# Chapter 17 notebook: Financial Mathematics: Comparing Money Across Time

**Goal:** Place hypothetical cash flows on one time and rate basis before comparing them.

**Start with:** Rules 17.2.2, 17.2.3, 17.3.1. Use a calculator or the optional Python lab. Programming is optional for the workbook; the Jupyter version requires a Python 3 kernel. All lab numbers are constructed practice inputs, not observed data.

**Source:** [Finished Chapter 17](../skills/math-thumb-financial-mathematics/references/chapter.md) from *Mathematical Rules of Thumb*. Numbered rules, classifications, and the decision path come from the book. The lab, exercises, answer key, and worksheet prompts are companion additions.

## 1. Frame your decision

Write a question in which an answer would change something you do. Gather: Cash-flow amounts, signs, dates and currency; valuation date; effective period rate; nominal versus real basis; payment timing.

- My question and intended decision:
- Known inputs and units:
- Required accuracy or threshold:
- What I expect before calculating:
- What I still need to find out:

Use the lab as a worked starting point if you do not yet have your own problem. For notation or prerequisites, ask the chapter skill to explain only the concept blocking the next step.

## 2. Choose a route

1. Write the objective: quick scale estimate, valuation, affordability, or contract comparison.
2. Draw a timeline and mark every amount, sign, date, and currency.
3. Put rates on a common effective-period basis and decide whether amounts and rates are nominal or real.
4. Move every cash flow to one valuation date. Use a geometric closed form only after recognizing an exact pattern.
5. Check domain conditions: payment timing, \(r>0\), \(r>g\), fixed count, and compatible periods.
6. Stress-test cash flows and rates. Add taxes, fees, default, liquidity, and constraints when relevant.
7. Treat the number as a model output, not personal financial advice; escalate high-stakes decisions to contract-specific analysis and qualified professionals.

**Chapter-specific stop check:** Treat exercise rates as explicit assumptions. Match rate periods and cash-flow timing; do not convert a classroom valuation into personalized investment or borrowing advice.

## 3. Work the lab

A hypothetical project costs 1,000 currency units now and returns 400 at each of the next three year ends. At an assumed effective annual discount rate of 8%, present value is 400/1.08+400/1.08^2+400/1.08^3≈1,030.84, giving NPV≈30.84. Positive NPV passes this specified model's screen. The example's rate is an assumption, not a current market quote or a recommendation.

Predict the sign and scale before running the code. Then change one input and explain why the result moves. The code checks the constructed example; it does not prove the rule for every possible input. Code assertions may describe the example's chosen regime, so inspect them before changing that regime.

```python
rate, payment, periods, initial_cost = 0.08, 400.0, 3, 1000.0
assert rate > -1 and isinstance(periods, int) and periods > 0
discounted = [payment/(1+rate)**t for t in range(1, periods+1)]
present_value = sum(discounted)
annuity_value = payment*periods if rate == 0 else payment*(1-(1+rate)**(-periods))/rate
print("Discounted payments:", [round(x, 2) for x in discounted])
print(f"Present value: {present_value:.2f}; NPV: {present_value-initial_cost:.2f}")
assert abs(present_value-annuity_value) < 1e-8
```

**My prediction, observed result, and explanation:**

_Record your work here._

## 4. Practise without the answers

### Exercise 1

After a 20% loss, what gain restores the starting value?

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

### Exercise 2

Convert a nominal annual rate of 12%, compounded monthly, to an effective annual rate.

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

### Exercise 3

A growing perpetuity has next payment 100, discount rate 4%, and growth rate 5%. Is 100/(0.04-0.05) a valid negative value?

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

**Coaching prompt:** “Use the Chapter 17 skill to help me with Exercise 2. Ask for my attempt, give one useful hint if I need it, and help me check the assumptions before showing the answer.”

## 5. Answer key and reasoning

Read this after attempting the exercises, or use it immediately if you prefer a complete walkthrough. An answer is complete only when its assumptions and stopping point are clear.

### Answer 1

0.20/(1-0.20)=0.25, or 25%. For example 100 falls to 80, then 80 × 1.25=100.

