Monte Carlo Simulation — Four Corners Case

Comparison matrix

From Who is Tom Gifford?: refill the Symbol column from what you know. The rest of the table is as it appeared.

Match the pairs

From The five tasks, three ideas — match each one to what it actually does. The descriptions have been shuffled.

• c1. Project
• c2. Randomize
• c3. Decide

• b1. A deterministic engine + Goal Seek (Tasks 1).
• b2. Two distributions → a Monte Carlo model (Task 2).
• b3. Risk metrics, recommendations, 25-yr horizon, template (Tasks 3–5).

Why: Project, Randomize, Decide are easy to tell apart while they are sitting next to their descriptions and much harder afterwards, which is what this checks.

Step through it

Step through Tom's worksheet: five known rows one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: Year is 1
2. Step 2: Year is 2
3. Step 3: Year is 3
4. Step 4: Year is 4
5. Step 5: Year is 5

Discussion prompt

Every row of the worksheet is the same four formulas. Subscript t is the year; Pₜ is the ending balance.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Put in order

Put the moves of Year 1 by hand — and the ½ convention into the order they have to happen.

1. New investment: 6% × $85,000 = $5,100
2. Earnings: ($50,000 + ½ × $5,100) × 10% = $52,550 × 10% = $5,255
3. Ending balance: $50,000 + $5,100 + $5,255 = $60,355

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Contribution is the rate times this year's salary; salary is still $85,000 in year 1.

Comparison matrix

From Year 1 by hand — and the ½ convention: every row here is a choice with a cost. Fill the Value column, then say which row you would actually pick and what you give up for it.

Predict first

A student writes this, and it looks reasonable:

Treat the whole new $5,100 as if it sat in the market all year.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: Assumes every contributed dollar earns a full 12 months — but Tom said they trickle in evenly.

Expose only half the new contribution to this year's return.

Why: Assumes every contributed dollar earns a full 12 months — but Tom said they trickle in evenly.

Put in order

These are the steps of The deterministic recipe, scrambled. Put them back in order before the next slide shows you.

1. Seed bal = P0, sal = S0.
2. Contribution = inv × sal.
3. Earnings = (bal + ½ × contribution) × g_port.
4. Ending balance = bal + contribution + earnings; record it.
5. Grow salary by g_sal only if another year follows; repeat.

Why: This is the order the recipe itself gives. Recalling the sequence without the slide in front of you is the difference between recognising the method and being able to run it — most of what goes wrong in practice is a step done out of turn.

Discussion prompt

Explain The question Goal Seek answers by analogy to something with no Business Analytics in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Eliminate the wrong options

Year 1 ended at $60,355. Year-2 salary grows to $89,250 (= 85,000 × 1.05). Using the ½-year earnings convention at 10% growth and a 6% contribution, what is the year-2 ENDING balance?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. $72,013
• B. $72,281
• C. $71,746
• D. $65,710

Survives elimination: A

Why: Contribution = 6% × 89,250 = $5,355. Earnings = (60,355 + ½ × 5,355) × 10% = 63,032.5 × 10% = $6,303. Ending = 60,355 + 5,355 + 6,303 = $72,013 — matching Tom's row 2.

Check your understanding

Year 1 ended at $60,355. Year-2 salary grows to $89,250 (= 85,000 × 1.05). Using the ½-year earnings convention at 10% growth and a 6% contribution, what is the year-2 ENDING balance?

• A. $72,013 (correct)
• B. $72,281
• C. $71,746
• D. $65,710

Answer: A

Why: Contribution = 6% × 89,250 = $5,355. Earnings = (60,355 + ½ × 5,355) × 10% = 63,032.5 × 10% = $6,303. Ending = 60,355 + 5,355 + 6,303 = $72,013 — matching Tom's row 2.

Why B tempts people

Charged the full $5,355 a whole year of growth: (60,355 + 5,355) × 10% = $6,571, giving $72,281. That ignores the ½-year convention.

Why C tempts people

Forgot to grow salary — reused $85,000, so contribution = $5,100 and earnings = $6,290, giving $71,746. Salary must rise 5% before year 2.

