How can we optimize this code using vectorization?

% initialize temporary array to store cumulative return factors

cumRetFact = zeros(13);
cumRetFact(13) = 1; % the index starts one year and one month ago

% build index forward using returns from the input variable 'returns'

for index = 12:-1:1
    cumRetFact(index) = cumRetFact(index + 1,1) * (1 + (returns{index ,2})/100);
end

For the purpose of performance optimization, it is better to replace the above "for loop" with the following piece of code:

cumRetFact(1:12,2) = cumRetFact(2:13,1).*((cell2mat(returns(1:12,2))/100)+1)

If not, how can we optimize the above "for loop" by using vectorization?

Answer 1

[Geoff Hayes]

Akash - if I understand your above code correctly, you may be able to use the cumprod function which can return the returns the cumulative product of an array starting at the beginning of the first element. For example (from the given link), if

 A = [1 2 3 4 5]';
 cumprod(A)

 ans =
     1
     2
     6
    24
   120

Suppose then that we extract the second column from your returns cell array, dividing each element by 100 and adding 1, as

 A = 1 + cell2mat(returns(1:12,2))/100;

This means that your above loop becomes

 for index = 12:-1:1
    cumRetFact(index) = cumRetFact(index + 1,1) * A(index);
 end

If we consider each index, this means that (the below isn't MATLAB code)

 cumRetFact(13) = 1
 cumRetFact(12) = cumRetFact(13)*A(12) = 1*A(12)
 cumRetFact(11) = cumRetFact(12)*A(11) = 1*A(12)*A(11)
 cumRetFact(10) = cumRetFact(11)*A(10) = 1*A(12)*A(11)*A(10)
 .
 .
 cumRetFact(1)  = cumRetFact(2)*A(1)   = 1*A(12)*A(11)*A(10)*..*A(2)*A(1)

This seems to suggest that you can use cumprod in order to update cumRetFact as

 cumRetFact          = zeros(13,1);
 cumRetFact(13)      = 1;
 A                   = 1 + cell2mat(returns(1:12,2))/100;
 cumRetFact(12:-1:1) = cumRetFact(13)*cumprod(flipud(A));

Note that we are updating cumRetFact in the order 12:-1:1 (instead of the usual 1:12) just to keep it more similar to your for loop. That means that we have to flip our column A in the up and down direction (so reverse the elements) using flipud.

For example,

 returns = cell(12,2);
 for k=1:12
    returns{k,2} = randi(255,1,1);
 end;

The above initialized returns to

 returns = 

    []    [178]
    []    [ 81]
    []    [243]
    []    [  9]
    []    [112]
    []    [ 98]
    []    [196]
    []    [203]
    []    [ 48]
    []    [125]
    []    [114]
    []    [165]

We then extract the second column as

 A = 1 + cell2mat(returns(1:12,2))/100;

so that A is

 A =
         2.78
         1.81
         3.43
         1.09
         2.12
         1.98
         2.96
         3.03
         1.48
         2.25
         2.14
         2.65

THis means that cumRetFact can be initialized to

 cumRetFact          = zeros(13,1);
 cumRetFact(13)      = 1;
 cumRetFact(12:-1:1) = cumRetFact(13)*cumprod(flipud(A));

and so

 cumRetFact =

        13375
         4811
         2658
       774.94
       710.95
       335.35
       169.37
        57.22
       18.884
        12.76
        5.671
         2.65
            1