## Horizon Present Value

10 Annual Payments of \$10 8% Discount Rate

Present Cumulative Horizon

Value of Each Present Value Present Value,

Present Cumulative Horizon

Value of Each Present Value Present Value,

 Horizon, H Payments Payment, (H,H ) PV(1,H ) HPV(H+1,10) 0 \$0 \$0.00 \$0.00 \$67.10 1 10 9.26 9.26 62.47 2 10 8.57 17.83 57.47 3 10 7.94 25.77 52.06 4 10 7.35 33.12 46.23 5 10 6.81 39.93 39.93 6 10 6.30 46.23 33.12 7 10 5.83 52.06 25.77 8 10 5.40 57.47 17.83 9 10 5.00 62.47 9.26 10 10 4.63 67.10 0.00 11 0 0.00 67.10 0.00 12 0 0.00 67.10 0.00 13 0 0.00 67.10 0.00 14 0 0.00 67.10 0.00 15 0 0.00 67.10 0.00

calculation. At the outset, H = 0 and the entire 10-year annuity remains ahead of us, so that the HPV(1,10) is just the same as the PV(1,10) for the entire annuity. In Table 1, PV(1,10) = \$67.10, and in Table 3, HPV(1,10) is also \$67.10. However, at the end of the first year, for H =1, there are only 9 remaining payments. In other words, Table 3's HPV(2,10) = \$62.47 is the same as Table 1's PV(1,9), that is, the PV for a level flow with 9 annual payments. We can continue in this fashion until we reach a horizon H = 9, at which point there is only the one remaining \$10 payment to be received. Thus, Table 3's HPV(10,10) = \$9.26 is just the discounted value of a \$10 payment one year forward, PV(1,1), which we could have read from the third column of Table 1.

The exceptionally simple relationship between Tables 1 and 3 holds only for level cash flows. For more complex cash flows, the HPV(H+1, M ) must be adjusted to reflect the PV of the cash flow's remaining payments from year (H + 1) to the maturity year M.