CH 8 return on capital page 9

[kiki]

Hi,

Can someone please help me understanding the following core reading about coherence of risk measures or statistics :

"Linear homogeneity" , it suggested that if the constant were to represent a bigger portfolio of risks then the linear homogeneity condition would not allow for any extra diversification benefits of writing a bigger book of business"

is that means if the portfolio is expanding by 50% , then the risk should by increased by 50% ?

if it is the case , is that counter intuitive as writing more risks clearly will increase the diversification benefit ? or is that means we can allow the diversification benefit via adjusting the volatility parameter eg decrease the CoV for attritional loss ?

thank you

 

[CapitalActuary]

Your interpretation is basically right. In practice if a book of business gets significantly bigger you would probably change the distribution you're using to model the portfolio, by doing something like decreasing the CoV, to account for the fact the book is likely more diversified when it's bigger.

Note it's not necessarily the case that a bigger portfolio means more diversification benefit though. For example, if you add risk to the portfolio which is 100% correlated with the existing risk in the portfolio, then theoretically the book isn't any more diversified. In this case it if you use 'X' to model the original portfolio it would be reasonable to model the new portfolio using '2 * X'.

It would actually be pretty unintuitive if positive homogeneity did not hold - that's why it's a good property for a risk measure, and why it's a property of 'coherent' risk measures. If you hold N shares in a company, then the risk should intuitively be N times as big as having just one share. i.e. the risk scales linearly with position size.

The weird thing, as you hit upon, would be to choose to model a more diversified portfolio as a straightforward multiple of a less diversified portfolio. That's not really to do with the risk measure though - that's an issue to do with your choice of modelling distribution.

On another practical note: one usually models risks in reasonably homogeneous groups. For example you would model 'US employer's liability', 'marine', and 'Japan property' portfolios as separate random variables. If I use the random variable E to model my US employer's liability losses and the portfolio gets 10% bigger, it might be reasonable to assume it's that not much more diversified (because it's all US EL business, after all, and it's not a huge amount bigger). So I could model the new portfolio using the random variable 1.1 * E. Then yes, a coherent risk measure would tell me the risk was 1.1x as big.

 

[kiki]

Thank you , CapitalActuary

one more question in the next bullet point for coherence of risk measures or statistics
"translational invariance, adding a portfolio with a known , guaranteed outcome to an existing portfolio reduces the risk by the guaranteed outcome"

is that means if "a portfolio with a known , g'teed outcome" with risk X, and "an existing portfolio" with risk Y, when two adding together , the risk will be Y-X?

is that right by assuming the X and Y is negatively correlated ?

thank you again

 

[CapitalActuary]

I think you’re confusing terms here.

You’ve said that X and Y are the “risks” of two portfolios, i.e. the results of calculating a coherent risk measure on two different portfolios. But that means X and Y are both just real numbers. It doesn’t make sense to talk about the correlation between two real numbers.

Translational invariance says if I have a portfolio I model with random variable A, and it has risk R_A, and another portfolio with deterministic payout B, then my risk is R_A - B. Since B is deterministic it still doesn’t make sense to talk about the correlation between A and B.

Let’s think about why translational invariance should hold. I could have a portfolio paying out an amount B equal to R_A. Our risk measure says the risk on A is R_A, so if we know we’re getting paid R_A then that should perfectly cancel our risk. So our risk should be zero according to our coherent risk measure. Translational invariance makes sure this is true.

Alternatively, if we have our random payout A, and in addition we *know* we’re going to lose B (or equivalently, get paid -B), then clearly our risk should be the risk on A, plus the B we’ll lose. i.e. risk(A - B) = R_A + B. Then if you let C = -B, we recover translational invariance: risk(A + C) = R_A - C.

 

[kiki]

Hi,

I have re-read the conditions for "coherent" , somehow i feel two of the conditions are contradict to each other , may be i have missed something here

"Monotonicity" : if one portfolio is worth more than another , it cannot be riskier . -> it implies if one portfolio (y) is worth $50m , vs identical portfolio (x) worth $60m. then risk of x should be lower than risk of Y?

"linear homogeneity : scaling a portfolio by a constant , will change the risk by the same proportion
therefore for portfolio worth $50m (y) , when scaling by 20% -> $60m (ie portfolio x) , then the risk level of X should increase by 20%. but monotonicity suggest risk of x should be less than risk of Y? i am really confused

thank you so much for your time

 

[CapitalActuary]

These conditions are all about modelling the future returns of the portfolio as random variables. Your mistake is only thinking about the initial values of the portfolio.

If you have two portfolios worth $50 and $60 both with the same future return distributions in % terms, and these portfolios will only ever lose money (I.e. have negative returns) then the risk on the $60 portfolio has to be at least as high due to monotonicity, and linear homogeneity tells you it is in fact exactly 1.2x as high.

