π PMP Math & Formulas
Complete Interactive Study Guide with Interpretation Tables, Scenarios & Exam Tips β Score 100%
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π EVM Overview β The Big 3 Values
What is EVM? Earned Value Management integrates Scope + Schedule + Cost into a single performance picture. Traditional tracking tells you only how much you spent. EVM tells you what you got for what you spent.
| Value | Abbrev. | Old Name | Key Question It Answers | Formula |
|---|---|---|---|---|
| Planned ValuePV is the authorized budget for work that was SUPPOSED to be done by a certain date. Think of it as your "target odometer" for spending. Also called BCWS (Budgeted Cost of Work Scheduled). | PV | BCWS | "How much SHOULD we have spent?" | % Planned Γ BAC |
| Earned ValueEV is the dollar value of work ACTUALLY completed. If you planned $100K of work and finished 60% of it, your EV = $60K regardless of what you spent. Also called BCWP (Budgeted Cost of Work Performed). | EV | BCWP | "What is the work we DID worth?" | % Complete Γ BAC |
| Actual CostAC is what you literally spent β real invoices, real salaries, real expenses. It has nothing to do with how much work was done. Also called ACWP (Actual Cost of Work Performed). | AC | ACWP | "How much DID we actually spend?" | Given in problem |
π― Memory Rule: EV is always the FIRST number in every formula. CV = EV β AC. SV = EV β PV. CPI = EV Γ· AC. SPI = EV Γ· PV.
Starting Formula β Always Calculate EV First
Earned Value
EV = % Complete Γ BAC
BAC = Budget at Completion (total original project budget β always GIVEN)
Quick Example: BAC = $300,000 | Project is 45% complete β EV = 0.45 Γ $300,000 = $135,000
π Core EVM Formulas
Planned Value (PV)
PV = % Planned Γ BAC
Example: If 30% of work was PLANNED to be done and BAC=$500K β PV = $150,000
Earned Value (EV)
EV = % Complete Γ BAC
Example: Only 20% is ACTUALLY done β EV = $100,000
Budget at Completion (BAC)
BAC = Total authorized project budget (given)
You NEVER calculate BAC β it is always stated in the problem. It is the original plan total.
π― Exam Tip: If BAC is not given but total work breakdown is, sum all activity budgets: BAC = sum of all activity planned costs.
π Variances β CV & SV
Cost Variance (CV)
CV = EV β AC
Are we getting value for money? Positive = GOOD (under budget). Negative = BAD (over budget).
Schedule Variance (SV)
SV = EV β PV
Are we ahead or behind on work? Positive = GOOD (ahead). Negative = BAD (behind schedule).
β CV & SV Interpretation Table
| Metric | Value | Meaning | Real-World Translation | Action |
|---|---|---|---|---|
| CV (Cost Variance) | Positive (+) | UNDER Budget β | Getting more value than spent β efficient spending | Maintain pace |
| CV | = Zero (0) | Exactly On Budget | Spending exactly as planned | Continue |
| CV | Negative (β) | OVER Budget β | Spent more than work is worth β overspending | Corrective action |
| SV (Schedule Variance) | Positive (+) | AHEAD of Schedule β | More work done than planned β working faster | Maintain pace |
| SV | = Zero (0) | Exactly On Schedule | Completing work exactly as planned | Continue |
| SV | Negative (β) | BEHIND Schedule β | Less work done than planned β falling behind | Schedule recovery |
π Scenario 1 β Highway Project (Good Performance):
BAC=$2M | PV=$800K (40% planned) | EV=$900K (45% complete) | AC=$820K
CV = $900K β $820K = +$80,000 UNDER BUDGET
SV = $900K β $800K = +$100,000 AHEAD OF SCHEDULE
Interpretation: Team completed more work than planned AND spent less than earned. Excellent performance.
BAC=$2M | PV=$800K (40% planned) | EV=$900K (45% complete) | AC=$820K
CV = $900K β $820K = +$80,000 UNDER BUDGET
SV = $900K β $800K = +$100,000 AHEAD OF SCHEDULE
Interpretation: Team completed more work than planned AND spent less than earned. Excellent performance.
π Scenario 2 β IT Implementation (Struggling):
BAC=$500K | PV=$200K | EV=$150K | AC=$230K
CV = $150K β $230K = β$80,000 OVER BUDGET
SV = $150K β $200K = β$50,000 BEHIND SCHEDULE
Interpretation: Both cost and schedule are in trouble β double problem. Needs immediate escalation.
BAC=$500K | PV=$200K | EV=$150K | AC=$230K
CV = $150K β $230K = β$80,000 OVER BUDGET
SV = $150K β $200K = β$50,000 BEHIND SCHEDULE
Interpretation: Both cost and schedule are in trouble β double problem. Needs immediate escalation.
π Scenario 3 β Bridge Inspection (Mixed Results):
BAC=$1.2M | PV=$400K | EV=$450K | AC=$490K
CV = $450K β $490K = β$40,000 OVER BUDGET
SV = $450K β $400K = +$50,000 AHEAD OF SCHEDULE
Interpretation: Team is working fast (ahead of schedule) but overspending to do it β perhaps overtime costs. Schedule is good but burning through budget too quickly.
BAC=$1.2M | PV=$400K | EV=$450K | AC=$490K
CV = $450K β $490K = β$40,000 OVER BUDGET
SV = $450K β $400K = +$50,000 AHEAD OF SCHEDULE
Interpretation: Team is working fast (ahead of schedule) but overspending to do it β perhaps overtime costs. Schedule is good but burning through budget too quickly.
π Scenario 4 β Exam Trap! Behind schedule but UNDER budget:
BAC=$300K | PV=$120K | EV=$90K | AC=$80K
CV = $90K β $80K = +$10,000 UNDER BUDGET
SV = $90K β $120K = β$30,000 BEHIND SCHEDULE
Trap: "We're under budget β great!" BUT the team is behind schedule because they simply haven't done enough work yet. Being behind means less work = less spending. Not efficient β just slow!
BAC=$300K | PV=$120K | EV=$90K | AC=$80K
CV = $90K β $80K = +$10,000 UNDER BUDGET
SV = $90K β $120K = β$30,000 BEHIND SCHEDULE
Trap: "We're under budget β great!" BUT the team is behind schedule because they simply haven't done enough work yet. Being behind means less work = less spending. Not efficient β just slow!
π Performance Indexes β CPI & SPI
Cost Performance Index (CPI)
CPI = EV Γ· AC
"For every $1 I spend, how much value do I get?"
Schedule Performance Index (SPI)
SPI = EV Γ· PV
"For every planned $1 of work, how much did I actually complete?"
