Present Worth Analysis - Theory & Concepts

Present Worth Analysis

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=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.

Checklist

Independent Single Project: A single, stand-alone project is economically justified if its $PW \ge 0$ when discounted at the MARR. This indicates the project meets or exceeds the minimum required return.
Mutually Exclusive Alternatives: When choosing among several competing projects where only one can be selected, choose the alternative with the largest (most positive) Present Worth or the least negative Present Worth (if only analyzing costs).

Present Worth Analysis Simulator

Initial Cost ($): 10000

Annual Benefit ($): 3000

Salvage Value ($): 2000

Life (years): 5

MARR (%): 8

Total Present Worth (PW)

$3339.30

Project Justified (PW ≥ 0)

PW = -P + A(P/A, i, n) + S(P/F, i, n)

Cumulative Present Worth over Time

Loading chart...

Comparing Alternatives with Unequal Lives

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:

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. For example, comparing a 3-year asset and a 4-year asset requires evaluating both over a 12-year horizon.
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.

Key Takeaways

Standardization: PW effectively standardizes varying cash flows to time $t=0$ for direct economic comparison.
The MARR Hurdle: A project is justifiable only if its $PW \ge 0$ 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.

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.

Capitalized Cost Formula

The capitalized cost represents the total present sum needed to construct the asset and maintain it perpetually.

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

Where:

$P$ = Initial, first cost
$A$ = Annual maintenance or operating cost (recurring every year forever)
$i$ = Interest rate (MARR)

If there is an additional major recurring cost

F

(e.g., rebuilding) required every

k

years forever, first convert that future cost into an equivalent uniform annual amount (

A

) using the Sinking Fund factor, then divide by

i

CC_{total} = P + \frac{A_{annual}}{i} + \frac{F \times (A/F, i, k)}{i}

Which simplifies algebraically to:

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

Key Takeaways

Infinite Horizon: Capitalized cost ( $CC$ ) is the present worth of perpetual, never-ending service.
Primary Formula: $CC = P + \frac{A}{i}$ .
Application: Common in governmental infrastructure projects (bridges, highways, dams) intended for permanent use, or university endowments.

Valuation of 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.

Bond Terminology

Face Value or Par Value ( $Z$ ): The stated denomination of the bond, typically the amount paid at maturity.
Coupon Rate ( $r$ ): The stated, fixed nominal interest rate paid on the face value. It determines the periodic uniform payment $A = Z \times r$ .
Yield Rate or Market Rate ( $i$ ): The actual interest rate earned by the investor based on current market conditions. This is the rate used for discounting.
Redemption Price ( $C$ ): The amount paid to retire the bond at maturity. This is usually equal to the face value ( $Z$ ), unless stated otherwise (e.g., callable bonds at a premium).

The value of a bond 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 yield rate (

i

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

Interactive Cash Flow Diagram

Visualize inflows, outflows, and calculate Present Worth.

Net Present Value (NPV)$0.00

Interest Rate (MARR) %

10%

Cash Flows

Key Takeaways

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 ( $r$ ) is only used to calculate the cash payment amount. The yield rate ( $i$ ) is the discount rate used to find the present worth.

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