For the variance- corvanriance method: let say 5% confident intervel,
VAR = expected return of the portfolio - 1.65( sd of the portfolio)
this means that the VAR is directly proportional to expected return and indirectly proportional to the s.d , seems not logical… please explain.
it is indirectly proportional to the mean and directly proportional to the SD.
reason -> VaR is a Loss Number.
Say Mean=10%, SD=10%
Var = 10-1.65 * 10 = -6.5%
on a 10M portfolio = this is 650K Loss.
Say mean becomes 11% - . Var Reduces to 550K - so indirectly proportional.
SD increases - VaR becomes a bigger Loss number. so directly proportional to the SD.
Just to clarify, we should be saying inversely here rather than indirectly.
Only if you want to be accurate.
Prob 23 on Risk Management Study session 14 Reading 26 ,2015
–VAR = μ_p_ – Z σ_p_
So the relation is inverse .
The increase in expected return would result in a lower calculated VAR (smaller losses).
So higher expected return~lower calculated VAR~Smaller losses.
Conceptually,that should be fine.
Of Course,higher S.D means higher volatility of returns which means higher risk and higher prospect of losses.
HOWEVER, in the readings, VAR = expected return on the portfolio - 1.65( S.D. of the portfolio) .
Looks counterintuitive,as Jess has concluded.Higher expected return should reduce the likelihood of loss.
But the above equation would mean higher ER would lead to higher VAR,mathematically…(SEE TABLE BELOW)
Besides,SD is a deduction for VAR number in this equation.Higher SD would mean higher negative result and hence higher VAR.However at very low SD(such that SD*Z
The only possible way this can be agreed with is,in future higher return will mean higher value at exposure,and higher amount subject to risk(VALUE,at risk).But VAR is a loss number.
@Jess,mathematically,keeping Z and S.D constant,the resulting number indeed increses as we increase expected return,if we take the readings as the basis:
ER Z S.D. Result 10 1.65 10 -6.5 12 1.65 10 -4.5 14 1.65 10 -2.5 16 1.65 10 -0.5 18 1.65 10 1.5 20 1.65 10 3.5 22 1.65 10 5.5 24 1.65 10 7.5 26 1.65 10 9.5 28 1.65 10 11.5 30 1.65 10 13.5 32 1.65 10 15.5
The result moves from -ve towards zero as Expected Return approaches z*standard deviation,and then starts giving a +ve result.
How to interpret a result of +15.5 using Expected Return-1.65 S.D.?The losses would be smaller,but VAR presumably cannot be a gain(-6.5 is a loss).We get similar results at different Z and SD levels,keeping them fixed and increasing the expected return.
MY UNDERSTANDING:As expected return increases,so would standard deviation,more return should come at higher ,not fixed, risk.Higher risk is compesated by higher return.
Am I correct?
Regret the error in pasting excel.Please read in the order
ER Z SD Result
Inverse relationship, ceteris paribus. Yep, I played that card.
For the context of the exam I believe you just have to assume the inverse relationship between VaR and S.D. Higher S.D. (risk) is lower VaR (value at risk). Dont think you have to “assume” that higher risk is compensated by higher expected rturn
thank you for the above reply
@ Analyst mamba,
(Higher correlation and hence)Higher S.D.would result in HIGHER (not lower) calculated VAR and ,larger losses.
That’s a Direct Relation.
Prob 23 asks you to ASSUME “Considered independently, and assuming that other variables are held constant”.And it asks you to explain the effect of increasing Expected Return on VAR.But keeping SD and Z value constant and increasing Expected Return would lead to a +ve value for VAR,as Expected Return value becomes > z*S.D.
This has been illustrated in excel(That is ,if you use the formula :Expected Return-Z*S.D. to calculate VAR).But conceptually that is difficult to comprehend.
Even if you use -VAR=Exp Return-z*S.D,you will start getting +ve values for Expected Return
The “assumption” of S.D. NOT REMAINING CONSTANT,as Expected return increases(or decreases),is an attempt to explain that positive result is unlikely, as Expected return becomes greater(smaller) than SD*Z value,and VAR becomes +ve.
The premise is,in real life,risk as measured by S.D., will NOT remain constant as Expected Return increases.
So presumably a +ve value for VAR will NOT result.
I was just checking if THAT is the explanation,or I am missing something.
And I would appreciate if this could be explained/confirmed by anyone on the forum.
Appreciate that discussing parametric var without plotting a few normal distributions is a huge undertake but does this need to be so complicated?
Not unless one chooses a confidence interval of (say) less than 50%, which, incidentally, wouldn’t be ‘confident’ at all i.e. would yield a var (defined as maximum loss expected within a given confidence level) that is far less useful as a measure of how much money can go down the drain than a 90% confidence interval.
For confidence intervals of 90% or above, monthly (or yearly) parametric var for a risky asset is bound to be a negative value. I.e. the higher the return one tries to acheive above what is delivered by the risk free asset, the higher the risk one is facing the higher the absolute var value (or amount).
Higher risk, here as elsewhere in CFA territory, can be graphically depicted (in comparison with lower risk) by a wider and squatter distribution of returns which on its left will extend into negative return territory beyond the reach of the distribution of a less risky asset.
Finding a risky asset whose parametric var @ 90% confidence is positive would be tantamount to finding a free lunch.
@Band 6,
Agree.
But I got VAR>0 WITHOUT CHANGING THE CONFIDENCE LEVEL,i.e.z=1.65throughout(even S.D.was kept constant as per Prob 23 requirement).That’s why i thought of “Considered independently, and assuming that other variables are held constant”not being realistic ,and logical,and tried to explain that to myself in terms of risk-return relationship.
Since there is no errata from CFA Institute,I guess i should be wrong and am missing something here.
In any case, we’ve got the basic relationship right -that serves the exam requirement.
Var against Expected return-Inverse
Var against Std Deviation-Direct.
You are welcome.
I don’t think you are.
Your are trying to figure out how the “Considered independently, and assuming that other variables are held constant,…" can be ‘realistic’.
It isn’t. In real life none would ‘consider independently’ risk and return to figure their effect on var.