β CPI & SPI Interpretation Table
| Metric | Value | Status | What It Means | Example Reading |
|---|---|---|---|---|
| CPI | > 1.0 | UNDER Budget β | Getting MORE value than spent | CPI=1.20: $1.20 value per $1 spent |
| CPI | = 1.0 | On Budget | Exactly as planned | CPI=1.0: $1.00 value per $1 spent |
| CPI | < 1.0 | OVER Budget β | Getting LESS value than spent | CPI=0.80: only $0.80 value per $1 spent |
| SPI | > 1.0 | AHEAD of Schedule β | More work done than planned | SPI=1.15: working 15% faster than planned |
| SPI | = 1.0 | On Schedule | Exactly on plan | SPI=1.0: exactly on schedule |
| SPI | < 1.0 | BEHIND Schedule β | Less work done than planned | SPI=0.75: only 75% efficiency, 25% behind |
Combined CPI + SPI Quick-Read Matrix
| CPI | SPI | Overall Status | Typical Cause |
|---|---|---|---|
| >1 | >1 | BEST β Under budget & Ahead of schedule | Efficient team, good estimates, favorable conditions |
| >1 | <1 | Under budget BUT Behind β Cautiously watch | Work not started yet, slow team (less spending because less done) |
| <1 | >1 | Over budget BUT Ahead β Costly acceleration | Overtime, extra resources used to speed up; burning cash fast |
| <1 | <1 | WORST β Over budget & Behind schedule | Underestimated complexity, poor execution, scope creep |
π Scenario 5 β Construction Project, Month 6:
BAC=$4M | PV=$1.6M | EV=$1.4M | AC=$1.8M
CPI = 1.4MΓ·1.8M = 0.778 OVER BUDGET ($0.78 value per $1 spent)
SPI = 1.4MΓ·1.6M = 0.875 BEHIND SCHEDULE (87.5% efficiency)
Double trouble: spending $1.28 to get $1 of work done AND only 87.5% as fast as planned.
BAC=$4M | PV=$1.6M | EV=$1.4M | AC=$1.8M
CPI = 1.4MΓ·1.8M = 0.778 OVER BUDGET ($0.78 value per $1 spent)
SPI = 1.4MΓ·1.6M = 0.875 BEHIND SCHEDULE (87.5% efficiency)
Double trouble: spending $1.28 to get $1 of work done AND only 87.5% as fast as planned.
π Scenario 6 β Software Sprint:
BAC=$200K | PV=$60K | EV=$75K | AC=$65K
CPI = 75Γ·65 = 1.154 UNDER BUDGET ($1.15 value per $1)
SPI = 75Γ·60 = 1.25 AHEAD OF SCHEDULE (25% faster)
Best case: team is delivering efficiently and faster than planned.
BAC=$200K | PV=$60K | EV=$75K | AC=$65K
CPI = 75Γ·65 = 1.154 UNDER BUDGET ($1.15 value per $1)
SPI = 75Γ·60 = 1.25 AHEAD OF SCHEDULE (25% faster)
Best case: team is delivering efficiently and faster than planned.
β Master EVM Interpretation Table β All Metrics at a Glance
| Metric | Formula | Value > 0 or > 1 | Value = 0 or = 1 | Value < 0 or < 1 |
|---|---|---|---|---|
| CV (Cost Variance) | EV β AC | UNDER budget β Value earned > money spent |
Exactly on budget | OVER budget β Spent more than value earned |
| SV (Schedule Variance) | EV β PV | AHEAD of schedule β More work done than planned |
Exactly on schedule | BEHIND schedule β Less work done than planned |
| CPI (Cost Index) | EV Γ· AC | >1: Under budget β e.g. 1.2 = $1.20 value per $1 |
1.0 = exactly on budget | <1: Over budget β e.g. 0.8 = $0.80 value per $1 |
| SPI (Schedule Index) | EV Γ· PV | >1: Ahead of schedule β e.g. 1.1 = 110% efficiency |
1.0 = exactly on schedule | <1: Behind schedule β e.g. 0.9 = 90% efficiency, 10% behind |
| EAC (Estimate at Completion) | BAC Γ· CPI | EAC < BAC: Finishing under budget β | EAC = BAC: Finishing exactly on budget | EAC > BAC: Finishing over budget β |
| VAC (Variance at Completion) | BAC β EAC | Positive: Expected under-run β Will finish UNDER budget |
Zero: Exactly on final budget | Negative: Expected over-run β Will finish OVER budget |
| TCPI (To-Complete Index) | (BACβEV)Γ·(BACβAC) | <1: Easier than current CPI β Can afford to be less efficient |
1.0: Must maintain current CPI exactly | >1: Harder than current CPI β Must improve efficiency to meet budget |
| ETC (Estimate to Complete) | EAC β AC | Always positive ($ still needed). Smaller = closer to done. Compare to remaining budget. | ||
π― TCPI Decision Rule: If TCPI > 1.2, the original budget goal is likely unachievable β recommend revising the budget (new EAC). If TCPI β€ current CPI, the goal is easy to achieve.
π Forecasting: EAC, ETC, VAC, TCPI
EAC β Formula 1 (MOST COMMON on exam)
EAC = BAC Γ· CPI
Use when: "Assume current performance continues for the rest of the project"
EAC β Formula 2 (Atypical variance)
EAC = AC + (BAC β EV)
Use when: "The past variance was unusual β remaining work will be done at original budget rate"
EAC β Formula 3 (Re-estimate)
EAC = AC + ETC
Use when: "The team has re-estimated the remaining work from scratch"
EAC β Formula 4 (Both CPI & SPI)
EAC = AC + [(BAC β EV) Γ· (CPI Γ SPI)]
Use when: "Both cost and schedule performance affect remaining work"
ETC (Estimate to Complete)
ETC = EAC β AC
How much MORE money is needed from today to finish
VAC (Variance at Completion)
VAC = BAC β EAC
Expected over/under-run at the end. Negative = over budget at completion.
TCPI β Based on BAC (original budget goal)
TCPI = (BAC β EV) Γ· (BAC β AC)
CPI efficiency needed on remaining work to hit original budget
TCPI β Based on EAC (revised budget goal)
TCPI = (BAC β EV) Γ· (EAC β AC)
CPI efficiency needed on remaining work to hit NEW revised budget
EAC Formula Decision Tree
| Exam Keyword / Phrase | Use This EAC Formula |
|---|---|
| "current performance will continue" / "trend continues" | EAC = BAC Γ· CPI |
| "variance was atypical" / "one-time event" / "remaining at budget rate" | EAC = AC + (BAC β EV) |
| "team re-estimated" / "new bottom-up estimate" | EAC = AC + ETC |
| "both cost and schedule affect future" / "schedule pressure increases cost" | EAC = AC + [(BACβEV) Γ· (CPIΓSPI)] |
π― TCPI Interpretation:
TCPI = 0.95 β Need to work at 95% efficiency on remaining work β Easier than current CPI if CPI=0.90
TCPI = 1.0 β Must maintain exact current performance β Neutral
TCPI = 1.15 β Need 15% better efficiency than current β Harder β likely unrealistic
TCPI = 1.30+ β Almost certainly impossible without scope reduction
TCPI = 0.95 β Need to work at 95% efficiency on remaining work β Easier than current CPI if CPI=0.90
TCPI = 1.0 β Must maintain exact current performance β Neutral
TCPI = 1.15 β Need 15% better efficiency than current β Harder β likely unrealistic
TCPI = 1.30+ β Almost certainly impossible without scope reduction
π EVM Full Scenarios β 8 Real Cases
Case 1 β Road Paving Project (Everything Fine)
BAC=$600K | PV=$180K | EV=$200K | AC=$190K
CV=+$10K UNDER BUDGET | SV=+$20K AHEAD | CPI=1.053 | SPI=1.111
EAC = $600KΓ·1.053 = $569,800 | VAC = $600Kβ$569,800 = +$30,200 savings
TCPI = (600β200)Γ·(600β190) = 400Γ·410 = 0.976 Easy to maintain
CV=+$10K UNDER BUDGET | SV=+$20K AHEAD | CPI=1.053 | SPI=1.111
EAC = $600KΓ·1.053 = $569,800 | VAC = $600Kβ$569,800 = +$30,200 savings
TCPI = (600β200)Γ·(600β190) = 400Γ·410 = 0.976 Easy to maintain
Case 2 β Data Center Build (Over Budget, Behind Schedule)
BAC=$1M | PV=$400K | EV=$320K | AC=$450K
CV=β$130K OVER BUDGET | SV=β$80K BEHIND | CPI=0.711 | SPI=0.80
EAC = $1MΓ·0.711 = $1,406,400 | VAC = β$406,400 MASSIVE overrun expected
TCPI = (1Mβ320K)Γ·(1Mβ450K) = 680Γ·550 = 1.236 23.6% improvement needed β unrealistic
Recommendation: Formally revise budget. Use TCPI based on EAC instead of BAC.
