Formula sheet

p

the win rate, as a share (0.9 for 90%); 1 − p is the loss rate

W

the average win, in R

L

the average loss, in R, as a positive size (10, not −10)

E

the expectancy: the average result per trade, in R

n

the number of trades

p*

the break-even win rate: the p at which E = 0

P_n

the price after n steps; P₀ is where it started

x₁ … x_n

the steps, each +s or −s

s

the step size

n

the number of steps (days)

d_n

the typical distance from the start after n steps: the square root of the average squared distance (lesson A9 gives it its textbook name, the standard deviation)

p

the win rate, as a share (0.5 for 50%); 1 − p is the loss rate

W

the average win, in R

L

the average loss, in R, as a positive size (1.1, not −1.1)

c

the cost per trade, in R

E_net

the expected value per trade after costs, in R

n

the number of trades

P(…)

the chance of what is in the brackets, from 0 (never) to 1 (always)

A, B

two independent events

p

the chance of one event, as a share: ½ for heads, 0.6 for a trading rule that wins 60% of its trades

k

how many in a row

p^k

p multiplied by itself k times: (½)³ = ½ × ½ × ½ = ⅛

n

the number of flips (or of trades that win or lose 50–50)

k

a streak length

log₂ n

the power of 2 that gives n: 2¹⁰ = 1,024, so log₂ 1,024 = 10

σ_total

the standard deviation of the total: how far luck typically puts it from the mean (σ is the Greek letter sigma)

σ

the standard deviation of one trade. A trade that is always +1 or −1 lands exactly 1 from its average of 0, so its σ is 1

n

the number of trades (pegs) added up

x

the result you are judging

μ

the mean of the no-edge bell: 0 here (μ is the Greek letter mu)

σ_total

the standard deviation of the total, from the last step

t

a draw from Student's t distribution with ν degrees of freedom

ν

"nu", the degrees of freedom, above 2: small for fat tails, large for close to a bell

σ

the standard deviation the days should have (1% in this lesson)

r

the day's return, with standard deviation σ

x

one return; there are n of them

μ

"mu", the mean (average) of the returns

σ

their standard deviation (given in this lesson's examples; a later lesson, Volatility, works it out)

z

the return's distance from the mean, in standard deviations

K

the excess kurtosis: 0 for a bell, above 0 for fatter tails

P₀, P₁

the price, or the account, at the start and at the end of one period (a day, or a trade)

P₁ ÷ P₀

the growth factor of that period

r

the simple return, as a decimal: 0.5 is +50%

ℓ

the log return of one period (ℓ is a handwritten l, for log). For small moves ℓ ≈ r: a +1% day (r = 0.01) has ℓ = 0.00995

ln

the natural logarithm: the ln key

ℓ₁ … ℓ_n

the log returns of n periods in a row

ℓ_total

the log return from the first price, P₀, to the last, P_n

e^x

the eˣ key, which undoes ln (e is a fixed number, about 2.718)

N

the number of shares bought

P_buy

the buy price

P_stop

the stop: the price at which the trade will sell if it goes wrong

risk

the dollars lost if the price falls to the stop and the trade sells there

result

what the trade made or lost in dollars: what came back minus what went in

result in R

the R-multiple: +2R made twice the risk; −1R lost exactly the risk

x₁ … x_n

the numbers, for example daily returns; n is how many there are

μ

"mu", their mean (average): add them up, divide by n

σ²

the variance, in squared units

σ

"sigma", the standard deviation, in the numbers' own unit

s

the sample standard deviation, for a few numbers standing in for many

σ_day

the daily volatility

252

trading days in a year, the count most people agree to use; √252 ≈ 15.87

σ_year

the yearly (annualised) volatility

error

the standard error: how far a simulated answer typically lands from the answer that endless runs of the same model would give

N

the number of runs

σ

the standard deviation of one run's score (σ is the Greek letter sigma); for a run that scores 1 or 0 it is √(p × (1 − p))

p

the chance of a hit, as a share: 0.518 for 51.8%

σ_average

the standard deviation of a sample's average: how far luck typically puts the average of n trades from the true average (textbooks call it the standard error)

