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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