Lecture 5: Present Worth

Introduction

Present Worth (PW) analysis resolves all monetary consequences of an alternative into a single equivalent present sum (Time 0). This sum is called Net Present Worth (NPW), Present Value (PV), or Net Present Value (NPV). PW analysis is ideal for valuing future costs and benefits, such as assets, stocks, or bonds.

I. Assumptions in Economic Analysis

  1. End-of-Year Convention: Present sums (\(P\)) at Time 0; future payments (\(A\), \(F\)) at end of period.
  2. Viewpoint: Analyze from the total firm/owner perspective.
  3. Sunk Costs: Ignore costs from past decisions; focus on differences between alternatives.
  4. Borrowed Money: Assume all money is borrowed at the defined interest rate (MARR).
  5. Inflation/Deflation: Assume stable prices initially; address inflation later.
  6. Income Taxes: Ignore taxes initially; address in later chapters.

II. Economic Criteria for Present Worth

Situation Criterion (PW)
Neither Input nor Output Fixed Maximize NPW (\(PW_{benefits} - PW_{costs}\))
Fixed Input Maximize PW of benefits
Fixed Output Minimize PW of costs

III. Time Period for Analysis

Define a common analysis period (planning horizon/project life) for all alternatives.

A. Equal Useful Lives

  • Compare NPW for each alternative over the same period.

B. Unequal Useful Lives

  • LCM of Lives: Use least common multiple of lives; assume identical replacement.
  • Fixed Study Period with Terminal Value: Use a practical analysis period; estimate market/salvage value for alternatives exceeding the period.

C. Infinite Analysis (Capitalized Cost)

  • For perpetual service (\(n=\infty\)):
    • \(A = P i\)
    • \(P = \frac{A}{i}\)
  • For recurring major costs, convert all to equivalent annual cost (\(A\)), then \(P=A/i\).

IV. Multiple Alternatives

Apply the same PW criteria across all options to identify the best alternative.

V. Spreadsheets in Present Worth Analysis

Spreadsheets simplify PW calculations, especially for irregular cash flows. Use Excelโ€™s NPV and PV functions, and always handle Time 0 cash flows correctly.

A. Direct NPW Calculation

  • Use compound interest factors or Excel PV/NPV functions

B. Year-by-Year Cash Flow Table

  • List cash flows, apply PV factors, sum for NPW

C. Excel NPV Function

  • Add Year 0 cash flow outside NPV function

D. Unequal Lives Comparison (LCM)

  • Calculate NPW for each cycle, discount future cycles, sum for total NPW

E. Sensitivity Analysis

  • Vary MARR to see impact on NPW and best alternative

F. Key Excel Tips

  1. Add Year 0 cash flow outside NPV
  2. Use named ranges for clarity
  3. Use Data Table for sensitivity
  4. Use Goal Seek for break-even MARR
  5. Create NPW vs. MARR charts
  6. Use absolute references for MARR

G. Summary: Excel Functions for PW

Function Purpose Example
NPV(rate, values) PV of future cash flows =NPV(12%, B2:B10)
PV(rate, nper, pmt, [fv]) PV of annuity =-PV(12%, 6, -15000)
(1+i)^n Discount factor =(1+0.12)^6
SUM(range) Total present values =SUM(D2:D10)

VI. Decision Summary

Alternative NPW (Own Life) NPW (LCM) Decision
A -$151,539 -$267,491 2nd
B -$192,465 -$229,851 3rd
C -$146,405 -$207,283 BEST โœ“

Criterion: For fixed output, minimize PW of costs. Recommendation: Select Alternative C โ€” lowest total cost on present worth basis.