Present Worth Analysis

Learning Objectives

  • Understand the Present Worth criterion and the Minimum Attractive Rate of Return (MARR).
  • Compare mutually exclusive alternatives using the Least Common Multiple (LCM) method.
  • Calculate Capitalized Cost for perpetual investments.
  • Evaluate the present worth of bonds.

Present Worth (PW) Analysis is one of the most widely used methods for evaluating investment alternatives. It converts all future cash flows (inflows and outflows) into a single equivalent lump sum at the present time (t=0t=0), using the Minimum Attractive Rate of Return (MARR).

The Present Worth Criterion

MARR

The Minimum Attractive Rate of Return (MARR) is the lowest interest rate that a company or individual is willing to accept for an investment. It reflects the opportunity cost of capital, the cost of borrowing, and the inherent risk of the project. A project must yield at least the MARR to be considered economically viable.

Opportunity Cost and MARR

The MARR is rarely a static, universal number. It varies significantly between companies and even between projects within the same company. A high-risk research and development project might require a MARR of 25%25\%, while a low-risk equipment replacement might only require a MARR of 10%10\%.

The MARR is inherently tied to opportunity cost—if a company uses its limited capital to fund Project A, it loses the opportunity to fund Project B or simply leave the money in a high-yield investment account.

Present Worth Analysis Criteria

Interactive Simulation

Use the simulation below to explore how the present worth of cash flows varies with time and interest rate.

Present Worth Analysis Simulator

Total Present Worth (PW)

$3339.30
Project Justified (PW ≥ 0)
PW=−P+A(P/A,i,n)+S(P/F,i,n)PW = -P + A(P/A, i, n) + S(P/F, i, n)

Cumulative Present Worth over Time

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Comparing Alternatives with Unequal Lives

Equal Time Period Rule

A critical fundamental rule in PW analysis is that mutually exclusive alternatives must be compared over the same time period (the analysis period or planning horizon).

Approaches to Equalize Lives

  1. Least Common Multiple (LCM) Method: Assume the services provided by the alternatives are needed for the LCM of their respective lives. This approach explicitly assumes that replacement assets will have exactly the same costs and performance profiles as the original assets.
    • Example: Comparing a Machine X (3-year life) and a Machine Y (4-year life) requires evaluating both over a 12-year horizon. Machine X will be purchased 12/3=412/3 = 4 times during this period, and Machine Y will be purchased 12/4=312/4 = 3 times.
  2. Study Period Method: Define a fixed study period (e.g., 5 years) regardless of the assets' lives. You must then estimate a salvage value or unrecovered investment value for any asset that outlives the study period at the end of that timeframe.

Capitalized Cost

Capitalized Cost

Capitalized Cost (CC) is the present worth of a project or asset that is assumed to have an infinite life. This is often used for evaluating public works projects (dams, bridges, pipelines, endowments) intended to last indefinitely.

Public Works and Permanent Endowments

The capitalized cost represents the total present sum needed to construct the asset and maintain it perpetually. For example, if a university wants to set up a permanent scholarship fund that pays out \10,000annuallyforever,andtheendowmentcanearnasteadyannually forever, and the endowment can earn a steady5%interest,theCapitalizedCost(theinitialprincipalrequired)isinterest, the Capitalized Cost (the initial principal required) is$10,000 / 0.05 = $200,000$. The principal is never touched; only the generated interest is spent.

Capitalized Cost Formula

Calculates the total present sum needed for perpetual construction and maintenance.

CC=P+AiCC = P + \frac{A}{i}

Variables

SymbolDescriptionUnit
CCCCCapitalized Cost-
PPInitial, first cost-
AAAnnual maintenance or operating cost (recurring every year forever)-
iiInterest rate (MARR)-

Major Recurring Costs

If there is an additional major recurring cost FF (e.g., rebuilding a bridge deck) required every kk years forever, first convert that future cost into an equivalent uniform annual amount (AA) using the Sinking Fund factor, then divide by ii:

Capitalized Cost with Major Recurring Cost

Calculates Capitalized Cost when there is a major recurring cost.

CCtotal=P+Ai+F(1+i)k−1CC_{total} = P + \frac{A}{i} + \frac{F}{(1+i)^k - 1}

Variables

SymbolDescriptionUnit
CCtotalCC_{total}Total Capitalized Cost-
PPInitial, first cost-
AAAnnual maintenance or operating cost (recurring every year forever)-
FFMajor recurring future cost-
iiInterest rate (MARR)-
kkNumber of years between major recurring costs-

Valuation of Bonds

Corporate and Government Bonds

A bond is a long-term financial instrument where a borrower (corporation or government) agrees to pay the bondholder a series of fixed interest payments (coupons) and return the face value at a specific maturity date.

The value of a bond to an investor is simply the present worth of its future cash flows (the series of coupon payments plus the final redemption price), discounted at the investor's required yield rate (ii).

Bond Valuation Formula

Calculates the present worth of a bond's future cash flows.

Vn=C(P/F,i,n)+(Z×r)(P/A,i,n)V_n = C(P/F, i, n) + (Z \times r)(P/A, i, n)

Variables

SymbolDescriptionUnit
VnV_nPresent value of the bond-
CCRedemption Price, the amount paid to retire the bond at maturity-
ZZFace Value or Par Value of the bond-
rrCoupon Rate, the stated nominal interest rate paid on the face value-
iiYield Rate or Market Rate used for discounting-
nnNumber of periods until maturity-

Interactive Simulation

Below is an interactive cash flow diagram tool to help visualize cash flows over time.

Interactive Cash Flow Diagram

Visualize inflows, outflows, and calculate Present Worth.

Net Present Value (NPV)$0.00
0$10001$3002$3003$3004$3005$500
10%
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Key Takeaways
  • Standardization: PW effectively standardizes varying cash flows to time t=0t=0 for direct economic comparison.
  • The MARR Hurdle: A project is justifiable only if its PW≥0PW \ge 0 when evaluated at the company's Minimum Attractive Rate of Return.
  • Equal Lifespans Required: When comparing mutually exclusive alternatives, they must absolutely be evaluated over the same time horizon, typically using the Least Common Multiple (LCM) of their lives.
  • Infinite Horizon: Capitalized cost (CCCC) is the present worth of perpetual, never-ending service. Common in governmental infrastructure projects intended for permanent use, or university endowments.
  • Primary Formula: CC=P+AiCC = P + \frac{A}{i}.
  • Bond Valuation Principle: A bond's current value is the Present Worth of all future coupon payments (an annuity) plus the Present Worth of the final face value maturity payment.
  • Rate Distinction: The coupon rate (rr) is only used to calculate the cash payment amount. The yield rate (ii) is the discount rate used to find the present worth.
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