When we move from the PV, which declines with higher discount rates, to the RFV, which increases with higher reinvestment rates, another "volatility" measure becomes important. The reinvestment volatility (RFV-VOL) is rarely characterized in the same quantitative way as the Duration concept, but doing so leads to some interesting results that should be more widely appreciated and that may be particularly useful for long-term holders of fixed-income exposures such as insurance companies and pension funds.

The first step in such a discussion is to focus on some prescribed future date as the RFV horizon. For bonds, as discussed in Inside the Yield Book, the reference RFV horizon H is almost always the bond's maturity date. We noted earlier how premium bonds with their higher coupon flows are more reinvestment-sensitive than discount bonds. In contrast, the zero-coupon bond is the ultimate in terms of reinvestment insensitivity: When its maturity is taken as the reference point, it will always have its maturity payment as the RFV regardless of the level of intervening interest rates. In other words, it has absolutely no sensitivity to changing reinvestment rates. (Incidentally, for all bonds having the same maturity date, the zero-coupon has the highest PV sensitivity to discount rate changes, but the lowest RFV sensitivity to reinvestment rates changes.)

It turns out that the percentage volatility RFV-VOL(1,H ) bears a very simple relation to the Duration value. For a cash flow stretching out to a given horizon H, the RFV volatility can be shown to be just the gap between the horizon and the flow's Macaulay Duration, adjusted by one plus the interest rate:

This result is developed in the Technical Appendix. As one might expect from the earlier discussion of the duration measure, this RFV-VOL finding is essentially derivative-based, that is, it acts as a better approximation for ever-smaller rate moves.

At the outset, it can be seen that for H = 0, before any payment whatsoever, there is no cash flow, and so the Duration D(1,0) has the trivial value of zero. Similarly, there is no reinvestment volatility:

Moreover, for the single lump-sum payment at the horizon, the Duration D(1,H ) just equals the horizon H:

D(1,H ) = D(H,H ) = H, and again there is no reinvestment sensitivity because

- H - H ]
- 1 + y) = 0.

However, moving from single lump-sum payments toward more general cash flows stretching out over time, the reinvestment volatility grows with longer horizons. Thus in Table 8, the annuity develops a significant exposure to changing reinvestment rates as the horizon lengthens. Taking the 7-year horizon as an example, Table 8 provides a value of .0(1,7) = 3.69, so that

To see how well this measure approximates an actual shift in reinvestment rates, refer to Table 2, which shows that RFV(1,7) = $89.23 at the 8% rate. If the reinvestment rate is raised to 9%, the RFV(1,7) becomes $92.10, a 3.22 percentage increase, which is reasonably approximated by the derivative-based value of 3.06.

Turning the above finding around, note that in general, when a flow's maturity M is taken as the FV horizon, the Duration and the RFV-VOL(1, M ) volatility add up to the flow's life,

Thus, for the annuity with M = 10, Table 8 provides values of D(1,10) = 4.87 and RFV-VOL(1,10) = 4.75, so that

M = D (1,10) + (1 + y) x [RFV- VOL( 1,10)] = 4.87 + 1.08 x 4.75 = 10.

It turns out that this relationship holds for any cash flow. For the 10-year annuity, the Duration and reinvestment volatility turn out to be nearly equal. However, this will not be true for more general cash flows, even though their sum will always equal the time to the last payment.

TABLE 8

Reinvestment Volatility

10 Annual Payments of $10 8% Discount Rate

Horizon, H |
Reinvested Future Value RFV(1,H ) |
Macaulay Duration D(1, H ) |
Horizon-to-Duration Gap H - D(1,H ) |
Reinvestment Volatility RFV-VOL(1,H ) = [H - 0(1, H ) / (1+y )] |

0 |
$0 |
0.00 |
0.00 |
0.00% |

1 |
10.00 |
1.00 |
0.00 |
0.00 |

2 |
20.80 |
1.48 |
0.52 |
0.48 |

3 |
32.46 |
1.95 |
1.05 |
0.97 |

4 |
45.06 |
2.40 |
1.60 |
1.48 |

5 |
58.67 |
2.85 |
2.15 |
1.99 |

6 |
73.36 |
3.28 |
2.72 |
2.52 |

7 |
89.23 |
3.69 |
3.31 |
3.06 |

8 |
106.37 |
4.10 |
3.90 |
3.61 |

9 |
124.88 |
4.49 |
4.51 |
4.17 |

10 |
144.87 |
4.87 |
5.13 |
4.75 |

11 |
156.45 |
4.87 |
6.13 |
5.67 |

12 |
168.97 |
4.87 |
7.13 |
6.60 |

13 |
182.49 |
4.87 |
8.13 |
7.53 |

14 |
197.09 |
4.87 |
9.13 |
8.45 |

15 |
212.86 |
4.87 |
10.13 |
9.38 |

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