[R] the problem of buying and selling
Zhang Weiwu
zhangweiwu at realss.com
Sat Sep 14 06:01:59 CEST 2013
I own a lot to the folks on r-help list, especially arun who answered every
of my question and was never wrong. I am disinclined to once again ask this
question, since it is more arithmatic than technical. But, having worked 2
days on it, I realized my brain is just not juicy enough....
Here is the problem.
Trust not for freedom to the Franks---
They have a king who buys and sells.
- Lord Byron: The Isles of Greece
Suppose the French King commands you to buy and sell, and tells you only to
deal if the profit is higher than 2%. Question: how much quantity will be
dealt, and what is the actual profit? In fact, the King wants to see the
relationship between his minimum-profit requirement and your result, in
order to better his decision.
Let's look at the input data - a dump of which is attached to this mail.
Column 1 is the price of the market where you buy goods from, column 2 is
the quantity of goods that is being sold at that price.
Column 3 is the price of the market where you sell goods to, column 4 is the
quantity the buyers willing to buy at that price.
> cbind(t(to_buy_from), t(to_sell_to))
[,1] [,2] [,3] [,4]
[1,] 61.7050 190 63.170 2500
[2,] 61.7500 29 63.150 799
[3,] 61.8050 166 63.110 500
[4,] 61.8950 166 63.060 10000
[5,] 61.9450 166 63.020 7840
[6,] 61.9805 6150 62.995 2000
[7,] 62.0000 3069 62.930 2000
[8,] 62.0600 166 62.860 10811
[9,] 62.1100 166 62.780 18054
[10,] 62.1450 166 62.755 9000
[11,] 62.1750 166 62.690 10960
[12,] 62.2250 166 62.635 100
[13,] 62.2450 166 62.585 2380
[14,] 62.2720 100 62.550 2119
[15,] 62.2830 4000 62.525 108091
[16,] 62.2875 100 62.505 2000
[17,] 62.2955 100 62.485 816
[18,] 62.3250 307 62.435 600
[19,] 62.3800 2906 62.400 300
[20,] 62.3940 1969 62.375 4611
[21,] 62.4250 166 62.355 5111
[22,] 62.4505 2000 62.335 1969
[23,] 62.4700 259 62.315 500
[24,] 62.4755 50 62.250 5142
[25,] 62.4800 166 62.165 660
[26,] 62.4935 305 62.115 2428
[27,] 62.4975 7786 62.085 779
[28,] 62.4995 50049 62.050 12811
[29,] 62.5045 914 62.015 192
[30,] 62.5150 1110 61.975 1200
[31,] 62.5285 400 61.895 40000
[32,] 62.5500 6352 61.835 100
[33,] 62.5750 9 61.775 133
[34,] 62.6000 394 61.750 7723
For the simpliest case, if the King had commanded that the minimum profit
should be 2.3742%, which is equal to 63.170/61.7050 (look at the first row),
then you can easily project that 190 quantity of goods will be dealt (the
minmum of [1,2] and [1,4]), and that the actual profit is 2.3742%.
If the king, however, has commanded that a deal should only be carried out
if the profit is higher than 2%, the calculation will be more complicated. I
don't know the right method, but I can demonstrate the wrong method and
explain why it is wrong.
The wrong approach is the following:
The idea is to write a function that asks how much volume (total quantity)
you want to deal, and returns the profit. This generates a relationship
between volume and profit, and with interpolation you can get the volumen
for any given minimum-profit requirement.