CV=β$130K OVER BUDGET | SV=β$80K BEHIND | CPI=0.711 | SPI=0.80
EAC = $1MΓ·0.711 = $1,406,400 | VAC = β$406,400 MASSIVE overrun expected
TCPI = (1Mβ320K)Γ·(1Mβ450K) = 680Γ·550 = 1.236 23.6% improvement needed β unrealistic
Recommendation: Formally revise budget. Use TCPI based on EAC instead of BAC.
Case 3 β Building Inspection (Behind Schedule but Under Budget)
BAC=$800K | PV=$300K | EV=$240K | AC=$210K
CV=+$30K UNDER BUDGET | SV=β$60K BEHIND | CPI=1.143 | SPI=0.80
EAC = $800KΓ·1.143 = $700K | VAC = +$100K expected savings
Note: Being "under budget" is misleading β it's because LESS WORK was done. The real problem is schedule.
CV=+$30K UNDER BUDGET | SV=β$60K BEHIND | CPI=1.143 | SPI=0.80
EAC = $800KΓ·1.143 = $700K | VAC = +$100K expected savings
Note: Being "under budget" is misleading β it's because LESS WORK was done. The real problem is schedule.
Case 4 β Ahead of Schedule but Over Budget (Fast but Costly)
BAC=$500K | PV=$150K | EV=$180K | AC=$210K
CV=β$30K OVER BUDGET | SV=+$30K AHEAD | CPI=0.857 | SPI=1.20
EAC = $500KΓ·0.857 = $583,430 | VAC = β$83,430 expected overrun
Typical: Team added overtime or extra resources to go fast β burning budget. Discuss with sponsor whether schedule gain justifies cost overrun.
CV=β$30K OVER BUDGET | SV=+$30K AHEAD | CPI=0.857 | SPI=1.20
EAC = $500KΓ·0.857 = $583,430 | VAC = β$83,430 expected overrun
Typical: Team added overtime or extra resources to go fast β burning budget. Discuss with sponsor whether schedule gain justifies cost overrun.
Case 5 β Using Atypical EAC Formula
BAC=$400K | AC=$120K | EV=$100K | PV=$110K
CPI = 100Γ·120 = 0.833 | EAC (normal) = $400KΓ·0.833 = $480K
Scenario: The overrun was caused by a one-time hurricane. Remaining work at budget rate.
EAC = AC + (BACβEV) = $120K + ($400Kβ$100K) = $120K + $300K = $420K
Much better outcome ($420K vs $480K) because we don't project the bad performance forward.
CPI = 100Γ·120 = 0.833 | EAC (normal) = $400KΓ·0.833 = $480K
Scenario: The overrun was caused by a one-time hurricane. Remaining work at budget rate.
EAC = AC + (BACβEV) = $120K + ($400Kβ$100K) = $120K + $300K = $420K
Much better outcome ($420K vs $480K) because we don't project the bad performance forward.
Case 6 β Using Both CPI & SPI Formula
BAC=$750K | AC=$250K | EV=$200K | PV=$230K
CPI=0.80 | SPI=0.87
Exam states: Schedule pressure causes more cost inefficiency on remaining work.
EAC = $250K + [(750β200)Γ·(0.80Γ0.87)] = $250K + [550Γ·0.696] = $250K + $790K = $1,040K
The most pessimistic forecast β both problems compound each other.
CPI=0.80 | SPI=0.87
Exam states: Schedule pressure causes more cost inefficiency on remaining work.
EAC = $250K + [(750β200)Γ·(0.80Γ0.87)] = $250K + [550Γ·0.696] = $250K + $790K = $1,040K
The most pessimistic forecast β both problems compound each other.
Case 7 β TCPI vs CPI Comparison
BAC=$300K | EV=$100K | AC=$130K | CPI=0.769
TCPI (to meet BAC) = (300β100)Γ·(300β130) = 200Γ·170 = 1.176
Current CPI = 0.769. Need TCPI = 1.176. That's 53% better than current β NOT achievable.
Sponsor approves new EAC = $360K.
TCPI (to meet EAC) = (300β100)Γ·(360β130) = 200Γ·230 = 0.870
Now TCPI=0.870 vs current CPI=0.769 β only need to improve 13% β Achievable!
TCPI (to meet BAC) = (300β100)Γ·(300β130) = 200Γ·170 = 1.176
Current CPI = 0.769. Need TCPI = 1.176. That's 53% better than current β NOT achievable.
Sponsor approves new EAC = $360K.
TCPI (to meet EAC) = (300β100)Γ·(360β130) = 200Γ·230 = 0.870
Now TCPI=0.870 vs current CPI=0.769 β only need to improve 13% β Achievable!
Case 8 β Bridge Deck Replacement (Ahmad's World)
3-year project. BAC=$3.6M. After Year 1:
Planned 35% done β PV=$1.26M | Actually 30% done β EV=$1.08M | Spent: AC=$1.30M
CV = $1.08Mβ$1.30M = β$220K OVER BUDGET
SV = $1.08Mβ$1.26M = β$180K BEHIND SCHEDULE
CPI = 1.08Γ·1.30 = 0.831 | SPI = 1.08Γ·1.26 = 0.857
EAC = $3.6MΓ·0.831 = $4.33M (expected $730K overrun)
TCPI = (3.6Mβ1.08M)Γ·(3.6Mβ1.30M) = 2.52Γ·2.30 = 1.096 β need 9.6% improvement
Recommend: Review subcontractor performance, consider renegotiating contract terms.