When you are talking about your two portfolios you’re only referencing their initial values and ignoring what the distributions of returns will be, which is the crucial part for measuring the risk.

 

[kiki]

Hi

I dont think i have explained my question clearly .

Monotonicity : according to the core reading , it is suggesting if one portfolio is more valuable than the other , the risk will be less (as the example after the core readying, Value (y) = $50m and Value(x) =$60m , then risk (Y) > risk (X)

it seems contradict to "linear homogeneity" , as it suggested if value(x) is greater than value(Y) by 20% , then the risk should be higher by 20% . thus risk(Y) < risk (X)

is there some typo ? or i missed something here?

 

[kiki]

a separate note about "translational invariance" , page 10, it mentioned about standard deviation / variance are risk measure , but not satisfy monotonicity. am i right to say neither StDev / variance satisfy translation invariance ?

for a portfolio modelled as random variable X ~ N(2,100) , with another portfolio Y with return $10 as deterministic return , when you simulate for 10000 sims, the combined portfolio Z (X+Y) , standard deviation still equal to 100 . it is contradict to translational invariance as R(X+Y) = 100 - 10 =90 ?

 

[CapitalActuary]

Hi

I dont think i have explained my question clearly .

Monotonicity : according to the core reading , it is suggesting if one portfolio is more valuable than the other , the risk will be less (as the example after the core readying, Value (y) = $50m and Value(x) =$60m , then risk (Y) > risk (X)

it seems contradict to "linear homogeneity" , as it suggested if value(x) is greater than value(Y) by 20% , then the risk should be higher by 20% . thus risk(Y) < risk (X)

is there some typo ? or i missed something here?

I don't know what the core reading says, but monotonicity does not say that if one portfolio starts more valuable than another then the risk for that portfolio is lower or higher.

Let X and Y be two random variables representing the profits and losses from two portfolios A and B respectively (equivalently, X and Y are the changes in the values of A and B over some time period). Monotonicity says that if X>=Y with probability 1, i.e. if the portfolio A almost surely makes more money (or loses less money) than portfolio B, then risk(X)<=risk(Y), i.e. portfolio A should be considered no more risky than portfolio B.

When you say value(Y)=$50m, I don't understand what you mean. Y is a random variable, it is not a single value.

If the initial value of A is $50m and the initial value of B is $100m, this doesn't tell us what the distributions of X and Y will be. It's the distributions of X and Y which matter. For example: say half of the time portfolios A and B make -10% returns (so with probability 0.5, X=-$5m and Y=-$10m) and half the time they make a +10% return (so with probability 0.5, X=+$5m and Y=+$10m). Here it's actually the case that Y=2X. Therefore linear homogeneity tells us risk(Y)=2*risk(X), i.e. portfolio B is twice as risky, which is intuitive as it has a larger initial value and the same % return distribution.

But monotonicity doesn't tell us anything here, because half of the time we have Y>X (when X=5m and Y=10) and half of the time we have X>Y (when X=-5m and Y=-10m).

Consider another example, where half of the time portfolios A and B make -10% returns (so with probability 0.5, X=-$5m and Y=-$10m) and half the time they make a 0% return (so with probability 0.5, X=Y=$0m). Then we have Y=2X again, so linear homogeneity tells us that risk(Y)=2*risk(X). This time we have X>=Y as well though, so monotonicity tells us that risk(X)<=risk(Y) for a coherent risk measure. I don't see a contraction here.

As I said in my initial post, I think the cause of the confusion is mixing up the random variables with the 'initial values' of the portfolios. You might also be assuming that risk(Z)>=0 for any Z, which isn't true. Consider getting paid a positive guaranteed cash amount of 'z' - this will have risk of -z according to translational invariance.

 

[CapitalActuary]

a separate note about "translational invariance" , page 10, it mentioned about standard deviation / variance are risk measure , but not satisfy monotonicity. am i right to say neither StDev / variance satisfy translation invariance ?

for a portfolio modelled as random variable X ~ N(2,100) , with another portfolio Y with return $10 as deterministic return , when you simulate for 10000 sims, the combined portfolio Z (X+Y) , standard deviation still equal to 100 . it is contradict to translational invariance as R(X+Y) = 100 - 10 =90 ?

Yes you're correct, neither standard deviation nor variance satisfy translation invariance or monotonicity.

If X~N{2,100} then the standard deviation is 10 not 100. Otherwise I agree with your example as proof that standard deviation does not satisfy translation invariance.

 

[CapitalActuary]

Also, a tutor (the man, the myth, the @Ian Senator) can probably chip in with more authority here, but I don't think this is a great topic to spend too much time on for the purpose of passing SA3. My impression is that there are many other topics where there is better "bang for buck" available for your study time, so I wouldn't get too stressed if this one isn't the most coherent (pardon the pun).