Why D tempts people

Added the contribution but dropped the earnings term: 60,355 + 5,355 = $65,710. The year's $6,303 of growth is missing.

Discussion prompt

Explain Kate's two criticisms to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Discussion prompt

Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.

Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.

Sort into buckets

These are the pieces of Monte Carlo Simulation — Four Corners Case, out of order. Put each one back under the part of the lesson it belongs to.

Part 1 — The Deterministic Engine

Tom's worksheet: five known rows; What's actually happening each year; The recursion, in four lines

Goal Seek as Root-Finding

The question Goal Seek answers; Why a search, not a formula; uniroot finds 9.13%

Part 2 — Why One Number Lies

Kate's two criticisms; Input 1 — salary growth ~ Uniform(0%, 5%); Input 2 — portfolio growth ~ Normal(10%, 5%)

s1

Part 1 — The Deterministic Engine is where Monte Carlo Simulation — Four Corners Case puts Tom's worksheet: five known rows, What's actually happening each year, The recursion, in four lines. Knowing which part of the lesson a problem belongs to is most of knowing which method to reach for.

s2

Goal Seek as Root-Finding is where Monte Carlo Simulation — Four Corners Case puts The question Goal Seek answers, Why a search, not a formula, uniroot finds 9.13%. Knowing which part of the lesson a problem belongs to is most of knowing which method to reach for.

s3

Part 2 — Why One Number Lies is where Monte Carlo Simulation — Four Corners Case puts Kate's two criticisms, Input 1 — salary growth ~ Uniform(0%, 5%), Input 2 — portfolio growth ~ Normal(10%, 5%). Knowing which part of the lesson a problem belongs to is most of knowing which method to reach for.

Predict first

A student writes this, and it looks reasonable:

Carry the deterministic 5% raise straight into the random model.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: Confuses the deterministic assumption with the random one — and silently inflates every contribution.

Draw from the stated distribution: Uniform(0%, 5%).

Why: Confuses the deterministic assumption with the random one — and silently inflates every contribution.

Discussion prompt

The rule this trap just fixed:

Matches Kate and Tom's agreed assumption; contributions grow at half the deterministic pace.

Now break it on purpose. Build a case that violates it and follow the consequences until something visibly fails. Where does the failure first show up — and would you have noticed it if you had not been looking?

Hint: The dangerous rules are the ones whose violation still produces an answer. If yours fails loudly, try to find one that fails quietly.

Comparison matrix

From The simulation engine, line by line: refill the Ending balance column from what you know. The rest of the table is as it appeared.

Predict first

A student writes this, and it looks reasonable:

Treat a disappointing probability as a sample-size problem.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: Assumes more runs change the answer, as if 1% were just bad luck in the draw.

Separate the estimate's precision from the underlying probability.

Why: Assumes more runs change the answer, as if 1% were just bad luck in the draw.

Put in order

These are the steps of The simulation recipe, scrambled. Put them back in order before the next slide shows you.

1. Name the uncertain inputs and pick a distribution for each (Uniform raises, Normal returns).
2. Draw one random scenario for all inputs across all years.
3. Run the deterministic engine on that scenario to get one outcome.
4. Repeat thousands of times, storing every outcome.
5. Summarize the collected outcomes — probabilities, averages, and tail risk.

Why: This is the order the recipe itself gives. Recalling the sequence without the slide in front of you is the difference between recognising the method and being able to run it — most of what goes wrong in practice is a step done out of turn.

Discussion prompt

The simulation recipe works on the cases you have just seen. Push it to the edge: what is the most degenerate input it still handles — empty, zero, one item, everything equal — and what is the first case where it stops being true? Name the case, not just "it breaks".

Hint: Try the smallest legal input, then the largest, then the one where two things collide. Methods are specified at their edges; the middle takes care of itself.

Predict first

The deterministic worksheet says $772,722, and the simulation uses the SAME 10% average portfolio growth. Why does the simulation give Tom only about a 1% chance of reaching $1M, with a median (~$690k) BELOW the deterministic figure?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: Random raises average 2.5% (Uniform 0–5%), not 5%, so contributions are smaller; multiplicative random returns drag the median below the average-rate path; and the $773k path was already under $1M anyway.