σ

the standard deviation of one trade's result (σ is the Greek letter sigma)

n

the sample size: the number of trades in the sample

p

the true win rate, in percent

n

the sample size: the number of trades in the sample

100 ÷ √n

the margin of error, in points: how far either side of p the win rates of about 95 samples in 100 land (± means that far above and that far below). A rule of thumb for win rates from about 30% to 70%

x₁ … x_n, y₁ … y_n

the paired numbers, one pair a day; n is how many pairs

μ_x, μ_y

"mu", the average: their averages

s_x, s_y

their sample standard deviations: how far each typically lands from its average, dividing by n − 1

cov

the covariance: the average product of the two distances from the averages, dividing by n − 1

ρ

"rho", the correlation: from −1 to +1; it measures a straight-line link only

σ₁, σ₂

"sigma", the two assets' volatilities: the standard deviations of their returns

ρ

the correlation of their returns

σ_mix

the volatility of a mix holding half of each, if ρ is the same throughout

n

the number of trades in the window

p₁ … p_n

their prices, first to last: p₁ is the first trade's price, p_n the last one's

max, min

the highest and the lowest of the prices in the brackets

O, H, L, C

the window's open, high, low and close

a

the ask: the lowest price a waiting seller accepts, which you pay to buy now

b

the bid: the highest price a waiting buyer pays, which you get to sell now

s

the spread, per share

m

the mid, halfway between the bid and the ask

q

the number of shares bought and then sold

market order:
  fill now, at the other side's best price

limit order at L:
  if the other side's best price is L or better:
      fill, at L or better
  else: wait (it may never fill)

stop order at S:
  if the price has reached S:
      become a market order
  else: wait

the other side's best price

for a buy, the best ask; for a sell, the best bid

L

the limit price. "L or better" means L or lower for a buy, L or higher for a sell

S

the stop price. "Reached S" here means the best bid falls to S or below for a sell stop, or the best ask rises to S or above for a buy stop

fill

the price your order actually traded at

intended

the price you planned on: for a stop, the stop price; for a limit, the limit; for a market order, the best price on offer (the ask for a buy, the bid for a sell) when you decided. For a buy a positive slippage is the worse price; for a sell, a negative one

shares

how many shares the order traded

r₁ … r_n

each period's return as a decimal: +10% is 0.10, a 10% fall is −0.10

n

the number of periods (months, in the lesson's example)

L

the fall, as a decimal: a 30% fall is 0.30

G

the gain that gets back to the start, as a decimal: 1.00 is +100%

g

the geometric mean: the growth per period you actually get

end ÷ start

how many times the account multiplied over all n periods

^{1 ÷ n}

the n-th root: the number that, multiplied by itself n times, gives end ÷ start (on a calculator, the x^y key with y = 1 ÷ n)

a

the arithmetic mean: the plain average return per period

s

the swing: how far a typical period's return sits from the average. Its textbook name is the standard deviation (lesson A8, Volatility)

s²

s multiplied by itself

≈

about equal: the rule is an approximation, close when the swings are small

PV

the present value: the amount today

FV

the future value: the amount at the later date

r

the interest rate per year, as a decimal: 5% is 0.05

t

the number of years between today and the later date

(1 + r)^t

(1 + r) multiplied by itself t times: 1.05³ = 1.05 × 1.05 × 1.05 = 1.157625 (on a calculator, the x^y key)

P

the bond's price today

C

the coupon, paid at the end of each year

N

the face value: the amount repaid at the end, with the last coupon

T

the number of years left, so the number of payments

y

the yearly rate: the going rate when you work out a price; the yield when the price is given

n

the number of trades

g

the gross edge: the average profit of one trade before costs

p

the win rate: the share of trades that win

c

the cost of one trade, paid win or lose. At c = g the net is $0: the break-even cost

σ_total

luck: the standard deviation of a trader's total, how far it typically lands from the average net total

σ

the standard deviation (sigma) of one trade's result. For a trade that wins or loses the same amount, σ is about that amount (in the lesson's example, $100)

