Markets Arithmetic Without a ScreenSome background helps

Twelve approximations that turn a question about a price into ten seconds of arithmetic — each one derived here, and each one checked against the exact answer so you can see how far off it is.

1. A basis point, on a notional, for a period

• A basis point is one hundredth of a percentage point: 0.01% = 0.0001.
• One basis point on 1,000,000 for a full year is 100. That single fact anchors everything else: scale the notional, then scale the time.
• Worked example. One basis point on 10,000,000 for three months: \(10{,}000{,}000 \times 0.0001 \times 0.25 = 250\).
• Why it matters: it converts a rate argument into money instantly. “They are two basis points wide on ten million for three months” is a disagreement about 500, and knowing that stops you from spending an hour on it.

2. Percent, and percent of percent

• A rate moving from 5.00% to 4.75% has fallen 25 basis points, which is a 5% reduction in the rate. Both statements are true and they are not the same statement.
• The ambiguity is exactly why basis points exist. Use them for changes in a rate, and percentages for changes in a value.
• The trap: “fees went up 1%” — from 0.5% to 0.505%, or from 0.5% to 1.5%? Say “ten basis points” and nobody has to ask.

3. The rule of 72 — and the rule of 114

• Money doubles in roughly \(72 / r\) years at \(r\) percent a year, because doubling needs \(\ln 2 \approx 0.693\) of continuous growth, and 72 is the numerator that best absorbs the discrete-compounding error across ordinary rates.
• Check. At 6%: the rule says 12 years; the exact answer is \(\ln 2 / \ln 1.06 = 11.90\). At 9%: the rule says 8 years, exact 8.04.
• Tripling needs \(\ln 3 \approx 1.099\), so the same trick gives \(114 / r\). At 6%: 19 years against an exact 18.85.
• Run it backwards for inflation. At 3% a year, prices double in 24 years — which is the honest way to read any long-horizon figure quoted in today's money. Why money grows and still buys less.

4. Duration: what a rate move does to a bond price

• Modified duration \(D\) is defined so that, for a small yield change \(\Delta y\):

• Worked example. A bond with \(D = 8.5\), yields rise 1%: the price falls about \(8.5 \times 1\% = 8.5\%\).
• The sanity check that catches most errors: duration is roughly, for an ordinary coupon bond, a bit less than its maturity. A ten-year bond with a duration of 15 is not a bond you have understood yet.
• Why a bond falls when rates rise, if the sign itself is not yet automatic.

5. DV01, in one multiplication

• DV01 is simply duration applied to one basis point:

• Worked example. \(D = 8.5\), price 100 per 100 face, notional 1,000,000: \(8.5 \times 1{,}000{,}000 \times 0.0001 = 850\) per basis point.
• Why the desk uses it instead of notional: 1,000,000 of two-year and 1,000,000 of thirty-year are not the same risk, and DV01 says so in one number. The rates desk talks in it constantly.
• The same trick in credit: CS01, the value of one basis point of spread, using spread duration in place of yield duration.

6. Convexity: the correction that matters on big moves

• Duration is the first term of a Taylor expansion. The second term is convexity \(C\):

• Worked example. \(D = 8.5\), \(C = 80\), yields up 1%: \(-8.5\% + \tfrac{1}{2}(80)(0.01)^2 = -8.5\% + 0.4\% = -8.1\%\).
• Read the sign. The correction is positive for an ordinary bond in both directions — it loses a little less than duration says and gains a little more. That asymmetry is worth paying for, which is why convexity has a price.
• And when it is negative — callable bonds, mortgage-backed securities, a short option position — the correction works against you on both sides. MBS is the standard example.

