Chapter 2: Financial Math I — Spot
This is the book's first "primer" chapter — the toolkit Chapter 1's examples borrowed without deriving. It's also where Chapter 1's illustrative duration-matching example (the $100M/$102M bond pair, with made-up DV01 numbers) gets its real formula: this chapter shows you exactly how duration is computed, with the book's own fully worked table.
- Source
- Extracted directly from the chapter text in /references
- Chapter type
- Financial Math primer — deliberately calculus-free, built on numerical examples
- Connects to
- Formalizes the duration concept used illustratively in Ch. 1
Discounting: the one idea everything else builds on
The book's opening claim, stated flatly: cash-flow discounting is the basis of all securities valuation. A stock's value is the present value of its future dividends and capital gains; a bond's value is the present value of its coupons and principal. You'd always pay less than $100 today for a promise of $100 later — not only because the promise carries risk, but because $100 today, invested, would grow to more than $100 by then. Whether you use your own cash or borrowed cash doesn't change this: borrowed cash costs explicit interest, your own cash costs the interest you gave up by not investing it elsewhere (an opportunity cost — just as real a cost as interest paid).
Present value, worked from scratch
Invest $500 at 5% for 1 year: $500(1.05) = $525. For 2 years, you earn interest on interest: $500(1.05)² = $551.25. General formula: FV = PV₀(1+r)ⁿ. Reversed: how much would you need today to have $500 in 2 years at 5%? PV₀ = 500/(1.05)² = $453.51 — the two amounts are equivalent at that rate; you'd be indifferent between them.
For a stream of cash flows — say $500 a year for 4 years — you discount and sum each one separately: PV₀ = 500 × [1/(1+r) + 1/(1+r)² + 1/(1+r)³ + 1/(1+r)⁴] = $1,772.98. A constant cash flow like this is an ordinary annuity, and the bracketed term is its present-value annuity factor.
Compounding frequency changes the real return — a lot
A quoted rate is always per annum, but how often it compounds changes what you actually earn. The book's own rollover example: invest €1,000 in a 3-month CD at 3.25% (p.a.), rolled over 4 times across a year.
| Convention | Formula | 1-year result |
|---|---|---|
| 30/360 (simplified quarters) | 1,000(1+0.0325/4)⁴ | €1,032.898 |
| Act/360 (real day counts: 92/91/90/92) | 1,000·∏(1+0.0325×days/360) | €1,033.361 |
What this means: the exact same 3.25% quoted rate produces two different real amounts — €0.463 apart — purely from which day-count convention is used. That's not rounding noise; the book calls this a real risk: "many unsuspecting investors have been burnt in the past by ruthless dealers playing day-count tricks." The equivalent annual rate (EAR) — 3.2898% for the 30/360 case — is how you make different compounding conventions comparable at all: (1+r/m)^m = 1+EAR, where m is the number of compounding periods per year.
Rate vs. yield — a distinction the market blurs but shouldn't
The book's own line: "rate" is the stated percentage used to compute the actual interest paid; "yield" is the percentage actually earned on what you invested. A 5% coupon bond selling at par (100% of face value) yields exactly 5%. The same bond selling below par yields more than 5%, because you paid less than the principal for the same coupon stream. The two numbers coincide only by coincidence, not by definition.
Duration — turning bond-market chaos into one comparable number
Where this connects: Chapter 1's duration-matched bond trade (long $100M 30-year, short $102M 26-year) used illustrative DV01 figures to make the point concrete. This section is where that number actually comes from.
Any single issuer can have wildly different bonds outstanding — different maturities, coupons, structures. Macaulay duration is defined as the present-value-weighted average time to a bond's cash flows — it collapses all that heterogeneity into one number, in years, that lets you compare completely different bonds on equal footing. Modified duration reframes the same idea as a price sensitivity: the % change in price per unit change in yield, ModD = −(ΔP/P)/Δy.
The book's own worked example: a 6-year, 7% semi-annual coupon bond yielding 8% — a $100 face bond paying $3.50 every 6 months plus $100 principal at year 6, currently worth $95.3075.
| Time (yrs) | Cash flow | PV of CF | % of total PV | t × %PV |
|---|---|---|---|---|
| 0.5 | 3.50 | 3.365 | 3.53% | 0.0177 |
| 3.0 | 3.50 | 2.766 | 2.90% | 0.0871 |
| 6.0 (final) | 103.50 | 64.646 | 67.83% | 4.0697 |
| Total (all 12 periods) | 95.3075 | 100.00% | 4.9720 | |
What this means: Macaulay duration = 4.9720 years — note the final principal repayment alone accounts for 67.83% of the bond's present value, which is why duration (4.97 years) comes out so much shorter than the bond's 6-year maturity: the coupons pull the weighted-average time to cash flows earlier. Modified duration = 4.9720/(1+0.08/2) = 4.7807. If yield rises from 8.00% to 8.15% (+15bp), predicted price change = −4.7807 × 0.15% = −0.7171%, or $95.3075 → $94.6240. The exact recalculated price is $94.6270 — duration is a very close linear approximation for a small yield move like this one.
