## Bond Price and Yield Volatility

Table 13-2 compares the RiskMetrics volatility forecasts for U.S. bond prices. The data are recorded as of December 31, 2002 and December 31, 1996. The table includes Eu-rodeposits, fixed swap rates, and zero-coupon Treasury rates, for maturities ranging from 30 day to 30 years. Volatilities are reported at a daily and monthly horizon. Monthly volatilities are also annualized by multiplying by the square root of twelve.

Short-term deposits have very little price risk. Volatility increases with maturity. The price risk of 10-year bonds is around 10% annually, which is similar to that of floating currencies. The risk of 30-year bonds is higher, at 20-30%, which is similar to that of equities.

Risk can be measured as either return volatility or yield volatility. Using the duration approximation, the volatility of the rate of return in the bond price is a

Type/ |
Code |
Yield |
End 2002 |
End 1996 | ||

Maturity |
Level |
Daily |
Mty |
Annual |
Annual | |

Euro-30d |
R030 |
1.360 |
0.002 |
0.012 |
0.04 |
0.05 |

Euro-90d |
R090 |
1.353 |
0.005 |
0.030 |
0.10 |
0.08 |

Euro-180d |
R180 |
1.348 |
0.009 |
0.064 |
0.22 |
0.19 |

Euro-360d |
R360 |
1.429 |
0.030 |
0.188 |
0.65 |
0.58 |

Swap-2Y |
S02 |
1.895 |
0.110 |
0.634 |
2.20 |
1.57 |

Swap-3Y |
S03 |
2.428 |
0.184 |
1.027 |
3.56 |
2.59 |

Swap-4Y |
S04 |
2.865 |
0.257 |
1.429 |
4.95 |
3.59 |

Swap-5Y |
S05 |
3.224 |
0.329 |
1.836 |
6.36 |
4.70 |

Swap-7Y |
S07 |
3.815 |
0.454 |
2.535 |
8.78 |
6.69 |

Swap-10Y |
S10 |
4.434 |
0.643 |
3.613 |
12.52 |
9.82 |

Zero-2Y |
Z02 |
1.593 |
0.107 |
0.631 |
2.18 |
1.64 |

Zero-3Y |
Z03 |
1.980 |
0.172 |
0.999 |
3.46 |
2.64 |

Zero-4Y |
Z04 |
2.372 |
0.248 |
1.428 |
4.95 |
3.69 |

Zero-5Y |
Z05 |
2.773 |
0.339 |
1.935 |
6.70 |
4.67 |

Zero-7Y |
Z07 |
3.238 |
0.458 |
2.603 |
9.02 |
6.81 |

Zero-9Y |
Z09 |
3.752 |
0.576 |
3.259 |
11.29 |
8.64 |

Zero-10Y |
Z10 |
3.989 |
0.637 |
3.600 |
12.47 |
9.31 |

Zero-15Y |
Z15 |
4.247 |
0.894 |
5.018 |
17.38 |
13.82 |

Zero-20Y |
Z20 |
4.565 |
1.132 |
6.292 |
21.80 |
17.48 |

Zero-30Y |
Z30 |
5.450 |
1.692 |
9.170 |
31.77 |
23.53 |

Here, we took the absolute value of duration since the volatility of returns and of yield changes must be positive.

Price volatility nearly always increases with duration. Yield volatility, on the other hand, may be more intuitive because it corresponds to the usual representation of the term structure of interest rates.

When changes in yields are normally distributed, the term a (Ay) is constant: This is the normal model. Instead, RiskMetrics reports a volatility of relative changes in yields, where a(Ay) is constant: This is the lognormal model. The RiskMetrics forecast can be converted into the usual volatility of yield changes:

Table 13-3 displays volatilities of relative and absolute yield changes. Yield volatility for swaps and zeros is much more constant across maturity, ranging from 0.9 to 1.2 percent per annum.

It should be noted that the square root of time adjustment for the volatility is more questionable for bond prices than for most other assets because bond prices must converge to their face value as maturity nears (barring default). This effect is important for short-term bonds, whose return volatility pattern is distorted by the

Type/ |
Code |
Yield |
a (dy/y ) |
a (dy ) | ||||

Maturity |
Level |
Daily |
Mty |
Annual |
Daily |
Mty |
Annual | |

Euro-30d |
R030 |
1.360 |
1.580 |
9.584 |
33.20 |
0.021 |
0.130 |
0.45 |

Euro-90d |
R090 |
1.353 |
1.240 |
7.866 |
27.25 |
0.017 |
0.106 |
0.37 |

Euro-l80d |
Rl80 |
1.348 |
1.267 |
8.321 |
28.83 |
0.017 |
0.ll2 |
0.39 |

Euro-360d |
R360 |
1.429 |
1.883 |
11.177 |
38.72 |
0.027 |
0.160 |
0.55 |

Swap-2Y |
S02 |
1.895 |
2.546 |
13.993 |
48.47 |
0.048 |
0.265 |
0.92 |

Swap-3Y |
S03 |
2.428 |
2.264 |
12.247 |
42.42 |
0.055 |
0.297 |
l.03 |

Swap-4Y |
S04 |
2.865 |
2.061 |
11.158 |
38.65 |
0.059 |
0.320 |
l.ll |

Swap-5Y |
S05 |
3.224 |
1.901 |
10.370 |
35.92 |
0.061 |
0.334 |
l.l6 |

Swap-7Y |
S07 |
3.815 |
l.6l9 |
8.883 |
30.77 |
0.062 |
0.339 |
l.l7 |

Swap-l0Y |
Sl0 |
4.434 |
1.409 |
7.827 |
27.ll |
0.062 |
0.347 |
l.20 |

Zero-2Y |
Z02 |
1.593 |
2.916 |
16.576 |
57.42 |
0.046 |
0.264 |
0.9l |

Zero-3Y |
Z03 |
1.980 |
2.583 |
14.681 |
50.86 |
0.051 |
0.291 |
l.0l |

Zero-4Y |
Z04 |
2.372 |
2.384 |
13.541 |
46.91 |
0.057 |
0.321 |
l.ll |

Zero-5Y |
Z05 |
2.773 |
2.263 |
12.847 |
44.50 |
0.063 |
0.356 |
l.23 |

Zero-7Y |
Z07 |
3.238 |
l.9l3 |
10.825 |
37.50 |
0.062 |
0.351 |
l.2l |

Zero-9Y |
Z09 |
3.752 |
1.650 |
9.309 |
32.25 |
0.062 |
0.349 |
l.2l |

Zero-l0Y |
Zl0 |
3.989 |
1.556 |
8.766 |
30.37 |
0.062 |
0.350 |
l.2l |

Zero-l5Y |
Zl5 |
4.247 |
1.376 |
7.694 |
26.65 |
0.058 |
0.327 |
l.l3 |

Zero-20Y |
Z20 |
4.565 |
1.223 |
6.776 |
23.47 |
0.056 |
0.309 |
l.07 |

Zero-30Y |
Z30 |
5.450 |
1.037 |
5.603 |
l9.4l |
0.057 |
0.305 |
l.06 |

convergence to face value. It is less of an issue, however, for long-term bonds, as long as the horizon is much shorter than the bond maturity.

This explains why the volatility of short-term Eurodeposits appears to be out of line with the others. The concept of monthly risk of a 30-day deposit is indeed fuzzy, since by the end of the VAR horizon, the deposit will have matured, having therefore zero risk. Instead this can be interpreted as an investment in a 30-day deposit that is held for one day only and rolled over the next day into a fresh 30-day deposit.

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