Why: Two forces lower the center — the average raise is 2.5% not 5% (smaller contributions), and volatility drag pulls the median of multiplicative growth below the mean-rate path. And the deterministic $773k was always short of $1M, so reaching the goal is a tail event, not the norm.

Check your understanding

The deterministic worksheet says $772,722, and the simulation uses the SAME 10% average portfolio growth. Why does the simulation give Tom only about a 1% chance of reaching $1M, with a median (~$690k) BELOW the deterministic figure?

• A. Random raises average 2.5% (Uniform 0–5%), not 5%, so contributions are smaller; multiplicative random returns drag the median below the average-rate path; and the $773k path was already under $1M anyway. (correct)
• B. The simulated portfolio growth has a lower mean than 10% once it's made random.
• C. 10,000 iterations is too few; with enough runs the probability would rise toward the deterministic result.
• D. The $772,722 figure is a spreadsheet error; with 10% growth Tom should clear $1M deterministically.

Answer: A

Why: Two forces lower the center — the average raise is 2.5% not 5% (smaller contributions), and volatility drag pulls the median of multiplicative growth below the mean-rate path. And the deterministic $773k was always short of $1M, so reaching the goal is a tail event, not the norm.

Why B tempts people

The portfolio return is Normal(10%, 5%) — its mean is exactly 10% by construction. The center didn't move from the return assumption; it moved from the salary mean (2.5%) and volatility drag.

Why C tempts people

More iterations shrink the estimate's noise (~1/√n); they don't change the true probability the assumptions imply. 1% is the honest answer, not a sampling artifact.

Why D tempts people

$772,722 is correct — it was verified against the case to the dollar. Because it's already below $1M, even the no-risk path misses the goal.

Discussion prompt

Explain The same story as a cumulative curve to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Discussion prompt

Explain Uncertainty grows with the horizon by analogy to something with no Business Analytics in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Comparison matrix

From Four numbers a manager actually wants: every row here is a choice with a cost. Fill the What it answers column, then say which row you would actually pick and what you give up for it.

Predict first

A student writes this, and it looks reasonable:

Read CVaR as a probability, or as a guaranteed floor.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: That conflates CVaR with VaR. The 5% cutoff is the VaR ($526k); CVaR is a dollar average of the tail, not a probability.

Keep the two roles distinct: VaR is a cutoff, CVaR is a tail average.

Why: That conflates CVaR with VaR. The 5% cutoff is the VaR ($526k); CVaR is a dollar average of the tail, not a probability.

Sort into buckets

Some of these hold up and some are the exact mistakes this lesson is built to prevent. Sort them.

Holds up

Four Corners' HR asked Tom to build a financial-planning model — and to test it on himself first. Every number in this deck traces back to this one profile.; The managerial report asks five questions, but they collapse into three analytical moves you'll build in order:; Every row of the worksheet is the same four formulas. Subscript t is the year; Pₜ is the ending balance.

Breaks

Treat the whole new $5,100 as if it sat in the market all year.; Carry the deterministic 5% raise straight into the random model.

sound

These are stated as this lesson states them — each one survives the edge cases Monte Carlo Simulation — Four Corners Case puts it through.

flawed

Each of these is lifted from a trap in this deck: reasonable-sounding, and wrong in a way that only shows up once you rely on it.

Predict first

Which statement correctly describes the 5% CVaR of $492,000?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: It is the average ending wealth across the worst 5% of simulated paths — a measure of how bad the tail is on average, beyond just its edge.

Why: CVaR (expected shortfall) averages the outcomes in the worst 5% tail. VaR only marks the tail's edge ($526k at the 5th percentile); CVaR summarizes the severity of everything beyond that edge, which is why it's $492k < $526k.

Check your understanding

Which statement correctly describes the 5% CVaR of $492,000?

• A. It is the average ending wealth across the worst 5% of simulated paths — a measure of how bad the tail is on average, beyond just its edge. (correct)
• B. It is the probability that Tom's portfolio ends below $526,000.
• C. It is larger than the VaR, giving an optimistic upper bound on losses.
• D. It guarantees Tom will finish with at least $492,000.

