## C [ yH

lxiii

Reinvested Future Value (RFV) at Horizon H

RFV (1, H) = accumulated reinvested value from payments in years 1 through year H (after the Hth year payment has been received).

The last equality is intended to relate the RFV back to the earlier PV concepts. We shall try to show these "throwback" relationships whenever possible. Also, for the special case of a level annuity,

Horizon Present Value (HPV) of "Tail" Cash Flow at Horizon H

HPV(H+1, M) = PV of flows in years H+1 to M with discounting to end of Hth year (after Hth year payment). • M-H . „
• 1 + y )H X Ci+H (1 + y)-i-H i=1

where PV (H+1, M) = the PV of flows received in years (H+1) through M evaluated as of the beginning of the first year.

For the special case of a level annuity,

### M - H

• C X (1 + y)-' i=1
• PV (1, M - H)

Total Future Value (TFV) at Horizon H

In general, when the pre-# reinvestment rate y1 and the post-# discount rate y2 differ,

For the case when the reinvestment rate and going-forward discount rate both equal y,

TFV (H ) = (1 + y )H PV (1, H) + (1 + y )H PV (H +1, M) = (1 + y)H [PV (1, H) + PV (H +1, M)] = (1 + y)H PV (1, M)

### For the special case of a level annuity, TFV (H ) = (1 + y )H PV (1, M) \H [f C

• v y j[ c)[.-(.+v)-«]|
• 1 + v)H -(1 + v f ]

Macaulay Duration D (1, H )

PV Volatility PV-VOL (1,H )

Reinvestment Volatility RFV-VOL (1, H )

The assumption here is that a single immediate rate move impacts all reinvestment throughout the horizon period.

For H > M, there are no future payments and so HPV and HD both have a value of zero.

When the same discount/reinvestment rate applies from year 1 through year M, the Horizon Duration can be expressed as a "throwback" to the earlier Duration expressions, but the resulting general formula is somewhat more complicated:

HD (H +1, M ) =

1

HPV (H +

1, M)]

1

HPV (H + 1

1, M)]

HPV (H + 1

1M)]

HPV (H +

HP D^H^M) _ PV (1, H) [V (1, H) _ V (1,M)]] _ H X HPV (H + 1,M)}

and since HPV (H +1, M) = (1 + y)H [PV (1, M)- PV (1, H)], we obtain the following general expression in Duration terms, 1

_ RFV (1, H) [V (1, H) _ V (1, M)] _ H X HPV (H +1, M)} RFV (1, H)

The above formulation holds for all forms of cash flow. However, for the special case of a level annuity,

and one obtains the following far simpler but narrower result,

Horizon Present Value Volatility HPV-VOL(H+1,M )

Total Future Value Volatility TFV-VOL(H )

The assumption here is a simple immediate rate move that impacts reinvestment throughout the horizon period and also affects the tail pricing at the horizon.

or finally,

This result could also be obtained using the earlier expressions for HD(H+1, M) and RFV-VOL(\, H) under the same assumption of a single rate movement,

### TFV (H) x [TFV-VOL (H )] = RFV (1, H) x [RFV -VOL (1, H)] +

• HPV (H +1, M) x [ HPV -VOL (H +1, M)]
• 0. H)]-

### (1 + v)

• D)1,M)-D(1, H )]
• P-(1, H)x . HPV1H + 1,M)r . . n

### "CTTyr (H-M M]--(W-^^^^1,^)^ H

• PN (1,H)
• D (1,M)-D(1,H1]

### RPT (1,/-)r ( HPV(H +1,MM) r .

• px+yN-Z[N/-Xy(NM+1] + (1 + +) ) [N-D(1'M)]
• RXX(1,H) + HPV(H + 1,M)] " H - D (1, M)

or finally,

The Babcock Rule

Under the earlier assumption that interest rates move from y1 to y2 at the very outset, with rates remaining stable thereafter, the total TFV volatility is needed.

### TFV (H) = TFVyi (H )[1 + TFV -VOLy (H) x (y2 - y1)]

• H - D(1,M)]
• TFVV1 (H) x +

Immunization

Under the same assumption as the Babcock rule, immunization is said to occur when TFV remains stable under an immediate interest rate shift from y1 to y2, i.e.,

or from the Babcock rule,

In other words, immunization occurs when D (1, M ) = H

If the liabilities consist of a stream of payouts, then the asset portfolio will immunize the liability flow when

The above are first-order conditions. There are various second-order conditions that must be applied under a broader range of assumptions regarding the structure of rate movements.

### Horizon Analysis

In one form of horizon analysis, the pre-horizon rate y1 is typically assumed to be stable, and the movement to a second rate y2 is assumed to be concentrated at the horizon. Thus, the only volatility term involved is HPV.

TFV (H) = RFVyi (1, H) + HPVy (H +1, M)[l + HPV-VOLy (H +1, M) X

« TFVyi (H) + HPVyi (H +1, M) X HPV-VOLy (H +1, M) X 