The Total Future Value Volatility at Longer Horizons
The preceding development of an RFV-VOL(1,H ) volatility also provides an answer to the question of the volatility TFV-VOL(H ) of a cash flow's TFV(H) with a horizon H that coincides with the flow's last payment M, that is, where H = M. With horizons that match the flow's last payment, there are by definition no tail flows and so the total future value, TFV, consists of just the reinvestment-driven RFV(1,H ). Because the reinvestment effect is always positive,
(1 + y) , higher rates will always lead to higher TFVs (except for the case of a single lump-sum payments in the Mth year).
Even when the FV horizon is extended beyond the last payment date, this relationship continues to hold. The Duration value remains stable, but the horizon gap increases by the exact length of the extension. For example, if we look at a 12-year horizon with the 10-year annuity, the Duration D(1,12) = D(1,10) remains unchanged at 4.87, but the RFV-VOL(1,12) volatility now increases to 6.60:
Moreover, this TFV volatility result will hold for any horizon longer than the last payment date:
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