Answer: A

Why: CVaR (expected shortfall) averages the outcomes in the worst 5% tail. VaR only marks the tail's edge ($526k at the 5th percentile); CVaR summarizes the severity of everything beyond that edge, which is why it's $492k < $526k.

Why B tempts people

That probability is fixed at 5% by the definition of the 5% VaR ($526k). CVaR is a dollar average of the tail, not a probability.

Why C tempts people

For a wealth (left-tail) distribution CVaR is LESS than VaR, because it averages outcomes even worse than the cutoff. It's the more conservative number, not an optimistic bound.

Why D tempts people

CVaR is the mean of the worst 5% — individual paths in that tail finish below $492k. An average guarantees no floor.

Discussion prompt

At 6% for 20 years, $1M is a ~1% event. The model also shows the levers — so the advice is concrete, not "save more, good luck":

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Put in order

Put the moves of Task 4 — what 25 years buys into the order they have to happen.

1. Deterministic 25-yr balance jumps to ≈ $1,339,000
2. Simulated P(success) climbs from ≈ 1% to ≈ 79%
3. Required contribution rate falls from 9.13% to ≈ 3.45%

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Compounding is exponential: the last five years act on the largest balance, so they add the most dollars of the whole plan.

Discussion prompt

Work backwards. The example finished here:

Required contribution rate falls from 9.13% to ≈ 3.45%

What was it asked to do, and what must it have been given? Reconstruct the problem from its answer.

Hint: Every quantity in the result had to enter somewhere. Account for each one.

Eliminate the wrong options

Working 25 years instead of 20 (still 6%) raises simulated P($1M) from ~1% to ~79% and the deterministic value to ~$1.34M. What best explains why five extra years is so disproportionately powerful?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. Growth is multiplicative, so the final years compound on the largest balance, and the extra years also let bad early draws recover — both the center and the success rate jump.
• B. Five years is 25% more time, so the portfolio grows about 25% more — roughly proportional.
• C. The extra years shrink the standard deviation of returns, so the risk essentially disappears.
• D. It only works because the required contribution rate drops to 3.45%, which is what produces the 79%.

Survives elimination: A

Why: Compounding is exponential: the last years operate on the biggest balance and contribute the most dollars, and more years give the portfolio time to recover from poor early returns. So the whole distribution slides past $1M — far beyond a linear 25% gain.

Check your understanding

Working 25 years instead of 20 (still 6%) raises simulated P($1M) from ~1% to ~79% and the deterministic value to ~$1.34M. What best explains why five extra years is so disproportionately powerful?

• A. Growth is multiplicative, so the final years compound on the largest balance, and the extra years also let bad early draws recover — both the center and the success rate jump. (correct)
• B. Five years is 25% more time, so the portfolio grows about 25% more — roughly proportional.
• C. The extra years shrink the standard deviation of returns, so the risk essentially disappears.
• D. It only works because the required contribution rate drops to 3.45%, which is what produces the 79%.

Answer: A

Why: Compounding is exponential: the last years operate on the biggest balance and contribute the most dollars, and more years give the portfolio time to recover from poor early returns. So the whole distribution slides past $1M — far beyond a linear 25% gain.

Why B tempts people

That assumes linear growth. Compounding is exponential, so the deterministic balance rises from $773k to $1.34M (~73%), not 25%, and the success rate leaps far more.

Why C tempts people

More years add more random draws, so the dispersion of terminal wealth actually grows. The success rate rises because the center moves past $1M, not because risk vanishes.

Why D tempts people

The 3.45% is the deterministic rate to exactly hit $1M; the 79% figure is computed at the ORIGINAL 6%. The horizon itself does the work.

Comparison matrix

From Task 5 — one model, every employee: refill the Tom's value column from what you know. The rest of the table is as it appeared.

Draw it

One page, no notation unless you need it: draw how these connect — Part 1 — The Deterministic Engine · Goal Seek as Root-Finding · Part 2 — Why One Number Lies · Part 3 — The Monte Carlo Engine · Part 4 — Reading the Output · Part 5 — Decisions & Generalization. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.