{"categories":["Risk Management"],"contentHtml":"

Exam Review

Chapter 2: Fundamentals of Probability

For a

𝑍=π‘Žπ‘‹+π‘π‘Œ

then

𝔼[𝑍]=π‘Žπ”Ό[𝑋]+𝑏𝔼[π‘Œ]

and for variance,

𝕍[𝑍]=π‘Ž2𝕍[𝑋]+𝑏2𝕍[π‘Œ]+2π‘Žπ‘Cov(𝑋,π‘Œ)

We can actually do this from a portfolio perspective, it’s the same idea. We have some weight

𝑍=𝑀1⋅𝑅1+𝑀2𝑅2

What’s the variance? 𝕍[𝑍]=𝑀1𝜎12+𝑀22𝜎22+2𝑀1𝑀2𝜎12

It’s the same in this exercise where we substitute the weights for the coefficients and the random variables for X and Y.

Another way to look at this problem is through matrix algebra.

We have a vector v,

𝑀⃗=[π‘Žπ‘]

We can rewrite 𝑍 as

𝑍=𝑀⃗𝑇𝑅⃗

Then the expectation and variance becomes

𝔼[𝑍]=𝑀⃗𝑇𝔼[𝑅]

This is nice because if you have a covariance matrix you don’t have to look at all the pairwise. The question gives us that this X and Y have covariance 0.35. Be mindful of the difference between covariance and correlation.

The formula for covariance is that

𝜎12=𝜌𝜎1𝜎2

so that’s where correlation comes into it. That 𝜌 is correlation.

We can then rewrite our problem into

𝑀′Σ𝑀

based on considering this as matrix.

If you $100 today, it can either go to $110 or $90. Up is 0.3, Down is 0.7. We’ll have the same increase and decrease amounts and probabilities at each step.

We’re actually interested in the value of

𝔼[𝑆2|𝑆0]

But in this case we have discrete outcomes, not like in GBM. If we have a random variable, it’s either going to be 120, 100, or 80. There is a probability assigned to each of these for a given 𝑆0.

We know that the probability for 120 is 0.3 * 0.3 or 0.09, for 80 it’s 0.7 * 0.7 = 0.49. For 110 we have to consider both the possibility of 0.3 * 0.7 up and down and 0.7 * 0.3 down and up, which becomes 0.42. And we can show that these all add to 1.

So how do we actually calculate probability? We think about it simply as

𝔼[𝑋]=βˆ‘βˆ€π‘₯β„™(𝑋=π‘₯)Γ—π‘₯

and

𝔼[π‘₯2]=βˆ‘βˆ€π‘₯β„™(𝑋=π‘₯)Γ—π‘₯2

And we can generalize this to any function 𝑔(𝑋).

Therefore the simple calculation is multiplying each probability of each outcome by its value and adding them together.

For variance we can use the same logic.

𝕍[𝑆2]=𝔼[𝑆22]βˆ’π”Ό[𝑆2]2

For the second one it’s simple, it must be 922. And for the expectation of squares we can use the 𝑔(𝑋) property.

Then afterwards we take the square root to get the volatility.

What about ten steps ahead? It’s not as intuitive if we continue to use the tree model, there are too many possibilities.

We can actually solve this problem using binomial process.

One way you can think about it is success and no success. So, we could think about it as, we have a process

𝐡2∼ Binom(2,0.3)

And the question is of how many possibilities do we have, and what’s the probability of success? Let’s use some reasoning

When 𝐡2=0, we know that 𝑃2=80. So we can think about it as

𝑃2=0β‹…20

As this is a random variable which modifies our answer. And we get either 80, 100, or 120 depending on the value 𝐡2. And we can in fact say that

β„™2=80+𝐡2β‹…20 Therefore,

𝔼[𝑃2]=80+20𝔼[𝐡2]

𝕍[𝑃2]=0+202𝕍[𝐡2]

So now let’s try to understand the distribution of our random variable 𝐡2.

We can actually leverage some common knowledge which tells us that the answer is based on the formulas.

If we have a general

π‘‹βˆΌπ‘(πœ‡,𝜎2)

and we want to know

β„™(𝑋<π‘Ž)

We know this is the same thing as asking

πœ™(π‘Žβˆ’πœ‡πœŽ)

And then you can use the Excel formula

NORMDIST((a - mu)/ sigma)

Remember that in order to solve this, you can consult the table or Excel. There is a standardized reference.

5.

If we have stock returns iid, with annual vol of 30% what is daily VaR at 99% confidence level.

This is tricky because we have to convert between annual and daily. If we have iid variables, than the VaR can be written as

π‘‰π‘Žπ‘…(𝑋,𝛼)=πœŽπ‘1βˆ’π›Ό

If we consider daily, we have

VaR𝑑=πœŽπ‘‘β‹…π‘1βˆ’π›Ό

and we are given that 𝛼 is 0.01.

We are not given information about πœŽπ‘‘, but we know 𝜎𝐴=0.3. To some extent, we are looking at the aggregation. If we have log returns, the annual returns of A is the summation of these returns from days until 252. So

𝜎𝐴2=252β‹…πœŽπ‘‘

That means that actually

πœŽπ‘‘=1252β‹…πœŽπ΄

We can plug this value in to get πœŽπ‘‘.

That’s how we move between periods. For the 𝑍, we just refer to the normal distribution table, which is the 99th percentile.

6.

In order to get the 10 day VaR, we use the formula

π‘‰π‘Žπ‘…π‘‘=π‘‘β‹…π‘‰π‘Žπ‘….

7.

What does value at risk mean?

In 1 out of however many days (represented by confidence level), the portfolio will decline by the value at risk or more.

e.g. for 95% confidence level, this is 1 out of 20 days.

