PE Exam Exam > PE Exam Notes > Engineering Fundamentals Revision for PE > Cheatsheet: Present Worth and Future Worth
Cheatsheet: Present Worth and Future Worth
1. Time Value of Money Fundamentals
1.1 Core Concepts
| Term | Definition |
|---|---|
| Present Worth (PW) | Current value of future cash flows discounted at the interest rate |
| Future Worth (FW) | Value of current or series of cash flows at a specified future time |
| Interest Rate (i) | Rate of return or discount rate, expressed as decimal or percentage per period |
| Number of Periods (n) | Time duration over which compounding or discounting occurs |
| Cash Flow Diagram | Visual representation showing timing and direction of cash flows |
1.2 Sign Conventions
- Cash outflows (costs, investments): Negative values
- Cash inflows (revenues, receipts): Positive values
- Time zero: Present time (now)
- End-of-period convention: Cash flows occur at period end unless stated otherwise
2. Single Payment Formulas
2.1 Present Worth of Single Payment
| Formula | Description |
|---|---|
| P = F(1 + i)-n | Present worth given future amount |
| P = F(P/F, i, n) | Factor notation form, where (P/F, i, n) is the single payment present worth factor |
| (P/F, i, n) = 1/(1 + i)n | Single payment present worth factor |
- P = Present worth ($)
- F = Future amount ($)
- i = Interest rate per period (decimal)
- n = Number of compounding periods
2.2 Future Worth of Single Payment
| Formula | Description |
|---|---|
| F = P(1 + i)n | Future worth given present amount |
| F = P(F/P, i, n) | Factor notation form, where (F/P, i, n) is the single payment compound amount factor |
| (F/P, i, n) = (1 + i)n | Single payment compound amount factor |
- (F/P, i, n) and (P/F, i, n) are reciprocals: (F/P, i, n) = 1/(P/F, i, n)
3. Uniform Series Formulas
3.1 Present Worth of Uniform Series
| Formula | Description |
|---|---|
| P = A[(1 + i)n - 1]/[i(1 + i)n] | Present worth given uniform series |
| P = A(P/A, i, n) | Factor notation form |
| (P/A, i, n) = [(1 + i)n - 1]/[i(1 + i)n] | Uniform series present worth factor |
- A = Uniform amount per period ($)
- First payment occurs at end of period 1
- Last payment occurs at end of period n
- P is located one period before first A
3.2 Future Worth of Uniform Series
| Formula | Description |
|---|---|
| F = A[(1 + i)n - 1]/i | Future worth given uniform series |
| F = A(F/A, i, n) | Factor notation form |
| (F/A, i, n) = [(1 + i)n - 1]/i | Uniform series compound amount factor |
- F is located at same time as last A (end of period n)
3.3 Uniform Series from Present Worth
| Formula | Description |
|---|---|
| A = P[i(1 + i)n]/[(1 + i)n - 1] | Uniform series given present worth |
| A = P(A/P, i, n) | Factor notation form (capital recovery factor) |
| (A/P, i, n) = [i(1 + i)n]/[(1 + i)n - 1] | Capital recovery factor |
- (A/P, i, n) and (P/A, i, n) are reciprocals
3.4 Uniform Series from Future Worth
| Formula | Description |
|---|---|
| A = F[i]/[(1 + i)n - 1] | Uniform series given future worth |
| A = F(A/F, i, n) | Factor notation form (sinking fund factor) |
| (A/F, i, n) = i/[(1 + i)n - 1] | Sinking fund factor |
- (A/F, i, n) and (F/A, i, n) are reciprocals
4. Gradient Series Formulas
4.1 Arithmetic Gradient Present Worth
| Formula | Description |
|---|---|
| P = G[(1 + i)n - in - 1]/[i2(1 + i)n] | Present worth of arithmetic gradient only |
| P = G(P/G, i, n) | Factor notation form |
| (P/G, i, n) = [(1 + i)n - in - 1]/[i2(1 + i)n] | Arithmetic gradient present worth factor |
- G = Constant arithmetic change per period ($)
- Gradient starts at end of period 2 (zero at end of period 1)
- Cash flow at end of period t = (t - 1)G
- For complete series: Ptotal = A1(P/A, i, n) + G(P/G, i, n)
4.2 Arithmetic Gradient Uniform Series
| Formula | Description |
|---|---|
| A = G[(1/i) - n/((1 + i)n - 1)] | Equivalent uniform series of arithmetic gradient |
| A = G(A/G, i, n) | Factor notation form |
| (A/G, i, n) = (1/i) - n/[(1 + i)n - 1] | Arithmetic gradient uniform series factor |
4.3 Geometric Gradient
| Formula | Description |
|---|---|
| P = A1[1 - (1 + g)n(1 + i)-n]/(i - g) | Present worth of geometric gradient (i ≠ g) |
| P = A1[n/(1 + i)] | Present worth of geometric gradient (i = g) |