Planned 35% done β PV=$1.26M | Actually 30% done β EV=$1.08M | Spent: AC=$1.30M
CV = $1.08Mβ$1.30M = β$220K OVER BUDGET
SV = $1.08Mβ$1.26M = β$180K BEHIND SCHEDULE
CPI = 1.08Γ·1.30 = 0.831 | SPI = 1.08Γ·1.26 = 0.857
EAC = $3.6MΓ·0.831 = $4.33M (expected $730K overrun)
TCPI = (3.6Mβ1.08M)Γ·(3.6Mβ1.30M) = 2.52Γ·2.30 = 1.096 β need 9.6% improvement
Recommend: Review subcontractor performance, consider renegotiating contract terms.
ποΈ EVM Cheat Sheet
β‘ All EVM Formulas
- EV = % Complete Γ BAC
- CV = EV β AC β +Good/βBad | CPI = EVΓ·AC β >1 Good/<1 Bad
- SV = EV β PV β +Good/βBad | SPI = EVΓ·PV β >1 Good/<1 Bad
- EAC = BACΓ·CPI (default) | EAC = AC+(BACβEV) (atypical) | EAC = AC+ETC (re-est)
- ETC = EAC β AC | VAC = BAC β EAC β +Good/βBad
- TCPI = (BACβEV)Γ·(BACβAC) β <1 Easy/>1 Hard
π Critical Path Method (CPM)
CPM identifies the critical pathThe critical path is the LONGEST sequence of activities from project start to finish. It determines the minimum project duration. Any delay to a critical path activity delays the ENTIRE project by that same amount. β the longest path through the network.
Forward Pass β Calculate ES and EF
EF = ES + Duration
Start from the beginning. EF of predecessor = ES of successor (with Finish-to-Start dependency)
Backward Pass β Calculate LS and LF
LS = LF β Duration
Start from the end. LF of last activity = Project End Date. Work backward.
| Term | Abbrev | Definition | Calculated How |
|---|---|---|---|
| Early Start | ES | Earliest an activity CAN start | Forward pass: = EF of predecessor |
| Early Finish | EF | Earliest an activity CAN finish | ES + Duration |
| Late Start | LS | Latest it can start WITHOUT delaying project | LF β Duration |
| Late Finish | LF | Latest it can finish WITHOUT delaying project | Backward pass: = LS of successor |
| Total Float | TF | Delay allowed without delaying project end | LSβES or LFβEF |
| Free Float | FF | Delay allowed without delaying successor | ES(next) β EF(current) |
π Float Formulas
Total Float (TF)
TF = LS β ES OR TF = LF β EF
Both give same result. Critical path activities always have TF = 0.
Free Float (FF)
FF = ES of next activity β EF of current activity
Always β€ Total Float. Free Float cannot exceed Total Float.
β Float / Schedule Interpretation Table
| Value | Total Float = 0 | Total Float > 0 | Total Float < 0 |
|---|---|---|---|
| Critical Path? | YES β on critical path β | NO β has buffer β | YES β project already late β |
| If delayed? | Project end date moves | Project end date unaffected (within float) | Project end date moves further out |
| Action | Monitor closely β no slack | OK β can use float strategically | URGENT β immediate recovery needed |
| Free Float Interpretation | |||
| FF = 0 | Delaying this activity immediately delays the next activity's early start | ||
| FF > 0 | Activity can slip by FF days without affecting the next activity at all | ||
π CPM Scenarios
Scenario 1 β Find Critical Path:
Path 1: A(5)βB(8)βE(4) = 17 days
Path 2: A(5)βC(10)βE(4) = 19 days
Path 3: A(5)βD(6)βF(9) = 20 days β Critical Path (longest)
Float of Path 1 = 20β17 = 3 days | Float of Path 2 = 20β19 = 1 day
Path 1: A(5)βB(8)βE(4) = 17 days
Path 2: A(5)βC(10)βE(4) = 19 days
Path 3: A(5)βD(6)βF(9) = 20 days β Critical Path (longest)
Float of Path 1 = 20β17 = 3 days | Float of Path 2 = 20β19 = 1 day
Scenario 2 β Forward & Backward Pass:
Activity B: Duration=6 | ES=4 | EF=10 | LF=13 | LS=7
TF = LSβES = 7β4 = 3 days
Next activity C: ES=12
FF = ES(C)βEF(B) = 12β10 = 2 days
B can slip 3 days total, but only 2 days before it affects C's early start.
Activity B: Duration=6 | ES=4 | EF=10 | LF=13 | LS=7
TF = LSβES = 7β4 = 3 days
Next activity C: ES=12
FF = ES(C)βEF(B) = 12β10 = 2 days
B can slip 3 days total, but only 2 days before it affects C's early start.
π Exam Trap β Negative Float:
A project has a mandatory finish date of Day 30. The calculated critical path = 35 days.
Total Float = 30β35 = β5 days
This means the project is ALREADY 5 days behind before it even starts. Must crash or fast-track.
A project has a mandatory finish date of Day 30. The calculated critical path = 35 days.
Total Float = 30β35 = β5 days
This means the project is ALREADY 5 days behind before it even starts. Must crash or fast-track.
π Multiple Critical Paths:
If two paths both equal 20 days (the longest), BOTH are critical paths.
Any delay on either path delays the project. Crashing must address BOTH paths simultaneously.
If two paths both equal 20 days (the longest), BOTH are critical paths.
Any delay on either path delays the project. Crashing must address BOTH paths simultaneously.
π Analogous & Parametric Estimating
| Type | Formula | Accuracy | Speed | Cost | Best Used |
|---|---|---|---|---|---|
| Analogous | Based on similar past projects | β25% to +75% (ROM) | Fast | Low | Early phases, limited info |
| Parametric | Unit Rate Γ Quantity | β10% to +25% | Moderate | Moderate | When reliable unit rates exist |
| Three-Point (PERT) | (O+4M+P)Γ·6 | β5% to +10% | Moderate | Moderate | Uncertain activities |
| Bottom-Up | Sum of all WP estimates | β5% to +10% | Slow | High | Detailed planning phase |
Parametric Estimate
Cost = Unit Rate Γ Quantity
Example: $65/sq ft Γ 12,000 sq ft = $780,000
π Parametric Scenarios:
Bridge deck: $130/sq ft Γ 9,200 sq ft = $1,196,000
Concrete curb: $48/LF Γ 2,400 LF = $115,200
Structural steel: $3.20/lb Γ 180,000 lb = $576,000
Bridge deck: $130/sq ft Γ 9,200 sq ft = $1,196,000
Concrete curb: $48/LF Γ 2,400 LF = $115,200
Structural steel: $3.20/lb Γ 180,000 lb = $576,000
π Three-Point Estimates β PERT
PERT Expected Value (Beta Distribution)
tE = (O + 4M + P) Γ· 6
O=Optimistic | M=Most Likely | P=Pessimistic | M weighted Γ4 because it is most realistic
Triangular Average (simpler version)
tE = (O + M + P) Γ· 3
Use ONLY when exam specifies "triangular distribution" β no weight on Most Likely
Standard Deviation (Ο) per Activity
Ο = (P β O) Γ· 6
Measures uncertainty. Larger range (PβO) = more uncertainty = larger Ο
Variance (ΟΒ²) per Activity
ΟΒ² = [(P β O) Γ· 6]Β²
Variances ADD together across a path. NEVER add Ο values directly β always add ΟΒ².