7. The at-the-money option, in one line

• Take an option struck at the forward, with rates set aside. Black–Scholes gives \(C = S[N(d_1) - N(d_2)]\) with \(d_1 = \sigma\sqrt{T}/2\) and \(d_2 = -\sigma\sqrt{T}/2\).
• For small \(x\), \(N(x) - N(-x) \approx 2\varphi(0)\,x\), and \(\varphi(0) = 1/\sqrt{2\pi} \approx 0.399\). Substituting \(x = \sigma\sqrt{T}/2\):

• Check. \(S = 100\), \(\sigma = 20\%\), \(T = 1\): the rule gives 8.00, Black–Scholes gives 7.97. At three months: rule 4.00, exact 3.99. At one month and 30 vol: rule 3.46, exact 3.45.
• What it buys you: an implied volatility from a price, in your head. A three-month at-the-money option costing 4 on a spot of 100 implies about 20 vol, because \(4 = 0.4 \times 100 \times \sigma \times 0.5\).
• Its limits: at the money, short-dated, no dividends, no rates. Away from the strike it degrades quickly. Say which one you are using. The exact version is the Black–Scholes calculator.
• The companion fact: an at-the-money option has a delta of about 0.5, so a small move in the underlying moves the option about half as much.

8. Volatility across time: divide by 16

• Volatility scales with the square root of time, and there are about 252 trading days in a year, so:

• So: divide annual volatility by 16 to get a typical daily move. A 16-vol index has typical daily moves of about 1%. A 32-vol one, about 2%.
• And multiply to go the other way. A market that has been moving 0.5% a day is running at about 8 vol, whatever anybody says it is.
• The one-sigma habit: a 2% day in a 16-vol market is a two-standard-deviation move. Saying that out loud is more useful than saying it was a big day. Volatility.

9. Forward points are arithmetic, not a forecast

• An FX forward is priced so that borrowing one currency and lending the other leaves nobody better off:

• Worked example. Spot 1.1000, quote-currency rate 4.5%, base-currency rate 2.5%, six months: \(1.10 \times (1 + 0.02 \times 0.5) = 1.1110\). Eleven hundred points, entirely from the rate difference.
• The point to be able to make: the forward is not the market's expectation of the future spot rate. It is today's spot plus the cost of carrying the position, and confusing the two is the single most common FX misunderstanding. FX forward.
• Cross rates are multiplication: EUR/USD at 1.10 and USD/JPY at 150 imply EUR/JPY of 165. If the quoted cross is not 165, one of the three is about to move.

10. Drawdown recovery is not symmetric

• To recover from a fall of \(d\), you need a gain of \(d/(1-d)\).

• The consequence people skip: a sequence of +50% then −50% is not flat. It leaves you down 25%, and averaging the two percentages gives zero, which is why an average return quoted without saying which kind of average is not a number at all.
• The arithmetic of drawdowns and reading a market number.

11. Fees, over a lifetime

• An annual fee \(f\) held for \(n\) years leaves you with roughly \(e^{-fn}\) of what you would otherwise have had.
• Worked example. 1% for 30 years: \(e^{-0.30} = 0.74\) — about a quarter of the final pot, gone. The exact figure for a 7% market is 57,400 against 76,100 on an initial 10,000, which is 24.5%.
• The reason it is so large is not that the fee is large; it is that it is charged on the whole balance every year, including on the gains that earlier fees already removed. What the fees actually cost and the fee-drag calculator.

12. Two shortcuts for valuing a stream

• A perpetuity is worth \(C/r\). A perpetual paying 4 a year, discounted at 5%, is worth 80 — which is also why a perpetual bond's price is roughly its coupon divided by the required yield, and why small yield changes move it so much.
• A discount factor: \(1/(1+r)^n\). For quick work, \(e^{-rn}\) is close and easier — at 5% over ten years it gives 0.607 against an exact 0.614, about one percent low. Where one percent matters, use the exact form; where it does not, you have the answer before anybody has opened a spreadsheet. IRR and NPV.

How to practise these so they are actually available

• One at a time, out loud, until it takes under ten seconds. An approximation you have to think about is not an approximation; it is a calculation with an error in it.
• Always say the units and the assumption. “About 850 a basis point, using a duration of eight and a half” is a professional answer. “About 850” is a guess that happens to be right.
• Check yourself against the exact tool. Every rule above has a calculator on this site that will do it properly; run each rule against the exact answer a few times and you will learn where it breaks down, which is the part that matters.
• Then throw the number away when precision is required. Nobody is impressed by a mental estimate defended past the point where a screen is available.