Yield Volatility

[jjtejadad]

Hi,

When we are calculating the parametric VaR for a Bond we use the volatility of the Yield and the DV01. My confussion is about the volatility of the Yield, should I use the standard deviation of the Yield or the standard deviation of the daily % change in the yield. If i use the daily % change then how can I multiply it by the DV01 and the Z value to obtain the VaR.

Thanks,

José

 

[David Harper CFA FRM]

Hi @Jose (@jjtejadad )

In general we are using (and any good FRM question should use) what Tuckman calls the basis point volatility, and further it most cases it should be expressed in per annum terms (aka, annual basis point volatility). That means if we consider the classic duration relationship:

ΔP/P = -D*Δy

The VaR is concerned with the worst case %PΔ as estimated by the duration's transmission (multiplier) on the worst expected yield "shock", which is a change in the units of the interest rate (aka, yield in the single-factor duration relationship). So we want:

ΔP/P = -D*[σ(y)*α(z)], where if we assume yield is normally distributed, then maybe Z(α) = 1.645 at α = 0.050.

And σ(y) wants to represent, as you say in the former case, the standard deviation of the yield but not the standard deviation of the daily percentage change in the yield. It is not a percentage, and is nearer to a percentage point except that's (IMO) not quite accurate because yields don't run 0 to 100%.

The key is that, as Tuckman says, "basis point volatility is in the units of an interest rate (e.g., 100 basis points)." (Tuckman Chapter10).

So, if the yield volatility is σ(y) = 1.0% = 0.010, then this means σ(y) = 100 basis points, as illustrated by y(0) = 3.0% jumping up 100 basis points to y(1) = 4.0%; or y(0) = 4.5% jumping up to y(1) = 5.5%, which is smaller in percentage terms but nevertheless is still + 100 basis point. It does not mean 6.0% up to 6.0%*1.01 = 6.06% or 6.0%*2 = 12%.

The reason is that the Price/Yield relationship has yield on the X-axis, so the relationship concerns a change in the units of this X-axis. The X-axis is not, to use your latter phrase, a percentage change in the yield; the units just happen to be yield. To understand what the yield volatility of σ(y) = x represents, we can literally imagine any shift of x basis points on the yield axis (X axis) of the bond P/Y plot.

DV01 gives us the dollar change for a 1 bps change in the yield, such that ΔP (per $100 face value) = DV01*Δy*10,000; if we want VaR(ΔP), then:

VaR($ΔP, per $100 face value) = DV01*[σ(y)*α(z)]*10,000; for example, if the σ(y) = 0.10% = 0.0010, then
95% $ΔP VaR = DV01*[0.0010*1.645]*10,000 = DV01 * 16.450. This is based on the similarity between:

• ΔP = -D*P*[σ(y)*α(z)]
• DV01 = D*P/10,000 = "dollar duration;" i.e., DV01 is simply a re-scaled dollar duration which itself is the slope of the tangent line to the P/Y curve; a single unit is 100% which is 10,000 basis points. I hope that's not too much, thanks!

 

[jjtejadad]

Thanks David! This really answers my question. And thanks for the excellent study material on your webpage!

 

[rmdfra]

Hi David. Your explanation makes much sense to me, but I can't ignore Fabozzi taking the standard deviation of log returns of a time series of interest rates here, and calling it volatility. Am I missing something?
Using the same data from Table A.1., if I take the difference of consecutive rates (in percentage points) and then take the standard deviation, I get a much different measure.

 

[ptf]

@David Harper CFA FRM , I second @rmdfra question about Fabozzi. I checked recently math behind Bloomberg terminal YLD_VOLATILITY_30D field for some bonds and it is exacty what rmdfra stated: annualized (Bloomberg takes 260D) standard deviation of yield log returns.

I built excel to check this field, and as I couldn't get the number Bloomberg provided I contacted help, and they gave me exact definition. The way I calculated yield volatility was to calculate standard deviation of changes in yields in bps with no annualization, as yields are already annualized. What's more, I also contanted guys from our trading desk and they think of yield volatility the same way and were surprised how Bloomberg calculates it.

Bloomberg also provides price volatility for bonds, calculated as annualized std deviation of bond prices daily returns, which is of no practical use, as with no yield change the price still goes up adding volatility - surprisingly I found out that this is the way one company I know providing VAR calculates volatility for bonds.

 

[David Harper CFA FRM]

Hi @ptf Thank you truly for that realistic (i.e., Bloomberg, trading desk!) and informative feedback. I happen to be currently updating the Tuckman notes (including an OAS model) and I will try to get this into my XLS update. A few thoughts:

• As a practical matter, we (an FRM EPP) can't vary the basic approaches utilized for the fixed income short-rate models, as largely reflected in assigned Tuckman. I'm sure you realize that, I get that's not your point at all. The entire front-end of Tuckman is based on what I would call arithmetic rate trees where (eg), we measure a jump from initial 3.0% to 3.20% as a +20 bps change.
• The latter part of Tuckman (aka, more sophisticated and it sounds like more realistic) introduce the lognormal-type models where my same example would generate a return of LN(3.20/3.00) = +6.45%, rather than the +0.20%, as the return's input into a series vector (of returns) that informs the volatility as a standard deviation of the return vector. I'm glad Fabozzi must use 6.694 rather than the more natural 0.06694 or 6.694 because I can never seem to use the latter!
• The either/or of this difference, to me, is not even a matter of right/wrong. Its analogous to the situation in equities where we can choose a naïve model of normal arithmetic returns, or a more elegant model of normal log returns. But I would just point out that, to my thinking, this choice of "how do we compute the return vector based on an observed interest rate series" is the prior step to computing the standard deviation of that vector. So Tuckman's well-established definition of "annual basis point volatility" is arguably independent. We can plug (eg) 37.5% annual basis point volatility into any of his models (I think!?). Also, I don't see the scaling issue--e.g., monthly to annual--as any issue. I hope that's helpful, I'll try to come back after the Tuckman update, maybe even with an XLS snippet Thank you again for realistic feedback from the trenches, I always appreciate that!

P.S. I just realized you said your guys at the trading desk disagreed with Bloomberg's approach and instead follow your/Tuckman's approach of simple yield change in bps (I'm ignoring the annualization, as I have not experienced that to present any special challenge; of course the log returns are time additive but delta bps is also!). That's fascinating, I would have assumed Bloomberg is synched with your desks' approach!!?

 

[ptf]

Hi @David Harper CFA FRM - indeed, they disagreed with Bloomberg. We discussed the topic of volatility levels and the levels my colleague from trading desk were talking about were what they observe daily and what is intuitive for them, and this happens to be the numbers I get using Tuckman approach. And the guy I am talking with on the daily basis is really experienced fixed income trader.

What I am trying to investigate now is how this approach work while building variance - covariance matrix. Let's say we have EUR bond, and our portfolio is priced in USD. Using simplistic approach we need to build matrix with EUR yield curve tenors as risk factors + one currency risk factor (EURUSD spot price). So, for EURUSD volatility input is easy - std dev of returns. For yield curve tenors - Tuckman approach. Than You map the CFs, discount them using USD DFs, calculate VAR yield volatility effect using modified duration, calculate VAR ccy volatility effect and then just multiply both effects by EURUSD spot.

The question is - what about combined effect (covariance / correlation) from both yield volatility and currency volatility?

Btw I really appreciate Bionic Turtle, I just hope I will manage to squeeze enough time to prepare for the exam (work...), but You are doing really good job here