Dr. Van Tharp’s book introduces the concept of “R multiples” to represent the characteristics of a trading/investment system.

(1) To rephrase the concept mathematically, R multiples can be derived as following:

Gain = f * (Wp * Ws – Lp * Ls)

where,

f = frequency of trades, trades/year

Wp/Lp = win/loss percentage

Ws/Ls = average win/loss size, Ls = R (i.e., Risk)

With, Lp = 1 – Wp and Ws = n * R, we got:

Gain = f * (Wp * n * R – (1 – Wp) * R)

Gain = f * (Wp * (n+1) – 1) * R

**Gain = f * M * R**

where,

n = Average win to loss ratio

M = (Wp * (n+1) – 1) = R multiples

(2) With above formula, R multiples can be used to explain the importance of frequent rebalancing.

- Using a mutual fund buy-and-hold model of 50 stocks in a good year, f = 6, n = 6.46, Wp = 65%, R = 0.4%, we have:

Annual return = f * M * R = 6 * (65%* (6.46+1) – 1) * 0.4% = 9.25%

- Instead of buy-and-hold stocks, if rebalancing frequently and without shooting for large n, f = 250, n = 1.5, Wp = 60%, R = 0.4%, we have:

Annual return = f * M * R = 250 * (60%* (1.5+1) – 1) * 0.4% = 50%

(3) We often see people comparing performance in turns of percentage without talking about risk. Comparing return percentage is meaningless if using different R.

- Higher R means higher drawdown.

For example, for a 50R system if drawdown is 10% when R = 1%, then drawdown is 100% when R = 10%. 100% drawdown is not a feasible system for most of people.

- Higher R means higher chance of ROR (risk of ruin), where ROR = [(1 – Edge)/(1+ Edge)]^(1/R)

For example, for the same 50R system with Edge = 20% (60% win, 40% loss):

Annual return = 50% and ROR = 0%, if R = 1%

Annual return = 500% and ROR = 1.73%, if R = 10%

(4) R multiples can also explain why high-frequency day trading can be very profitable. For example, f = 1000, n = 2, Wp = 40%, R = 0.5%, we have:

Annual return = f * M * R = 1000 * (40%* (2+1) – 1) * 0.5% = 100%

f = 1000 means around 5 trades a day, and Wp = 40% means the system only has to be right 4 out of 10 trades. For $200,000 account, n = 2 and R = 0.5% are equivalent to risk $1,000 per trade while shooting for $2,000 profit per trade on average.

f is one of the most important factors in any trading/investment system. The importance of f can also be illustrated from S&P500 index.

In 2007, if using buy-and-hold (f = 1), the return is 3.53% = (1468.36-1418.3)/1418.3. It is a gain of around 50 points even the index was up and down many times in the following graph:

Using zig-zag lines on the daily chart, we can see S&P 500 index has 30 zig-zag lines (f = 30) in the following graph, totaling around 1500 points. Even capturing one third of the points (500 points), the return is 10 times more than buy-and-hold.

In the following intraday chart, we can see S&P 500 can have 100 points up and down swing within a day. Only 5 trades a day is needed to make f = 1000.

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