- Context
- The book pushes the same bond to an extreme: what if yield fell all the way to 0%?
- Algo-relevant?
- Yes — this is exactly where a duration-only risk model breaks, and where a real fixed-income algorithm needs convexity too.
- What's applied
- Duration predicts: −4.7807 × (−8%) = +38.2456%, i.e. $95.3075 → $131.7584. But the true price at 0% yield is just the sum of undiscounted cash flows: $142 — duration's linear approximation misses by over $10.
- Action
- An algorithm sizing a hedge using duration alone (as Ch. 1's illustrative DV01 example did, for a small parallel move) is implicitly assuming small moves. For large moves, or for options-embedded bonds, it needs the second-order term (convexity) or full repricing, not just the first derivative.
- Why
- Duration is a first derivative — the slope of a tangent line to a curved price-yield relationship. A tangent line is a good local approximation and a bad global one. The gap between the linear prediction ($131.76) and the true price ($142) is convexity, quantified.
Portfolio duration has one very convenient property the book highlights: it's simply the weighted average of the individual bonds' durations, weighted by their share of the portfolio — because duration is a first derivative, and first derivatives are additive. This is exactly why duration-matching (Ch. 1) works as a hedge: you don't need to model each bond separately, just match the weighted-average sensitivity.
Equity math — why stock valuation can't be as clean as bond valuation
Bonds have contractually guaranteed cash flows (barring default); stocks don't. A company owes shareholders nothing — dividends and capital gains happen only if the company chooses and is able to deliver them. The dividend discount model handles this by discounting all expected future dividends: P₀ = D₁/(1+r) + D₂/(1+r)² + ... to infinity — notably, this holds regardless of your own holding horizon, since a future buyer's price already embeds their own expected future dividends.
| Company | Dividend pattern | Formula | Fair value |
|---|---|---|---|
| ABC-NoGrowth | constant $6/yr forever | 6/0.15 | $40.00 |
| ABC-ConstGrowth | $6 next year, +5%/yr forever | 6/(0.15−0.05) | $60.00 |
| ABC-NonConstGrowth | $6 flat for 3 yrs, then +5%/yr | 3-yr PV + terminal value | $55.12 |
What this means: the exact same $6 near-term dividend produces fair values ranging from $40 to $60 — a 50% spread — purely from the assumed long-run growth rate, not the current payout. This is the book's own point: equity valuation is far more sensitive to a subjective growth assumption than bond valuation is to anything comparable, which is why "complicating the math further doesn't make it more accurate" — the uncertainty is in the input, not the formula.
Currency quote conventions — a real, recurring source of errors
Unlike a barrel of oil, currencies are quoted both ways — GBP costs USD, and USD costs GBP, and the two are just reciprocals of each other. Convention varies by currency: the euro and former-Commonwealth currencies (AUD, NZD) are typically quoted in American terms — USD per unit of that currency. Most others (CHF, JPY, HKD) are quoted in European terms — units of that currency per USD. The book's specific warning: a "cable rate" is often written GBP/USD followed by a number like 1.65 — which actually means 1.65 USD per GBP, not 1.65 GBP per USD, despite the ordering. Getting this backwards is a real, recurring, entirely avoidable error.
Before moving to Chapter 3
- 1
Pick a real bond you can find data for (coupon, maturity, yield) and compute its Macaulay and modified duration by hand, the way the book's Table 2.5 does — then check it against whatever duration figure your broker or a bond screener reports.
- 2
Revisit Ch. 1's duration-matched pairs trade with real durations from step 1 instead of the illustrative $180,000/bp DV01 figure — recompute the parallel-move vs. curve-steepening P&L with real numbers.
- 3
Pick a stock you follow, and compute its fair value under two different long-run growth assumptions (e.g. 3% vs. 6%) using the constant-growth formula — see how sensitive the "fair value" really is.
- 4
Chapter 3 moves to fixed income securities themselves (money markets, government/corporate bonds, mortgages) — the instruments this chapter's math actually gets applied to.
Duration and DDM as the backbone of two algo strategy types
This chapter's two big tools map onto two different systematic-strategy families:
Duration is the actual risk metric a rates/bond algorithm hedges on — Ch. 1's duration-matched pairs trade, now with a real formula and a worked table instead of an assumed DV01.
A systematic strategy that estimates "fair value" for a stock and trades deviations from it needs exactly the dividend discount model's structure — and needs to know how sensitive that fair value is to its own growth assumption, since a small assumption change swung fair value from $40 to $60 in the book's own example.