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Yield Volatility

jjtejadad

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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

David Harper CFA FRM
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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!
 
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