Path Standard Deviation
Ο_path = β(sum of all variances on path)
Take square root of SUM of variances (not sum of standard deviations)
β PERT / Sigma Confidence Interpretation Table
| Range | Confidence % | Meaning | Example (tE=20, Ο=3) |
|---|---|---|---|
| tE Β± 1Ο | 68.27% | Likely range (low confidence) | 17 to 23 days β 68% chance |
| tE Β± 2Ο | 95.45% | Good confidence range | 14 to 26 days β 95% chance |
| tE Β± 3Ο | 99.73% | High confidence (standard control limit) | 11 to 29 days β 99.7% chance |
| tE Β± 6Ο | 99.9997% | Six Sigma β near perfection | Used in quality targets |
| What Ο SIZE tells you | |||
| Small Ο (PβO small) | Low uncertainty β estimate is reliable, estimates are close together | ||
| Large Ο (PβO large) | High uncertainty β wide range, estimates vary greatly, more risk | ||
π― Key Rule: Add VARIANCES (ΟΒ²) not standard deviations. If Activity A has Ο=2 and Activity B has Ο=3: Path Ο β 2+3=5. Path Ο = β(4+9) = β13 = 3.61
π PERT Scenarios
Scenario 1 β Single Activity:
O=8 days | M=12 days | P=22 days
tE = (8+4Γ12+22)Γ·6 = (8+48+22)Γ·6 = 78Γ·6 = 13 days
Ο = (22β8)Γ·6 = 14Γ·6 = 2.33 days
95% range: 13Β±2(2.33) = 8.34 to 17.66 days
O=8 days | M=12 days | P=22 days
tE = (8+4Γ12+22)Γ·6 = (8+48+22)Γ·6 = 78Γ·6 = 13 days
Ο = (22β8)Γ·6 = 14Γ·6 = 2.33 days
95% range: 13Β±2(2.33) = 8.34 to 17.66 days
Scenario 2 β Path of 3 Activities (PERT Network):
Act A: O=4, M=6, P=8 β tE=6 | Ο=0.67 | ΟΒ²=0.44
Act B: O=2, M=5, P=14 β tE=6 | Ο=2.0 | ΟΒ²=4.0
Act C: O=1, M=3, P=5 β tE=3 | Ο=0.67 | ΟΒ²=0.44
Path Expected Duration = 6+6+3 = 15 days
Path Variance = 0.44+4.0+0.44 = 4.88 | Path Ο = β4.88 = 2.21 days
99.7% confidence range: 15Β±3(2.21) = 8.37 to 21.63 days
Act A: O=4, M=6, P=8 β tE=6 | Ο=0.67 | ΟΒ²=0.44
Act B: O=2, M=5, P=14 β tE=6 | Ο=2.0 | ΟΒ²=4.0
Act C: O=1, M=3, P=5 β tE=3 | Ο=0.67 | ΟΒ²=0.44
Path Expected Duration = 6+6+3 = 15 days
Path Variance = 0.44+4.0+0.44 = 4.88 | Path Ο = β4.88 = 2.21 days
99.7% confidence range: 15Β±3(2.21) = 8.37 to 21.63 days
π PERT vs Triangular Trap:
O=10, M=14, P=18
PERT: (10+56+18)Γ·6 = 84Γ·6 = 14.0
Triangular: (10+14+18)Γ·3 = 42Γ·3 = 14.0
Same answer here ONLY because distribution is symmetric. With skewed data they differ!
O=5, M=8, P=20: PERT=(5+32+20)Γ·6=9.5 | Triangular=(5+8+20)Γ·3=11.0 β DIFFERENT!
O=10, M=14, P=18
PERT: (10+56+18)Γ·6 = 84Γ·6 = 14.0
Triangular: (10+14+18)Γ·3 = 42Γ·3 = 14.0
Same answer here ONLY because distribution is symmetric. With skewed data they differ!
O=5, M=8, P=20: PERT=(5+32+20)Γ·6=9.5 | Triangular=(5+8+20)Γ·3=11.0 β DIFFERENT!
Scenario 3 β Which path is riskier?
Path X: tE=20 days, Ο=1.5 β Β±3Ο range = 15.5 to 24.5 days
Path Y: tE=18 days, Ο=4.0 β Β±3Ο range = 6 to 30 days
Path X is the critical path (longer). But Path Y is riskier (wider range). Monitor both!
Path X: tE=20 days, Ο=1.5 β Β±3Ο range = 15.5 to 24.5 days
Path Y: tE=18 days, Ο=4.0 β Β±3Ο range = 6 to 30 days
Path X is the critical path (longer). But Path Y is riskier (wider range). Monitor both!
π Expected Monetary Value (EMV)
EMV
EMV = Probability Γ Impact
Threats = negative impact | Opportunities = positive impact | Sum all EMVs for total exposure
| EMV Result | Type | Meaning | Action |
|---|---|---|---|
| Negative EMV (β) | Threat | Expected financial loss from this risk | Plan risk response (avoid/mitigate/transfer) |
| Positive EMV (+) | Opportunity | Expected financial gain from this chance | Plan to exploit or enhance |
| Net EMV = 0 | Balanced | Threats and opportunities cancel out | Monitor; net neutral position |
| Net EMV very negative | High risk project | Large contingency reserve needed | Reassess project viability |
π Risk Register with EMV:
Contingency Reserve needed β $27,500
| Risk | Type | Prob. | Impact | EMV |
|---|---|---|---|---|
| Supplier delays | Threat | 35% | β$60,000 | β$21,000 |
| Permit rejection | Threat | 20% | β$80,000 | β$16,000 |
| Weather delays | Threat | 25% | β$40,000 | β$10,000 |
| Early completion bonus | Opportunity | 30% | +$50,000 | +$15,000 |
| Scope reduction | Opportunity | 15% | +$30,000 | +$4,500 |
| TOTAL | β$27,500 |
π Decision Tree Analysis
Scenario β Build In-House vs. Outsource:
Option A (Build): 55% success β +$300K value | 45% fail β β$100K
EMV(A) = (0.55Γ300K) + (0.45Γβ100K) = 165Kβ45K = +$120,000
Option B (Outsource): 80% success β +$200K | 20% partial β +$50K
EMV(B) = (0.80Γ200K) + (0.20Γ50K) = 160K+10K = +$170,000
Decision: Choose Option B β higher EMV ($170K vs $120K)
Option A (Build): 55% success β +$300K value | 45% fail β β$100K
EMV(A) = (0.55Γ300K) + (0.45Γβ100K) = 165Kβ45K = +$120,000
Option B (Outsource): 80% success β +$200K | 20% partial β +$50K
EMV(B) = (0.80Γ200K) + (0.20Γ50K) = 160K+10K = +$170,000
Decision: Choose Option B β higher EMV ($170K vs $120K)
π― Decision Rule: Always choose the HIGHEST EMV option. If all EMVs negative β choose least negative (minimize loss).
