Mathematical Rules of Thumb, illustrated reader · Chapter 17

17Financial Mathematics

Comparing Money Across Time

5 demonstrations follow the chapter's rules. Choose a value, watch the figure and the numbers change, and check your prediction. Every choice is precomputed from the notebook calculations.

Ask the chapter skill

“Help me use Chapter 17 for my question. Choose a rule, check its assumptions, and show how the result changes when an input changes.”

Use math-thumb-financial-mathematics from the companion's skill package. The demonstrations below also work on their own.

Examples use constructed inputs or the book's own values, disclosed in each panel. A picture illustrates a rule; its assumptions set its scope.

1Demonstration 1 of 5

Change the rate and recheck present value

Can payment totals alone establish positive NPV?

The discount-rate curve shows how a marginal present-value conclusion can reverse.

NPV=−1000+∑t=13400(1+r)t NPV=-1000+\sum_{t=1}^{3}\frac{400}{(1+r)^t}

Assumed annual discount rate r. Hypothetical dated cash flows, year-end payments, matching annual periods, no taxes or market claims.

Predict first. Can payment totals alone establish positive NPV?

Choose an example

Change the rate and recheck present value. Three year-end payments of 400 against cost 1000 have NPV 30.8388 at assumed rate 8%, so the project passes the positive-NPV screen (the break-even rate is about 9.7%). These hypothetical cash flows illustrate timing, not a current investment recommendation.
Assumed annual discount rate r: 0.08
Constructed teaching inputs; calculations executed locally. Supported menu choices are precomputed.

Calculated values

Initial cost
1000
Three year-end payments
400
Assumed annual rate
0.08
NPV
30.8388

Three year-end payments of 400 against cost 1000 have NPV 30.8388 at assumed rate 8%, so the project passes the positive-NPV screen (the break-even rate is about 9.7%). These hypothetical cash flows illustrate timing, not a current investment recommendation.

Use the idea

Use rule 17.2.3 when its stated conditions fit. Compare the calculation with your own decision threshold; retain the relevant error or uncertainty.

Where the conclusion applies

Hypothetical dated cash flows, year-end payments, matching annual periods, no taxes or market claims.

Check your understanding: Can payment totals alone establish positive NPV?
No. Discounted timing and the chosen rate matter; 1200 received later is not 1200 today.

Book source: Rule 17.2.3: Positive NPV is the basic discounted-cash-flow screen. Demonstration C17-D01. Worked illustration.

2Demonstration 2 of 5

Use the correct base for recovering a loss

What gain recovers a 50% loss?

Recovery percentages are measured against a smaller base than the original loss.

required gain=ℓ1−ℓ \text{required gain}=\frac{\ell}{1-\ell}

Loss fraction ℓ. Loss strictly below 100%; hypothetical value changes without cash flows.

Predict first. What gain recovers a 50% loss?

Choose an example

Use the correct base for recovering a loss. A 30% loss leaves 70% of the original value. Recovering requires 42.8571% of the reduced base.
Loss fraction ℓ: 0.3
Constructed teaching inputs; calculations executed locally. Supported menu choices are precomputed.

Calculated values

Loss fraction
0.3
Required gain fraction
0.428571

A 30% loss leaves 70% of the original value. Recovering requires 42.8571% of the reduced base.

Use the idea

Use rule 17.1.3 when its stated conditions fit. Compare the calculation with your own decision threshold; retain the relevant error or uncertainty.

Where the conclusion applies

Loss strictly below 100%; hypothetical value changes without cash flows.

Check your understanding: What gain recovers a 50% loss?
A 100% gain on the remaining half restores the original value.

Book source: Rule 17.1.3: Loss recovery is mathematically asymmetric. Demonstration C17-D02. Worked illustration.

3Demonstration 3 of 5

Compare compounding conventions fairly

Is nominal 12% monthly compounding the same as effective 12% annually?

Hold the nominal annual rate at 12% and change how often interest compounds.

reff=(1+rnom/m)m−1 r_{eff}=(1+r_{nom}/m)^m-1

Compounding periods per year m. Equal periods, fixed nominal rate, no fees or irregular payment dates.

Predict first. Is nominal 12% monthly compounding the same as effective 12% annually?

Choose an example

Compare compounding conventions fairly. A nominal annual rate of 12% compounded 4 times per year gives effective annual rate 12.5509%. Compare rates only after matching the convention.
Compounding periods per year m: 4
Constructed teaching inputs; calculations executed locally. Supported menu choices are precomputed.

