MECH 431
# Future Value Analysis & Benefit Cost Ratio

Updated 2018-07-19

- One might want to look at the worth at some point in the future.
- Future worth analysis is identical to present worth analysis, except the the analysis is in the future.
- Analysis period for the alternatives must be the same (same as PWA). Extend analysis period to least common multiple (LCM) of all alternatives’ useful life time.

*Recall the definitions:*

**Net Present Value (NPV)**: equivalent value of discounted project cashflow to present time (often time zero).

**Net Future Value (NFV):** equivalent value of compounded project cashflow to a desinated future point in ime. It is the accumulation of unrecoverd capital and required returns throughout the project.

- If NFV > 0, then the net cashflow at the end is profit, so one should
**accept**the project. - If NFV < 0, then the net cashflow at the end is loss, so one should
**reject**the project. - If NFV is 0, then
*marginally*accept. (Perhaps one should consider additional factors that may influencce the projection). - If
**all alternatives**have NFV < 0, don’t forget the*do nothing*option.

Recall the *economic criterion* based on fixed input/output:

Situation | Criterion | |
---|---|---|

Fixed input |
Amount of capital is fixed | Maximize future value of benefits |

Fixed output |
Amount of benefit is fixed | Minimize the future value of costs |

Unconstrained |
Neither capital nor benefits are fixed | Maximize NFV (maximize future value of benefits and minimize future value of costs) |

Example: lecture slide example

A B C Investment $2.5k $3.5k $5.0k Annual cost $900 $700 $1000 with $100 increase each year Salvage value $200 $350 $600 Life 5 5 5 Annual revenue $1.8k $1.9k $2.1k with 15% growth rate MARR 10% 10% 10% We compute the NFV:

\[\text{NFV}_A=(-2500)(F/P, 10\%,5)+(1800-900)(F/A, 10\%, 5)+200\\ \text{NFV}_B=(-3500)(F/P, 10\%, 5)+(1900-700)(F/A, 10\%, 5)+350\\ \text{NFV}_C=(-5000)(F/P, 10\%, 5)+(2100)(F/g, g=15\%, 10\%, 5)\\+(-1000)(P/A, 10\%, 5)+(-100)(F/G, 10\%, 5)+600\]Once computed, NFV for option C is highest. Therefore, this should be the option that should be taken.

In simple terms, it is the ratio of the benefit over costs:

\[\text{Ratio}=\frac{\text{PW of benefit}}{\text{PW of costs}}=\frac{\text{FW of benefit}}{\text{FW of costs}}=\frac{\text{EUAB}}{\text{EUAC}}\]For a sensible decision, the benefit to cost ratio should be \(\geq 1\).