π Risk Math Scenarios β 5 Cases
Case 1 β What contingency reserve to recommend?
3 identified risks: R1: 30%Γ$50K=β$15K | R2: 15%Γ$120K=β$18K | R3: 40%Γ$25K=β$10K
Total contingency = $43,000. Request this from sponsor for the cost baseline.
3 identified risks: R1: 30%Γ$50K=β$15K | R2: 15%Γ$120K=β$18K | R3: 40%Γ$25K=β$10K
Total contingency = $43,000. Request this from sponsor for the cost baseline.
Case 2 β Opportunity reduces net risk:
Threats: β$30K + β$20K = β$50K net risk
Opportunity: +$18K (probability 60% Γ $30K value)
Net EMV = β$50K + $18K = β$32K contingency needed (reduced by opportunity)
Threats: β$30K + β$20K = β$50K net risk
Opportunity: +$18K (probability 60% Γ $30K value)
Net EMV = β$50K + $18K = β$32K contingency needed (reduced by opportunity)
Case 3 β Exam Trap: High probability but low impact vs low probability high impact:
Risk A: 90% Γ β$5,000 = EMV = β$4,500
Risk B: 5% Γ β$200,000 = EMV = β$10,000
Risk B has lower probability but HIGHER EMV risk impact β prioritize Risk B for response planning.
Risk A: 90% Γ β$5,000 = EMV = β$4,500
Risk B: 5% Γ β$200,000 = EMV = β$10,000
Risk B has lower probability but HIGHER EMV risk impact β prioritize Risk B for response planning.
Case 4 β Project selection with EMV:
Project X: 70% success ($500K) + 30% fail (β$200K) = 350Kβ60K = $290K
Project Y: 90% success ($300K) + 10% fail (β$50K) = 270Kβ5K = $265K
Risk-neutral PM chooses X ($290K). Risk-averse PM may choose Y (safer).
PMP exam expects you to choose HIGHEST EMV: Project X.
Project X: 70% success ($500K) + 30% fail (β$200K) = 350Kβ60K = $290K
Project Y: 90% success ($300K) + 10% fail (β$50K) = 270Kβ5K = $265K
Risk-neutral PM chooses X ($290K). Risk-averse PM may choose Y (safer).
PMP exam expects you to choose HIGHEST EMV: Project X.
Case 5 β Management Reserve vs Contingency:
BAC=$500K | Contingency (EMV analysis) = $35K β Cost Baseline = $535K
Management Reserve (8%) = 0.08Γ$535K = $42,800 β Project Budget = $577,800
PM can authorize contingency. Must get approval for management reserve.
BAC=$500K | Contingency (EMV analysis) = $35K β Cost Baseline = $535K
Management Reserve (8%) = 0.08Γ$535K = $42,800 β Project Budget = $577,800
PM can authorize contingency. Must get approval for management reserve.
π Communication Channels
Communication Channels
Channels = n Γ (n β 1) Γ· 2
n = number of people. Every pair = 1 channel. Grows exponentially with team size.
| Team Size (n) | Channels | Add 1 person β Channels | New channels added |
|---|---|---|---|
| 2 | 1 | β 3 people = 3 | +2 |
| 5 | 10 | β 6 people = 15 | +5 |
| 10 | 45 | β 11 people = 55 | +10 |
| 15 | 105 | β 16 people = 120 | +15 |
| 20 | 190 | β 21 people = 210 | +20 |
π― Pattern: Adding 1 person adds (n) channels where n = new team size β 1. Adding person #11 adds 10 channels.
Scenario 1 β 3 people added to team of 10:
Before: 10Γ9Γ·2 = 45 | After: 13Γ12Γ·2 = 78 | Added = 33 channels
Before: 10Γ9Γ·2 = 45 | After: 13Γ12Γ·2 = 78 | Added = 33 channels
Scenario 2 β Exam asks "how many channels with 8 stakeholders plus PM?":
Total people = 8+1 = 9 | Channels = 9Γ8Γ·2 = 36
Total people = 8+1 = 9 | Channels = 9Γ8Γ·2 = 36
π Exam Trap β "stakeholders" vs "people":
Always include the PM in the count unless told otherwise. "5 stakeholders" = 5+1 PM = 6 people = 15 channels.
Always include the PM in the count unless told otherwise. "5 stakeholders" = 5+1 PM = 6 people = 15 channels.
π Contract Type Math β FPIF & CPIF
FPIF β Seller Incentive Fee
Seller's Fee = Target Profit + (Seller% Γ Cost Savings)
Cost savings = Target Cost β Actual Cost. If actual cost > target, seller absorbs Seller% of overrun.
CPIF β Calculated Fee
Fee = Target Fee + [Seller% Γ (Target Cost β Actual Cost)]
Constrained by Min Fee floor and Max Fee ceiling. Fee outside this range = use the limit.
| Contract Type | Who Bears Cost Risk? | Cost Overrun Impact | Cost Savings Impact |
|---|---|---|---|
| FFP | 100% Seller | Seller absorbs all overrun | Seller keeps all savings |
| FPIF | Shared (by ratio) | Shared per agreed ratio (up to ceiling) | Shared β seller gets incentive |
| CPFF | 100% Buyer | Buyer pays all overruns | Buyer saves β fee stays fixed |
| CPIF | Mostly Buyer | Buyer mostly, seller shares some | Seller gets incentive fee |
| T&M | Mostly Buyer | Buyer pays all hours/materials used | Less time = less cost for buyer |
FPIF Example β Under budget:
Target Cost=$200K | Target Profit=$20K | Share: 80/20 | Ceiling=$250K
Actual Cost=$175K (saved $25K)
Seller Incentive = 20%Γ$25K = $5,000
Seller receives: $175K cost + $20K profit + $5K bonus = Total: $200K
Seller total profit = $25,000 (better than target profit of $20K)
Target Cost=$200K | Target Profit=$20K | Share: 80/20 | Ceiling=$250K
Actual Cost=$175K (saved $25K)
Seller Incentive = 20%Γ$25K = $5,000
Seller receives: $175K cost + $20K profit + $5K bonus = Total: $200K
Seller total profit = $25,000 (better than target profit of $20K)
FPIF Example β Over budget:
Same contract. Actual Cost=$220K (overran $20K)
Seller absorbs 20% Γ $20K = $4K β Seller Profit = $20Kβ$4K = $16K
Buyer pays: $220K+$16K = $236K (below ceiling of $250K) β
Same contract. Actual Cost=$220K (overran $20K)
Seller absorbs 20% Γ $20K = $4K β Seller Profit = $20Kβ$4K = $16K
Buyer pays: $220K+$16K = $236K (below ceiling of $250K) β
CPIF Example with Min/Max Fee:
Target Cost=$500K | Target Fee=$40K | Share 70/30 | Min=$15K | Max=$60K
Actual Cost=$460K (saved $40K) β Fee = $40K + 30%Γ$40K = $40K+$12K = $52K
$52K is between $15K and $60K β Fee = $52K β
Total payment = $460K+$52K = $512K
Target Cost=$500K | Target Fee=$40K | Share 70/30 | Min=$15K | Max=$60K
Actual Cost=$460K (saved $40K) β Fee = $40K + 30%Γ$40K = $40K+$12K = $52K
$52K is between $15K and $60K β Fee = $52K β
Total payment = $460K+$52K = $512K
π Point of Total Assumption (PTA)
PTA β FPIF Contracts Only
PTA = [(Ceiling Price β Target Price) Γ· Buyer's Share %] + Target Cost
Above PTA: seller bears ALL cost risk. Ceiling Price is max buyer ever pays.