Calculated values

Nominal annual rate
0.12
Periods per year
4
Effective annual rate
0.125509

A nominal annual rate of 12% compounded 4 times per year gives effective annual rate 12.5509%. Compare rates only after matching the convention.

Use the idea

Use rule 17.2.1 when its stated conditions fit. Compare the calculation with your own decision threshold; retain the relevant error or uncertainty.

Where the conclusion applies

Equal periods, fixed nominal rate, no fees or irregular payment dates.

Check your understanding: Is nominal 12% monthly compounding the same as effective 12% annually?
No. Its effective rate is (1.01)¹²−1, about 12.6825%.

Book source: Rule 17.2.1: Convert periodic rates to an effective annual rate. Demonstration C17-D03. Worked illustration.

4Demonstration 4 of 5

Check the rule of 72

When is the rule of 72 almost exact?

The chart shows the shortcut error in years. It crosses zero near 8%.

t2=ln⁡2ln⁡(1+r/100)≈72r t_2=\frac{\ln2}{\ln(1+r/100)}\approx\frac{72}{r}

Annual return r (%). Annual compounding, constant rate, no withdrawals.

Predict first. When is the rule of 72 almost exact?

Choose an example

Check the rule of 72. At 8% a year, money doubles in 9.01 years; the rule says 9. The shortcut is within a few days here, good enough for mental math.
Annual return r (%): 8
Constructed teaching inputs; calculations executed locally. Supported menu choices are precomputed.

Calculated values

Annual return (%)
8
Exact doubling time (years)
9.00647
Rule of 72 (years)
9
Rule error (years)
-0.00646834

At 8% a year, money doubles in 9.01 years; the rule says 9. The shortcut is within a few days here, good enough for mental math.

Use the idea

Use rule 17.1.1 when its stated conditions fit. Compare the calculation with your own decision threshold; retain the relevant error or uncertainty.

Where the conclusion applies

Annual compounding, constant rate, no withdrawals.

Check your understanding: When is the rule of 72 almost exact?
Near 8%. Below it the rule runs long; above it the rule runs short.

Book source: Rule 17.1.1: Rule of 72 for doubling time. Demonstration C17-D04. Worked illustration.

5Demonstration 5 of 5

Watch a growing perpetuity blow up

Why is a growing-perpetuity value with g close to r unreliable?

Hold the discount rate at 5% and move the assumed growth toward it. The value and its sensitivity to g explode as the spread shrinks.

V=C1r−g,r>g V=\frac{C_1}{r-g},\quad r>g

Growth rate g. Constant growth forever, next payment 100, r=5%, r>g.

Predict first. Why is a growing-perpetuity value with g close to r unreliable?

Choose an example

Watch a growing perpetuity blow up. With next payment 100, r=5% and g=3%, the value is 100/(r−g)=5,000. Raising g by a quarter of a percentage point changes the value by 14.3%. Compared with g=1%, halving the spread to 2 points doubles the value and makes it about twice as sensitive to g.
Growth rate g: 0.03
Constructed teaching inputs; calculations executed locally. Supported menu choices are precomputed.

Calculated values

Discount rate r (%)
5
Growth rate g (%)
3
Value (next payment 100)
5000
Value if g is 0.25 point higher
5714.29
Change from that 0.25 point (%)
14.2857

With next payment 100, r=5% and g=3%, the value is 100/(r−g)=5,000. Raising g by a quarter of a percentage point changes the value by 14.3%. Compared with g=1%, halving the spread to 2 points doubles the value and makes it about twice as sensitive to g.

Use the idea

Use rule 17.3.3 when its stated conditions fit. Compare the calculation with your own decision threshold; retain the relevant error or uncertainty.

Where the conclusion applies

Constant growth forever, next payment 100, r=5%, r>g.

Check your understanding: Why is a growing-perpetuity value with g close to r unreliable?
Because value depends on 1/(r−g). At a half-point spread, adding a quarter point of growth doubles the value.

Book source: Rule 17.3.3: Growing perpetuity value depends on the rate spread. Demonstration C17-D05. Worked illustration.

Bring the idea to a question of your own

Choose the relationship that answers your question, check its conditions, and compare the result with the accuracy or decision threshold you need.

The chapter skill can adapt these calculations to your inputs. It should name the assumptions, explain what the result supports, and say what still needs evidence. The chapter workbook adds a lab and three exercises with answers.