I have a question about how Lagrangian multipliers are applied in the exams and in the example questions in the CT8 notes. The covariances (in %%) and the expected values (in %) are both written down as integer values when writing out the Lagrangian function. I.e 12%% and 12% are both written down as 12 in the W function. I cant understand this, fair enough if 12%% was 0.12 and 12% was 12, but why are the both written in the same order of magnitude when they are off by 2 orders of magnitude? Sorry if I asking something which is in some fashion obvious but I cant see why this works.
Answer..... Answer to the above if anyone is interested, as supplied by an ActEd tutor (David Hopkins). Using Question 3.9 from Chapter 3 as an illustration. Expected returns are just "straight percentages" and an expected return of 4% is the same as 4/100 = 0.04. However, variances and covariances are "squared" quantities, derived by multiplying two percentages together. So they have units of "%%" and a variance of 4%% is the same as 4/100 /100 = 0.0004, ie you divide by 100 twice. Note that the square root of 4%%, ie the standard deviation, is 2%, which is the same as saying that the square root of 0.0004 is 0.02. Standard deviations are "straight" percentages. So when you are doing this type of question, there are two approaches you can use. You can either (a) convert everything to decimals or (b) work in terms of percentage units. With method (a), the Lagrangian function will be: W = 0.0004 xA^2 + 0.0036 xB^2 + 0.0018 xA xB - lambda (0.04 xA + 0.08 xB - EP) - mu (xA + xB - 1) Here the value of W is a "decimal" value. With method (b) -- as used in the notes -- the Lagrangian function will be: W = 4 xA^2 + 36 xB^2 + 18 xA xB - lambda (4 xA + 8 xB - EP) - mu (xA + xB - 1) Here the value of W is expressed in %% units (since it is a squared quantity). The values of lambda and mu using the two methods will differ by a factor of 100, but we're not really interested in their actual values anyway. The two methods are really just like measuring a new carpet. You can either measure in centimetres and say that your room is 400cm by 500cm, so you need 200,000cm^2 of carpet, or you can measure in metres and say that the room is 4m by 5m (ie each number is divided by 100) and you need 20m^2 (ie 200,000 divided by 10,000). Often using % units makes the numbers more manageable -- you don't get so many zeros in the calculations! -- but you can use either method in the exams.