| Cost Level | Who Bears Additional Costs? | Buyer Pays |
|---|---|---|
| Actual Cost < Target Cost | Seller benefits β gets incentive | Less than Target Price |
| Actual Cost = Target Cost | On target | Target Price |
| Target Cost < Actual < PTA | Shared per ratio (e.g. 80/20) | Between Target and Ceiling |
| Actual Cost > PTA | Seller bears ALL overrun beyond PTA | Ceiling Price (max) |
PTA Calculation:
Target Cost=$300K | Target Price=$330K | Ceiling=$390K | Share: 80/20
PTA = [(390Kβ330K)Γ·0.80] + 300K = [60KΓ·0.80] + 300K = 75K+300K = $375,000
If actual cost exceeds $375K β seller absorbs 100% of any additional cost.
Target Cost=$300K | Target Price=$330K | Ceiling=$390K | Share: 80/20
PTA = [(390Kβ330K)Γ·0.80] + 300K = [60KΓ·0.80] + 300K = 75K+300K = $375,000
If actual cost exceeds $375K β seller absorbs 100% of any additional cost.
π Sigma & Quality Control Charts
Control Limits
UCL = Mean + 3Ο | LCL = Mean β 3Ο
99.73% of all data falls within Β±3Ο of the mean in a normal distribution
β Sigma & Control Chart Interpretation Table
| Sigma Level | % In Range | Defects per Million | What It Means |
|---|---|---|---|
| Β±1Ο | 68.27% | 317,300 | Rough estimate β poor quality control |
| Β±2Ο | 95.45% | 45,500 | Better β typical for many processes |
| Β±3Ο | 99.73% | 2,700 | Standard control chart limits β acceptable |
| Β±6Ο | 99.9997% | 3.4 | Six Sigma excellence β near perfect |
| Control Chart Signal | Status | Meaning | Action |
|---|---|---|---|
| Point OUTSIDE UCL or LCL | Out of Control β | Special cause variation β something unusual happened | Investigate immediately |
| 7 points same side of mean | Out of Control β | Process is systematically drifting (Rule of 7) | Find and fix root cause |
| 7 points trending up/down | Out of Control β | Consistent trend β something is changing the process | Investigate the trend |
| All points within limits, random | In Control β | Only common cause variation β process is stable | No action needed |
| Points near control limits | Warning | Approaching out-of-control β monitor closely | Watch trend |
π― Rule of 7 Critical Exam Point: If 7 consecutive measurements fall on ONE side of the center line β even if ALL within control limits β the process is considered out of control. The pattern itself is the problem, not any single data point.
Scenario β Concrete Strength Testing:
Mean strength = 4,000 PSI | Ο = 200 PSI
UCL = 4,000 + 3(200) = 4,600 PSI | LCL = 4,000 β 3(200) = 3,400 PSI
Test result: 3,350 PSI β OUTSIDE LCL β out of control β reject batch
Test results for 7 straight batches: 3,900 / 3,880 / 3,870 / 3,860 / 3,850 / 3,840 / 3,820
All within limits but all BELOW mean for 7 consecutive tests β RULE OF 7 β out of control
Mean strength = 4,000 PSI | Ο = 200 PSI
UCL = 4,000 + 3(200) = 4,600 PSI | LCL = 4,000 β 3(200) = 3,400 PSI
Test result: 3,350 PSI β OUTSIDE LCL β out of control β reject batch
Test results for 7 straight batches: 3,900 / 3,880 / 3,870 / 3,860 / 3,850 / 3,840 / 3,820
All within limits but all BELOW mean for 7 consecutive tests β RULE OF 7 β out of control
π Depreciation Methods
Straight-Line (SL)
Annual Depr. = (Cost β Salvage) Γ· Useful Life
Same amount every year. Simplest method.
Double Declining Balance (DDB) β Accelerated
Depr. = Book Value Γ (2 Γ· Useful Life)
Applied to REMAINING book value each year. Never depreciates below salvage value.
Sum of Years Digits (SYD) β Accelerated
SYD = n(n+1) Γ· 2 | Year k = [(nβk+1) Γ· SYD] Γ (Cost β Salvage)
n = useful life. Year 1 gets highest fraction, decreasing each year.
| Method | Year 1 Depr. | Over Time | Best For | Tax Advantage |
|---|---|---|---|---|
| Straight-Line | Medium (equal) | Same every year | Stable assets, simple reporting | Spread evenly |
| DDB | HIGHEST | Rapidly decreasing | Tech assets, rapid obsolescence | Front-loaded benefit |
| SYD | High (less than DDB) | Gradually decreasing | Moderate acceleration needed | Front-loaded (less than DDB) |
π All 3 Methods β $60,000 machine, 5-year life, $5,000 salvage:
*DDB Year 5: Limited to reach salvage of $5,000 (book value $7,776 β $5,000 = $2,776)
SYD: SYD=5Γ6Γ·2=15 | Y1=(5/15)Γ55K=18,333 | Y2=(4/15)Γ55K=14,667 etc.
| Year | SL ($) | DDB ($) | SYD ($) |
|---|---|---|---|
| 1 | 11,000 | 24,000 | 18,333 |
| 2 | 11,000 | 14,400 | 14,667 |
| 3 | 11,000 | 8,640 | 11,000 |
| 4 | 11,000 | 5,184 | 7,333 |
| 5 | 11,000 | 2,776* | 3,667 |
| Total | 55,000 | 55,000 | 55,000 |
SYD: SYD=5Γ6Γ·2=15 | Y1=(5/15)Γ55K=18,333 | Y2=(4/15)Γ55K=14,667 etc.
π― Exam Tip: Accelerated depreciation (DDB, SYD) = higher deduction in early years = MORE tax savings early = better for company cash flow. All methods depreciate the SAME total amount over the asset's life.
π NPV, IRR & Present Value
Present Value (PV)
PV = FV Γ· (1 + r)^n
"What is this future amount worth TODAY?" r=discount rate, n=years
Future Value (FV)
FV = PV Γ (1 + r)^n
"What will this money be worth in the future?"