revenues <- function(open_orders, volumes) {
# calculate revenue using a list of open orders and desirable "volumes" of goods
# expecting volumnes as a vector, to test the revenue (total amont of money)
# for each volume (total amount of goods to deal) in the 'volumes'
volume <- sapply(1:length(open_orders[2,]),
function(x) { sum(open_orders[2,1:x])})
revenue <- sapply(1:length(open_orders[2,]),
function(x) { sum(open_orders[1, 1:x] * open_orders[2,1:x])})
i <- findInterval(volumes, c(0, volume))
c(0, revenue)[i] + c(open_orders[1,], 0)[i]*(
volumes - c(0, volume)[i])
}
data.frame(volume = volumes, profit = revenues(to_sell_to, volumes) /
revenues(to_buy_from, volumes) - 1)
With the above routine, let us test the profit with the following volumes:
> volumes = c(10, 100, 500, 1000, 5000, 10000, 30000, 50000, 70000, 90000)
And the result:
> data.frame(volume = volumes, profit = revenues(to_sell_to, volumes) /
+ revenues(to_buy_from, volumes) - 1)
volume profit
1 10 0.023741938
2 100 0.023741938
3 500 0.022424508
4 1000 0.020974612
5 5000 0.018972785
6 10000 0.018087976
7 30000 0.012223652
8 50000 0.009288480
9 70000 0.007729286
10 90000 0.006204251
So, by looking up the table, if the king requires minimum profit of 2%, the
volume (total quantity) of goods being dealt should be a bit more than 1000.
This answer is inexact, but our French King should get by with it. After
all, he remembers nothing more than the number of digits.
Now let's look at why it is wrong. This answer is, actually, correct, but
the method won't hold.
Suppose our greedy King asks what volume should be deal if he requires
ANY deal should be done as long as there is a tiny bit of profit to be made,
then, according to our lookup-table, we need to deal 90000 volume of goods,
which is about all the goods you can buy from the market (look at the
to_buy_from vector, the sum of all goods is some 90000). Now look at the
bottom rows:
> cbind(t(to_buy_from), t(to_sell_to))
[,1] [,2] [,3] [,4]
[1,] 61.7050 190 63.170 2500
[2,] 61.7500 29 63.150 799
[3,] 61.8050 166 63.110 500
...
[31,] 62.5285 400 61.895 40000
[32,] 62.5500 6352 61.835 100
[33,] 62.5750 9 61.775 133
[34,] 62.6000 394 61.750 7723
It suggests that if you buys up the whole market, the last a few hands of
deals are perhaps not profiting at all - they are losing money - since the
price you buy may be higher than the price you sell - consider for example
buying at 62.6000 and selling at 61.750. This can be verified.†
So what is the right approach? I don't know. I exhused my brain. Perhaps you
can shed some lights.
And in case you wonder why I do the calculation in R: that's because I am
studying hundreds of markets, having calculation in one place and statistics
in another is not convenient, besides noone said R isn't good for
calculation.
--
† To verify:
When you buy up the whole market, and sell all goods, the worse price you
sell at will be 62.525, at [15,3], the point where the market-to-sell-in
volume reaches that of the market-to-buy-from, this is lower than the
price you buy at 62.6000 ([34,1]), indicating that you are, indeed, in
this last hand of deal, buying at a higher price than you sell it. The
lose is small, but can be much wore with a different set of data.
-------------- next part --------------
to_sell_to <-
structure(c(63.1699981689453, 2500, 63.1500015258789, 799, 63.1100006103516,
500, 63.060001373291, 10000, 63.0200004577637, 7840, 62.9949989318848,
2000, 62.9300003051758, 2000, 62.8600006103516, 10811, 62.7799987792969,
18054, 62.7550010681152, 9000, 62.689998626709, 10960, 62.6349983215332,
100, 62.5849990844727, 2380, 62.5499992370605, 2119, 62.5250015258789,
108091, 62.5050010681152, 2000, 62.4850006103516, 816, 62.435001373291,
600, 62.4000015258789, 300, 62.375, 4611, 62.3549995422363, 5111,
62.3349990844727, 1969, 62.314998626709, 500, 62.25, 5142, 62.1650009155273,
660, 62.1150016784668, 2428, 62.0849990844727, 779, 62.0499992370605,
12811, 62.0149993896484, 192, 61.9749984741211, 1200, 61.8950004577637,
40000, 61.8349990844727, 100, 61.7750015258789, 133, 61.75, 7723
), .Dim = c(2L, 34L))
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