W, L

what a win makes and a loss takes, before costs

p*

the break-even win rate. After costs a win makes W − c and a loss takes L + c. Break-even: p × (W − c) = (1 − p) × (L + c); the c's cancel in the sum, (W − c) + (L + c) = W + L, so p* = (L + c) ÷ (W + L)

V_t

the account's value at step t: after month t, or after trade t

V₀

the account's starting value, before the first step, when there is one: the €10,000 before trade 1

n

how many steps there are

peak_t

the peak at step t: the highest value so far, the starting value and V_t included

DD_t

the drawdown at step t, as a decimal: 0.20 is 20%. It is 0 at a new peak

MDD

the maximum drawdown over all n steps

k

how many times as long as the backtest the trading after it lasts: 750 trades after a 250-trade backtest is k = 3. At k = 1 the chance is 1 ÷ 2: a coin flip

k + 1

the number of equal stretches, the backtest included; each is equally likely to hold the deepest drawdown

≈

about equal: after the backtest, drawdowns that run from one stretch into the next push the chance a little higher

V_t

the account after trade t, with no money added or taken out; V₀ is its starting value

V_t−1

the account before trade t

R_t

the result of trade t in R: −1 for a trade sold at its stop, +2 for one that made twice its risk

A

the fixed amount: the money risked on every trade

f

the fixed fraction: the share of the account risked on every trade, as a decimal: 2% is 0.02

N

the number of shares to buy, rounded down to a whole number so the trade never risks more than f × V

V

the account before the trade

f

the share of the account to risk on the trade, as a decimal: 1% is 0.01

f × V

the trade's risk in money

P_buy

the price paid for one share

P_stop

the stop, below the buy price: the price at which the trade is sold if it goes wrong

P_buy − P_stop

what one share loses if the trade is sold at its stop, with no gap and no cost

k

losses in a row, counted from the round's first 1R stake

L

what those k losses lose together, in R

P

the chance that a round starts with k losses in a row, on a fair coin

B

the account at the start of the round, in R

r

the rungs the account pays for in full

P_broke

the chance a round ends broke: its r rungs and the all-in trade all lose. (If B is exactly 2^r − 1, nothing is left for an all-in trade and P_broke = (½)^r.)

s

the stake of one trade, in R

V₀

the account's starting value

M

the maximum loss, in money

peak

the account's highest value so far, its starting value included; it never moves down

B

the account's value when the day began

D

the daily loss limit, in money

line

breached when the account, valued after every finished trade, is at or below it

P

the share of accounts that reach the target first, as a decimal (0.75 is 75 in 100), for a trading rule with no edge and no other way to stop

T

the gain that reaches the target, in money

M

the loss that reaches the maximum loss, in money

r

the result of one trade, in dollars (not a percentage)

S

the position size: dollars put into the trade

P_buy, P_sell

the price at the buy and at the sell

c

the costs of that trade

A₀

the account at the start

r₁ … r_k

the first trade's result, the second's, and so on up to the k-th

A_k

the account after k trades: one point of the account line

x, y

the paired numbers, n pairs (here the market's and the stock's returns)

x̄, ȳ

their averages

cov(x, y)

the sum of (x − x̄)(y − ȳ), divided by n − 1

var(x)

the sum of (x − x̄)², divided by n − 1: the square of x's standard deviation

b, a

the slope and the intercept of the least-squares line

ŷ

"y-hat", the line's value, a + b × x; so y − ŷ are its misses

y − ȳ

the distances from y's average: the flat line's misses

R²

for a least-squares line with an intercept, on the data it was fitted to: from 0 (no better than the average) to 1 (every dot on the line); with one input it equals the correlation squared

S

the price of the underlying on delivery day (forward) or at expiry (option)

F

the forward's agreed price

K

the option's strike: the price at which it lets its buyer buy (call) or sell (put)

premium

what the option's buyer paid the seller up front

max(a, b)

the larger of the two numbers

s, j

states: the situations that decide what can happen next

E_s

the expected waiting time from state s: the average number of tries still needed

p_s→j

the chance that one try takes you from state s to state j

Σ_j

add up over every state j

p

the chance of the event on each try