Present Worth Analysis

Present Worth Analysis

$Learning Objectives$

Compute net present worth from arbitrary discrete cash flows at a stated MARR.
Apply correct decision rules for independent and mutually exclusive alternatives.
Compare cost-only alternatives using present cost on a common service basis.
Include salvage, periodic costs, and recurring costs with correct timing and signs.
Value a conventional fixed-coupon bond by discounting coupons and maturity value at the market yield per coupon period.
Recognize when unequal lives require a fixed study period, repeatability assumption, or another equivalent-worth basis.

$Present Worth$

Present worth is the equivalent value at time zero of all cash flows in a study, discounted or compounded to the present at a stated interest rate.

Net Present Worth of Discrete Cash Flows

Sums all signed cash flows at time zero.

PW=\sum_{t=0}^{n}\frac{CF_t}{(1+i)^t}

Variables

Symbol	Description	Unit
$PW$	Net present worth at time zero	-
$CF_t$	Net cash flow at period t	-
$i$	MARR or effective discount rate per period	-
$n$	Study horizon in periods	-

$Independent Alternative$

An independent project can be accepted or rejected without requiring another project to be rejected solely because of mutual exclusivity.

$Independent-Project Present-Worth Rule$

Accept an economically independent project if $PW\ge0$ at the stated MARR, subject to technical and non-economic constraints.
Reject it on economic grounds if $PW<0$ .

$Mutually Exclusive Alternatives$

Mutually exclusive alternatives satisfy the same decision need such that selecting one precludes selecting the others for that need.

$Mutually Exclusive Present-Worth Rule$

Establish a common service requirement, study basis, and MARR.
Compute net PW for each feasible alternative.
Select the alternative with the largest PW when benefits and costs are represented with consistent signs.
For cost-only alternatives that provide the same required service, select the alternative with the smallest equivalent present cost.

$Interactive Alternative Comparison$

Use the comparator to change first cost, annual net cash flow, salvage, study life, and MARR for two alternatives on one consistent basis.

MARR10.00%

Common study life8 years

Alternative A

First costPHPAnnual net cash flowPHP/yearSalvagePHP

Alternative B

First costPHPAnnual net cash flowPHP/yearSalvagePHP

PW — Alternative A

₱1,741,625

Meets the MARR screen.

PW — Alternative B

₱1,968,658

Meets the MARR screen.

Preferred on this common basis

Alternative B

PW(B) − PW(A) = ₱227,033.

For unequal lives, do not force a common-life comparison without stating the repeatability assumption. Annual worth or a fixed study period is often clearer.

MARR per period10.00%

Cash flows

PeriodSigned amountPHP

This teaching diagram limits the editable horizon to 50 periods so very large accidental inputs cannot create an unbounded SVG/tick workload.

Net present worth

₱185,605

Net PW is nonnegative at the selected MARR.

Positive arrows are inflows and negative arrows are outflows from the defined viewpoint. A zero cash flow is shown as a point on the axis, not as an arrow. If two entries share a period, the calculation combines them before discounting.

$Salvage and Terminal Value$

A positive salvage value received at the end of the study is an inflow and is discounted from its actual disposal date. A disposal cost is an outflow. Never insert salvage at time zero merely because it offsets first cost algebraically in another formula.

$Capitalized Cost$

Capitalized cost is the present equivalent of a perpetual or indefinitely repeated service requirement. It is useful for infrastructure or endowment-type analyses only when the perpetuity assumptions are appropriate.

Capitalized Cost of a Perpetual Uniform Annual Cost

Present equivalent of an initial cost plus a perpetual annual cost when i is positive.

CC=C_0+\frac{A}{i}

Variables

Symbol	Description	Unit
$CC$	Capitalized cost	-
$C_0$	Initial cost at time zero	-
$A$	Perpetual uniform annual cost	-
$i$	Positive effective annual rate	-

$Bond Face or Par Value$

Face or par value $F$ is the amount conventionally repaid at bond maturity and the amount on which the stated coupon rate is commonly applied.

$Coupon Rate and Coupon Payment$

The coupon rate is the stated rate used to determine contractual coupon payments. If an annual coupon rate $r_c$ is paid in $m$ equal coupon periods per year, the periodic coupon is commonly $C=Fr_c/m$ .

$Bond Yield Rate$

The yield rate used for valuation is the investor's required market return on the bond cash flows. The effective yield per coupon period must be used with the number of coupon periods.

Present Value of a Conventional Fixed-Coupon Bond

Discounts the level coupon series and face value at the market yield per coupon period.

P_b=C(P/A,i,n)+F(P/F,i,n)

Variables

Symbol	Description	Unit
$P_b$	Bond value at the valuation date immediately before the modeled coupon series begins	-
$C$	Coupon payment per coupon period	-
$F$	Face or redemption value paid at maturity	-
$i$	Effective market yield per coupon period	-
$n$	Remaining coupon periods	-

$Premium, Par, and Discount Bonds$

For a standard fixed-coupon bond with the same coupon and yield frequency, a coupon rate above the required yield generally produces a price above par; a coupon rate below the required yield generally produces a price below par; equality gives a par-value price. This is a present-worth result, not a separate valuation principle.

$Match Coupon and Yield Periods$

A semiannual coupon stream has semiannual timing. Convert the quoted yield to the effective rate per semiannual period under the stated compounding convention and use the total number of semiannual coupon periods. Do not insert an annual rate directly into a half-year cash-flow series.

$Study Period$

The study period is the time horizon over which alternatives are evaluated. It may be set by project need, planning policy, asset service requirement, or another explicit decision boundary.

$Unequal Lives Need an Explicit Basis$

Do not automatically compare the raw PWs of alternatives with different service lives when one alternative simply stops providing service earlier. Use a common study period with stated terminal assumptions, a defensible repeatability assumption, or an equivalent annual-worth method.

$Common Present-Worth Errors$

Typical errors include discounting the initial cash flow, omitting salvage timing, mixing cost-positive and benefit-positive sign conventions, using different study periods without justification, mismatching coupon/yield periods in bond valuation, and selecting the smallest numerical PW when net cash flows use the standard benefit-positive convention.

$Present-Worth Comparison Procedure$

STEP-BY-STEP

Define the common service requirement and viewpoint.
Draw each alternative's cash-flow diagram over a common study basis.
State the MARR and confirm its period matches the cash flows.
Discount every future cash flow to time zero.
Sum signed equivalents to obtain net PW or, for cost-only studies, present cost.
Apply the decision rule appropriate to independent or mutually exclusive alternatives.
Test important uncertain assumptions before recommending a choice.

Key Takeaways

Present worth converts all cash flows to time zero at a stated MARR.
Independent projects with $PW\ge0$ pass the economic screen.
Mutually exclusive alternatives must be compared on a common service and study basis.
With standard signed net cash flows, choose the largest PW; with equivalent cost-only values, choose the smallest present cost.
Salvage belongs at its actual terminal date before discounting.
A conventional fixed-coupon bond is valued by discounting coupons and maturity value at the market yield per coupon period.
Coupon frequency, yield period, and number of periods must be consistent.
Unequal-life comparisons require explicit assumptions rather than automatic least-common-multiple repetition.