- g = Constant rate of change per period (decimal)
- A1 = First payment at end of period 1 ($)
- At = A1(1 + g)t-1 for period t
- Use special formula when i = g
5. Factor Relationships and Identities
5.1 Reciprocal Relationships
| Relationship | Expression |
|---|---|
| (F/P, i, n) = 1/(P/F, i, n) | Single payment factors |
| (A/P, i, n) = 1/(P/A, i, n) | Uniform series present worth factors |
| (A/F, i, n) = 1/(F/A, i, n) | Uniform series future worth factors |
5.2 Compound Relationships
| Relationship | Expression |
|---|---|
| (P/A, i, n) = (P/F, i, n)(F/A, i, n) | Present worth via future worth |
| (A/F, i, n) = (A/P, i, n) - i | Sinking fund via capital recovery |
| (A/P, i, n) = (A/F, i, n) + i | Capital recovery via sinking fund |
5.3 Special Cases
- When n = 1: (F/P, i, 1) = 1 + i; (P/F, i, 1) = 1/(1 + i)
- When n = 1: (F/A, i, 1) = 1; (A/F, i, 1) = 1
- When i = 0: (P/A, 0, n) = n; (F/A, 0, n) = n
- When n → ∞: (P/A, i, ∞) = 1/i (capitalized cost)
6. Present Worth Analysis Methods
6.1 Net Present Worth (NPW)
| Concept | Description |
|---|---|
| Definition | Sum of present worth of all cash inflows minus present worth of all cash outflows |
| NPW = PWbenefits - PWcosts | Basic formula |
| Decision Rule | Accept if NPW ≥ 0; Reject if NPW <> |
| Multiple Alternatives | Select alternative with highest NPW (if NPW ≥ 0) |
6.2 Present Worth of Perpetual Series
| Type | Formula |
|---|---|
| Uniform Perpetuity | P = A/i |
| Capitalized Cost | P = Initial cost + A/i (for infinite life projects) |
6.3 Present Worth Comparison Requirements
- Compare alternatives over equal time periods (common study period)
- Use least common multiple of lives if alternatives have different durations
- Or use repeatability assumption: replace at end of life with identical alternative
- All alternatives must use same interest rate (MARR)
7. Future Worth Analysis Methods
7.1 Net Future Worth (NFW)
| Concept | Description |
|---|---|
| Definition | Sum of future worth of all cash inflows minus future worth of all cash outflows |
| NFW = FWbenefits - FWcosts | Basic formula |
| Relationship to NPW | NFW = NPW(F/P, i, n) |
| Decision Rule | Accept if NFW ≥ 0; Reject if NFW <> |
7.2 Future Worth Applications
- Analysis when decision point is in the future
- Retirement planning and savings goals
- Bond maturity values
- NFW and NPW yield identical decisions (same ranking of alternatives)
8. Compounding Periods
8.1 Nominal vs Effective Interest Rates
| Term | Definition |
|---|---|
| Nominal Rate (r) | Annual interest rate without compounding adjustment; APR |
| Effective Annual Rate (ia) | Actual annual rate accounting for compounding frequency |
| Period Rate (i) | Interest rate per compounding period |
| Compounding Frequency (m) | Number of compounding periods per year |
8.2 Interest Rate Conversion Formulas
| Formula | Description |
|---|---|
| i = r/m | Period rate from nominal rate |
| ia = (1 + r/m)m - 1 | Effective annual rate from nominal rate |
| ia = (1 + i)m - 1 | Effective annual rate from period rate |
| ia = er - 1 | Effective annual rate for continuous compounding |
8.3 Common Compounding Frequencies
| Frequency | m Value |
|---|---|
| Annual | m = 1 |
| Semiannual | m = 2 |
| Quarterly | m = 4 |
| Monthly | m = 12 |
| Weekly | m = 52 |
| Daily | m = 365 |
| Continuous | m → ∞ |
9. Payment and Compounding Period Alignment
9.1 Matching Periods
| Situation | Solution |
|---|---|
| Payments match compounding | Use i = r/m and apply standard formulas directly |
| Payments do not match compounding | Convert to effective interest rate for payment period |
9.2 Effective Interest Rate for Payment Period
| Formula | Application |
|---|---|
| ipp = (1 + r/m)c - 1 | c = number of compounding periods per payment period |
- Example: Monthly payments with quarterly compounding (m = 4, c = 4/12)
- Use ipp in standard formulas with n = number of payments
10. Shifted Uniform Series
10.1 Deferred Annuities
| Concept | Approach |
|---|---|
| Series starts after period 1 | Calculate P at one period before first payment, then discount to time 0 |
| Deferral of k periods | P0 = A(P/A, i, n)(P/F, i, k) |
- k = number of periods before first payment
