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.

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