## Reinvested Future Value

10 Annual Payments of $10 8% Discount Rate

Horizon, H |
Payments |
Carryforward Amount |
New Interest |
Reinvested Future Value, RFV(1,H) |

0 |
$0 |
$0.00 |
$0.00 |
$0.00 |

1 |
10 |
0.00 |
0.00 |
10.00 |

2 |
10 |
10.00 |
0.80 |
20.80 |

3 |
10 |
20.80 |
1.66 |
32.46 |

4 |
10 |
32.46 |
2.60 |
45.06 |

5 |
10 |
45.06 |
3.60 |
58.67 |

6 |
10 |
58.67 |
4.69 |
73.36 |

7 |
10 |
73.36 |
5.87 |
89.23 |

8 |
10 |
89.23 |
7.14 |
106.37 |

9 |
10 |
106.37 |
8.51 |
124.88 |

10 |
10 |
124.88 |
9.99 |
144.87 |

11 |
0 |
144.87 |
11.59 |
156.45 |

12 |
0 |
156.45 |
12.52 |
168.97 |

13 |
0 |
168.97 |
13.52 |
182.49 |

14 |
0 |
182.49 |
14.60 |
197.09 |

15 |
0 |
197.09 |
15.77 |
212.86 |

- 9.26 is obtained. And as in the preceding section, this PV(1,1) = $9.26 grows to exactly the RFV(1,1) = $10 when invested at 8%. Similarly, for H = 7, Table 2 shows that the RFV(1,7) = $89.23. From Table 1, this 7-year flow has a cumulative PV(1,7) = $52.06. A simple computation shows that
- 52.06 x (1.08)7 = $89.23.

More generally, as demonstrated in the Technical Appendix,

Like the PV, the FV has the appeal of great simplicity. Rather than think through a complex pattern of payments, we can just observe that all investments with the same last payment date will have the same RFV( 1 ,H) for each dollar of PV( 1 ,H):

In other words, any investment's RFV(1,H) can be reproduced by simply deploying its PV amount into a savings account that is then compounded forward at the given rate.

In the preceding, the RFV's horizon was defined to coincide with the last payment date. Suppose that is not the case, that is, suppose we want to consider a 15-year horizon but the investment's payments only cover 10 years? There is an easy fix. After the last payment in the 10th year, the accumulated value RFV(1,H) would simply be reinvested and compounded forward at the market rate until the 15th-year horizon, thereby growing to the RFV(1,15) = $212.86 as shown in Table 2.

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