Algebra

Oh, OK.  So either

a) It's a secret, or
b) no-one understands the question, or
c) all 30 people who have viewed this thread are as tick as me.

B = Back Odds
C = Back Stake
D = Lay Odds
(There is no E)
F = Lay Stake
G = Lay Liability
H = Net P/L (as a percentage of C)
R = Commission Rate

We can work out F, G & H from known values of B, C, D & R thus:
[1] F = B*C/D
[2] G = (D-1)*F
[3] H = (F-C)*(1-R)/C

What I want to do is determine D, F and G from known values of B, C, H & R.

Can it be done?

Assuming that your exit trade is to an equal profit on all outcomes as implied, then:

[4] (F-C)*(1-R) = (1-R) * C * ((B/D)-1)

We know that,

H = (F-C)*(1-R)/C

Therefore,

[5] (F-C)*(1-R) = H * C

Therefore from equations [4] and [5],

H * C = (1-R) * C * ((B/D)-1)
H/(1-R) = (B/D)-1
1+ H/(1-R) = B/D

[6] D = B/(1+(H/(1-R)))

As D is now known, F can be calculated using equation,

[1] F = B*C/D

As D and F are now known, G can then be calculated using equation,

[2] G =[D-1]*F


Example:

Assume that prior to a coin toss you back 'Heads' @ 2.2 for £100. Strangely you also know that your Net P/L expressed as a percentage of the back stake is 9.5%. The commission rate is 5%. Assume the stated commission rate to be inclusive of any customer discount rate. The applicable commission rate would be known to be 0% if H was negative if the bets were struck on this exchange.

B = 2.2
C = £100
D =
F =
G =
H = 9.5% (0.095)
R = 5% (0.05)

Therefore to find the lay price D, use equation [6],

D = B/(1+(H/(1-R)))
D = 2.2/(1+0.095/(1-0.05))
D = 2.2/1.1
D = 2

To find the lay stake F, use equation [1],

F = B*C/D
F = 2.2*100/2
F = £110

To find the lay liability G, use equation [2],

G =[D-1]*F
G = (2-1)*110
G = £110

Alternatively you could observe that,

(F-C)*(1-R) = H * C
(F-C) = (H*C)/(1-R)
F = C +(H*C)/(1-R)

Having calculated F, use that value to calculate D using equation [1] and then G using equation [2]. That would be boring though.

Early night?

If not, you're welcome.

jt45

You were correct in your assumption - (I should have explained that).

Yep - all tested in Excel and spot on!  That's perfect!

Many, many thanks.

Steve

(Keeping odd hours whilst the missus is away).