubbish, it really is.

If you think this through, 1 day before a bond is due to mature, most of the coupons will have been paid and consequently the risk from interest rate volatility will have been reduced substantially. There simply will be no duration left on the bond to give us a connection to the interest rate vol surface.

Why on earth would historical VaR be meaningful in any sense given this case?

It wouldn’t, it would overestimate the potential loss value grossly.

PV01 gives us a precise value for interest rate volatility at a point in time as shown in step 2, if you want to dial up 10bps of interest rate volatility risk you simply lift that number into the calculation, if you want 100bps of interest rate shock, you simply insert that into the calculation.

If we look at historical MTM however, we can’t infer the impacts from forward interest rates as a shock on the portfolio because we are moving through the coupon payments. Historical MTM will have credit spread data, market vol and liquidity risk all captured in a price, a price yesterday that is meaningless tomorrow as the bond moves through its own par curve position.

So historical VaR on a fixed income portfolio is just such a load of rubbish, it can’t be shocked coherently apart from the fact that historical VaR captures risks on coupons that have already been paid!!!

There is a problem however, while PV01 is the true exposure of risk for a bond position from interest rate changes, the PV01 will change through time. Consequently, we need to bootstrap the par curve for the bond and an immunised par curve for the portfolio to understand the impacts of interest rate volatility on the portfolio overtime. This is computationally a little bit more complex than simply working out a spot PV01 measure in time and needs a more complex model. Then to top it all off, we need to show the true issuer risk on the bond, the credit spread component also needs to be measured overtime, that will make the calculation additionally a touch convoluted.

But for historical MTM, it’s toast in my opinion and meaningless in so many ways.

1. I completely disagree with you. PV01 cannot capture the convexity of the bond. It assumes the yield-price cover as linear. This curve is actually convex which can be captured only using full revaluation with historical var.

On the other hand, even if the bond is maturing after 1day, historical var will not over estimate the risk as revaluation will be done considering cash flow is happening the next day. So the revalued amount will not change significantly at all.

2. Historical VaR is erroneous for measuring the risk in these instruments. It is horrible and troublesome in most cases where it is applied and for many reasons but that is all for another debate.

However, In the coupon paying bond case, historical VaR is sensitive to volatility on previous time horizons and does not make any assumption to the cash flow structure of the bond or the swap being paid off

Very simple logic. How can historical VaR be accurate in this case?

There are many instruments that don't fit the historical VaR model, coupon paying bonds don't, Zero coupons do and so on. Anything that is not on a forward curve or perpetuity in nature is probably not suitable for historical VaR. Spot trades yes, futures and forwards okay, equities obviously, coupon paying instruments no.

To put it simply, a bond with a 1 year maturity is not going to have the same sensitivity to interest rates as a 10 year note, this we know. As bonds move to maturity, their sensitivity also changes which nullifies the usefulness of historical VaR. Large price / yield swings in the past will eventually become irrelevant but the historical VaR calculation does not include these dynamics.

The Potential Future Exposure (PFE) of a vanilla swap or a bond needs to be mathematically 'induced' or bootstrapped on a curve that is then recalculated against a volatility surface through recursive techniques such as Monte Carlo. There are other ways. This is a second order calculation, while historical VaR is simply a first order calculation based on price movements through time.

https://en.wikipedia.org/wiki/Bond_convexity

Finally, if you look at some of the strong VaR engines out there such as the Sungard products just as an example, you can see the PFE calculations are tenor sensitive while historical VaR calculations are not. What I am saying is that the good risk software products are actually doing this already, thank heavens for that.

3. In the end, this might need to be demonstrated with a spreadsheet and if that is the case, I will write another blog to explain how the spreadsheet works.