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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