- n = number of payments in series
- Standard (P/A) gives value one period before first A
10.2 Series Ending Before Period n
- Calculate F at end of last payment, then compound to final time
- Or calculate P at one period before first payment, then discount to time 0
11. Problem-Solving Strategy
11.1 Systematic Approach
- Draw cash flow diagram with timeline
- Identify all cash flows with amounts, timing, and direction
- Determine interest rate (i) and number of periods (n)
- Select appropriate formulas or factors
- Verify payment and compounding period alignment
- Calculate present worth or future worth
- Check sign conventions (costs negative, revenues positive)
11.2 Common Mistakes to Avoid
- Using wrong n (count periods carefully, not years if compounding is non-annual)
- Mixing nominal and effective rates
- Incorrect gradient starting point (gradient begins at period 2, not period 1)
- Wrong sign on cash flows
- Forgetting to locate P one period before first A in uniform series
- Comparing alternatives over unequal time periods
- Using annual rate when compounding is more frequent
11.3 Quick Checks
- Present worth < future="" worth="" when="" i=""> 0
- Higher interest rate → lower present worth for future receipts
- Higher interest rate → higher future worth for present investment
- NPW and NFW must agree on accept/reject decision
- Factors are always positive
12. Key Factor Table Values (Reference)
12.1 Factor Behavior Patterns
| Factor | As n increases |
|---|---|
| (F/P, i, n) | Increases exponentially |
| (P/F, i, n) | Decreases toward zero |
| (F/A, i, n) | Increases |
| (P/A, i, n) | Increases toward 1/i |
| (A/P, i, n) | Decreases toward i |
| (A/F, i, n) | Decreases toward zero |
12.2 Interpolation Between Table Values
- Linear interpolation: Factor = Factor1 + (x - x1)/(x2 - x1) × (Factor2 - Factor1)
- x = interest rate or number of periods being interpolated
- Use when exact value not in standard factor tables
The document Cheatsheet: Present Worth and Future Worth is a part of the PE Exam Course Engineering Fundamentals Revision for PE.
All you need of PE Exam at this link: PE Exam
About this Document
Apr 20, 2026
Last updated
Related Exams
Document Description: Cheatsheet: Present Worth and Future Worth for PE Exam 2026 is part of Engineering Fundamentals Revision for PE preparation.
The notes and questions for Cheatsheet: Present Worth and Future Worth have been prepared according to the PE Exam exam syllabus. Information about Cheatsheet: Present Worth and Future Worth covers topics
like and Cheatsheet: Present Worth and Future Worth Example, for PE Exam 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Cheatsheet: Present Worth and Future Worth.
Introduction of Cheatsheet: Present Worth and Future Worth in English is available as part of our Engineering Fundamentals Revision for PE
for PE Exam & Cheatsheet: Present Worth and Future Worth in Hindi for Engineering Fundamentals Revision for PE course.
Download more important topics related with notes, lectures and mock test series for PE Exam
Exam by signing up for free. PE Exam: Cheatsheet: Present Worth and Future Worth
Description
Cheatsheet: Present Worth & Future Worth of Engineering Fundamentals Revision to help you remember important concepts with short tricks. Start learning for PE Exam exam & improve retention with EduRev.
Information about Cheatsheet: Present Worth and Future Worth
In this doc you can find the meaning of Cheatsheet: Present Worth and Future Worth defined & explained in the simplest way possible. Besides explaining types of
Cheatsheet: Present Worth and Future Worth theory, EduRev gives you an ample number of questions to practice Cheatsheet: Present Worth and Future Worth tests, examples and also practice PE Exam
tests
Related Searches
Cheatsheet: Present Worth and Future Worth, Previous Year Questions with Solutions, shortcuts and tricks, pdf , Cheatsheet: Present Worth and Future Worth, study material, practice quizzes, video lectures, ppt, Summary, Extra Questions, MCQs, past year papers, Free, Important questions, Exam, Viva Questions, Semester Notes, Objective type Questions, Sample Paper, mock tests for examination, Cheatsheet: Present Worth and Future Worth;