Net Present Value (NPV)
NPV = Ξ£[CF_t Γ· (1+r)^t] β Initial Investment
Sum of all discounted future cash flows minus the upfront investment
| NPV Value | Decision | Meaning |
|---|---|---|
| NPV > 0 (Positive) | ACCEPT the project β | Project creates value β returns more than the cost of capital |
| NPV = 0 | Break-even β borderline | Returns exactly the cost of capital β neutral |
| NPV < 0 (Negative) | REJECT the project β | Project destroys value β returns less than cost of capital |
| When COMPARING projects | ||
| Highest NPV | Choose this project | Creates the MOST value |
| Lowest NPV | Last choice | Least value created (or most destroyed) |
| IRR Interpretation | ||
| IRR > Discount Rate | ACCEPT β | Project return exceeds required return |
| IRR < Discount Rate | REJECT β | Project return below required return β not worth it |
NPV Calculation (Discount rate = 10%):
Year 0: β$100,000 investment
Year 1: +$40,000 β PV = 40KΓ·1.10 = $36,364
Year 2: +$45,000 β PV = 45KΓ·1.21 = $37,190
Year 3: +$40,000 β PV = 40KΓ·1.331 = $30,052
Sum of PV = $103,606 β $100,000 investment = NPV = +$3,606 β ACCEPT
Year 0: β$100,000 investment
Year 1: +$40,000 β PV = 40KΓ·1.10 = $36,364
Year 2: +$45,000 β PV = 45KΓ·1.21 = $37,190
Year 3: +$40,000 β PV = 40KΓ·1.331 = $30,052
Sum of PV = $103,606 β $100,000 investment = NPV = +$3,606 β ACCEPT
Project Selection β Choose the Best:
Project A: NPV = +$85,000 | Project B: NPV = +$120,000 | Project C: NPV = β$15,000
Reject C (negative NPV). Between A and B β Choose Project B (highest NPV)
Project A: NPV = +$85,000 | Project B: NPV = +$120,000 | Project C: NPV = β$15,000
Reject C (negative NPV). Between A and B β Choose Project B (highest NPV)
π ROI, BCR & Payback Period
ROI (Return on Investment)
ROI = (Net Benefit Γ· Cost) Γ 100%
Net Benefit = Total Benefit β Total Cost
BCR (Benefit-Cost Ratio)
BCR = Total Benefits Γ· Total Costs
Payback Period
Payback = Initial Investment Γ· Annual Cash Inflow
β Financial Metrics β Master Interpretation Table
| Metric | Value | Status | Meaning |
|---|---|---|---|
| BCR | > 1.0 | ACCEPT β | Benefits exceed costs β worthwhile |
| BCR | = 1.0 | Break-even | Benefits = costs β neutral |
| BCR | < 1.0 | REJECT β | Costs exceed benefits β not worth it |
| ROI | High % | More return per dollar | Prefer highest ROI when comparing |
| Payback Period | Shorter | Better | Recover investment faster = less risk | Longer | Higher risk | Money tied up longer β ignores time value |
| Opportunity Cost | = Value of the best REJECTED alternative. Not a sum β just the single best forgone option. | ||
| Sunk Cost | Already spent β IGNORE for future decisions. Never use past spending to justify continuing a bad project. | ||
Full Financial Comparison β 3 Projects:
Best BCR: Alpha (1.70) | Best ROI: Alpha (70%) | Best Payback: Gamma (1.8yr)
Opportunity Cost of choosing Alpha = Gamma net benefit = $60K net benefit
| Project | Investment | Benefits | BCR | ROI | Payback |
|---|---|---|---|---|---|
| Alpha | $200K | $340K | 1.70 | 70% | 2.4 yr |
| Beta | $150K | $225K | 1.50 | 50% | 3.0 yr |
| Gamma | $100K | $160K | 1.60 | 60% | 1.8 yr |
Opportunity Cost of choosing Alpha = Gamma net benefit = $60K net benefit
🎭 Sunk Cost Exam Trap:
"We spent $3M already. It now costs $5M more to complete, total value = $6M. Continue?"
Ignore $3M sunk. Future: spend $5M, get $6M value = net +$1M β CONTINUE.
If future value was only $4M: spend $5M, get $4M = net -$1M β CANCEL.
"We spent $3M already. It now costs $5M more to complete, total value = $6M. Continue?"
Ignore $3M sunk. Future: spend $5M, get $6M value = net +$1M β CONTINUE.
If future value was only $4M: spend $5M, get $4M = net -$1M β CANCEL.
📝 Master Cheat Sheet β All Formulas
EVM
- EV=%CompleteΓBAC | PV=%PlannedΓBAC
- CV=EV-AC (+good/-bad) | CPI=EVΓ·AC (>1 good/<1 bad)
- SV=EV-PV (+good/-bad) | SPI=EVΓ·PV (>1 good/<1 bad)
- EAC=BACΓ·CPI (default) | AC+(BAC-EV) atypical | AC+ETC re-est
- ETC=EAC-AC | VAC=BAC-EAC (+good/-bad)
- TCPI=(BAC-EV)Γ·(BAC-AC) <1=easy >1=harder than current CPI
CPM / FLOAT / PERT / RISK
- EF=ES+Dur | LS=LF-Dur | TF=LS-ES | FF=ES(next)-EF(current)
- tE=(O+4M+P)Γ·6 | sigma=(P-O)Γ·6 | Variance=sigma^2 β ADD variances
- EMV=ProbΓImpact | Channels=n(n-1)Γ·2
- PTA=[(Ceiling-TargetPrice)Γ·Buyer%]+TargetCost (FPIF only)
- BCR=BenefitsΓ·Costs (>1 good) | ROI=(NetBenefitΓ·Cost)Γ100%
- PV=FVΓ·(1+r)^n | NPV>0=accept | SL=(Cost-Salvage)Γ·Life
🎯 Top 20 PMP Math Exam Tips
- EV always first in every EVM formula: EV = % Complete Γ BAC.
- Negative variance = Problem: CV<0 = over budget. SV<0 = behind schedule.
- Index < 1.0 = Problem: CPI<1 = over budget. SPI<1 = behind schedule.
- Default EAC = BACΓ·CPI unless exam says "atypical" or "re-estimated."
- Negative float = project already late. Requires crashing or scope reduction.
- Critical path TF = 0. Longest path = critical path. NEVER shortest.
- PERT weight = 4 on Most Likely: (O+4M+P)Γ·6.
- Add VARIANCES (sigma squared), NOT standard deviations for path totals.
- Rule of 7: 7 consecutive points one side of mean = out of control.
- Communication channels = n(n-1)Γ·2. Include PM. "New channels" = before minus after.
- EMV threats negative, opportunities positive. Sum = contingency reserve.
- NPV>0 = accept. Higher NPV = better choice.
- BCR>1 = accept. Higher BCR = better project.
- Sunk costs are irrelevant. Past spending never justifies future decisions.
- Opportunity cost = best forgone option only β not sum of all rejected.
- TCPI>1.2 = unrealistic β recommend revising budget (use EAC-based TCPI).
- PTA only in FPIF. Above PTA = seller absorbs all risk. Ceiling = buyer max.
- Accelerated depreciation = higher deductions early = better cash flow.
- EAC<BAC = finishing under budget. EAC>BAC = overrun expected.
- SPI<1 but CPI>1: "Under budget" misleading β team simply hasn't done enough work yet.