### Answer 2

Monthly rate is 0.12/12=0.01; effective annual rate is 1.01^12-1≈0.126825, or 12.6825%.

### Answer 3

No. The convergent growing-perpetuity formula requires r>g. Here discounted payments do not form a convergent series; choose a finite horizon or a defensible different model.

## 6. Build the complete chapter toolkit

The new lab samples the chapter; the following checklist covers all 12 rules. Study one thematic group at a time. A large group can take several sessions.

- **Portable Mental Arithmetic:** work with rules 17.1.1, 17.1.2, 17.1.3.
- **Put Rates and Cash Flows on One Basis:** work with rules 17.2.1, 17.2.2, 17.2.3.
- **Recognize Repeating Cash-Flow Patterns:** work with rules 17.3.1, 17.3.2, 17.3.3, 17.3.4, 17.3.5, 17.3.6.

For each selected rule, read its equation, explanation, and worked use in the source. Reproduce that example; change one input; then change one assumption so the rule is no longer justified. Record the result and what check catches the failure. Historical examples remain labeled and qualified as in the source.

Read each complete numbered profile in the [chapter reference](../skills/math-thumb-financial-mathematics/references/chapter.md) before applying it. The cues below abbreviate the graph metadata; they are not complete conditions. Change the status only after doing the practice described below.

| Rule | Book role | First assumptions to inspect | Practice status |
|---|---|---|---|
| 17.1.1: Rule of 72 for doubling time | Independent | consistent time unit;  stable rate | new |
| 17.1.2: Real return is approximately nominal return minus inflation | Independent | consistent time unit;  stable rate | new |
| 17.1.3: Loss recovery is mathematically asymmetric | Independent | positive reference value;  loss fraction between zero and one | new |
| 17.2.1: Convert periodic rates to an effective annual rate | Independent | consistent time unit;  stable rate | new |
| 17.2.2: Discount each future cash flow to the same date | Workflow | consistent valuation date;  constant periodic rate | new |
| 17.2.3: Positive NPV is the basic discounted-cash-flow screen | Workflow | complete cash flow forecast;  defensible discount rate | new |
| 17.3.1: Present value of an ordinary annuity | Independent | consistent valuation date;  constant periodic rate | new |
| 17.3.2: Constant perpetuity value is payment divided by rate | Independent | consistent valuation date;  constant periodic rate | new |
| 17.3.3: Growing perpetuity value depends on the rate spread | Independent | consistent valuation date;  constant periodic rate | new |
| 17.3.4: Future value of regular end-of-period deposits | Independent | consistent valuation date;  constant periodic rate | new |
| 17.3.5: Level payment for an amortizing loan | Independent | consistent valuation date;  constant periodic rate | new |
| 17.3.6: Annuity-due values shift payments one period earlier | Specialized | consistent valuation date;  constant periodic rate | new |

For a completed row, record: **rule number / my new input / mathematical claim type / verified assumptions / calculation / check / valid use / rejected use / next step**. “Practised” means you worked an example. “Demonstrated” means you can explain a valid use, transfer it, and reject a misuse without the answer key.

## 7. Apply it to your own problem

Return to your opening question. Choose the smallest rule set that can settle it. Use the book's independent/workflow/specialized classification separately from the claim type (exact, approximate, bound, diagnostic, or heuristic).

- Selected rule number(s) and reason:
- Assumptions that hold, fail, or remain uncertain:
- Substitution with units:
- Result and error, uncertainty, or bound:
- Independent check or limiting case:
- Decision this supports:
- Stop here, do a named next calculation, or gather missing information:

## 8. Transfer and continue

Useful nearby chapters: Chapter 1: algebra; Chapter 12: probability; Chapter 16: measurement. Bring the question, units, assumptions, result type, and uncertainty to the next chapter. Choose a bridge only when it supplies an operation you actually need.

**Completion check:** Explain this chapter's lab in your own words; solve one changed-input exercise; reject one invalid use; and produce a decision record for your own problem. If one check fails, revisit that part of the chapter rather than marking every rule complete.