Regression Analysis

There’s a portfolio on Canvas, an Excel spreadsheet in the FE_535_Session_01_Portfolio_CAPM.xlsx. The file looks at the composition of TSLA returns. That’s what regression does.

8.

What is the correlation between these two assets? What is 𝛽, what does it stand for? It is the derivative of the regression, how will covariance change x?

𝛽=Cov(𝑋,π‘Œ)𝕍[𝑋]

If 𝛽 is positive, that tells us correlation is positive, so we can consider them related. This is also

𝜎π‘₯π‘¦πœŽπ‘₯2

We can play a little trick here to get

𝜎π‘₯π‘¦πœŽπ‘₯πœŽπ‘¦β‹…πœŽπ‘¦πœŽπ‘₯

And that first expression is the correlation 𝜌π‘₯𝑦

So we have calculated

𝛽=𝜌π‘₯π‘¦πœŽπ‘¦πœŽπ‘₯

Let’s take the square here.

𝛽2=𝜌π‘₯𝑦2πœŽπ‘¦2𝜎π‘₯2

I can write the volatility of this asset as systematic volatility.

πœŽπ‘¦2=𝛽2𝜎π‘₯2𝜌π‘₯𝑦2

There’s a proportion of total volatility that will be explained by this variable x.

If we have

𝑅2=𝛽2𝜎π‘₯2πœŽπ‘¦2

this is the meaning of this statistical property.

We can actually explain the total volatility as the systematic volatility (𝛽) as

πœŽπ‘¦2=𝛽2𝜎π‘₯2+πœŽπ‘’2

When you look at

","contentSourceFormat":"typst","contentTypst":"=== Exam Review\n\n==== Chapter 2: Fundamentals of Probability\n\nFor a\n\n$Z = a X + b Y$\n\nthen\n\n$bb(E)\\[Z\\]= a bb(E)\\[X\\]+ b bb(E)\\[Y\\]$\n\nand for variance,\n\n$bb(V)\\[Z\\]= a^2 bb(V)\\[X\\]+ b^2 bb(V)\\[Y\\]+ 2 a b thin \"Cov\"\\(X\\,Y\\)$\n\nWe can actually do this from a portfolio perspective, it's the same idea. We have some weight\n\n$Z = w_1 dot.op R_1 + w_2 R_2$\n\nWhat's the variance? $bb(V)\\[Z\\]= w_1 sigma_1^2 + w_2^2 sigma_2^2 + 2 w_1 w_2 sigma_12$\n\nIt's the same in this exercise where we substitute the weights for the coefficients and the random variables for X and Y.\n\nAnother way to look at this problem is through matrix algebra.\n\nWe have a vector v,\n\n$arrow(w) = mat(delim: \"[\", a med b)$\n\nWe can rewrite $Z$ as\n\n$Z = arrow(w)^T arrow(R)$\n\nThen the expectation and variance becomes\n\n$bb(E)\\[Z\\]= arrow(w)^T bb(E)\\[R\\]$\n\nThis is nice because if you have a covariance matrix you don't have to look at all the pairwise. The question gives us that this X and Y have covariance 0.35. Be mindful of the difference between covariance and correlation.\n\nThe formula for covariance is that\n\n$sigma_12 = rho sigma_1 sigma_2$\n\nso that's where correlation comes into it. That $rho$ is correlation.\n\nWe can then rewrite our problem into\n\n$w' Sigma w$\n\nbased on considering this as matrix.\n\nIf you \\$100 today, it can either go to \\$110 or \\$90. Up is 0.3, Down is 0.7. We'll have the same increase and decrease amounts and probabilities at each step.\n\nWe're actually interested in the value of\n\n$bb(E)\\[S_2\\|S_0\\]$\n\nBut in this case we have discrete outcomes, not like in GBM. If we have a random variable, it's either going to be 120, 100, or 80. There is a probability assigned to each of these for a given $S_0$.\n\nWe know that the probability for 120 is 0.3 \\* 0.3 or 0.09, for 80 it's 0.7 \\* 0.7 = 0.49. For 110 we have to consider both the possibility of 0.3 \\* 0.7 up and down and 0.7 \\* 0.3 down and up, which becomes 0.42. And we can show that these all add to 1.\n\nSo how do we actually calculate probability? We think about it simply as\n\n$bb(E)\\[X\\]= sum_(forall x) bb(P)\\(X = x\\)times x$\n\nand\n\n$bb(E)\\[x^2\\]= sum_(forall x) bb(P)\\(X = x\\)times x^2$\n\nAnd we can generalize this to any function $g\\(X\\)$.\n\nTherefore the simple calculation is multiplying each probability of each outcome by its value and adding them together.\n\nFor variance we can use the same logic.\n\n$bb(V)\\[S_2\\]= bb(E)\\[S_2^2\\]- bb(E)\\[S_2\\]^2$\n\nFor the second one it's simple, it must be $92^2$. And for the expectation of squares we can use the $g\\(X\\)$ property.\n\nThen afterwards we take the square root to get the volatility.\n\nWhat about ten steps ahead? It's not as intuitive if we continue to use the tree model, there are too many possibilities.\n\nWe can actually solve this problem using binomial process.\n\nOne way you can think about it is success and no success. So, we could think about it as, we have a process\n\n$B_2 tilde.op upright(\" Binom\")\\(2\\,0.3\\)$\n\nAnd the question is of how many possibilities do we have, and what's the probability of success? Let's use some reasoning\n\nWhen $B_2 = 0$, we know that $P_2 = 80$. So we can think about it as\n\n$P_2 = 0 dot.op 20$\n\nAs this is a random variable which modifies our answer. And we get either 80, 100, or 120 depending on the value $B_2$. And we can in fact say that\n\n$bb(P)_2 = 80 + B_2 dot.op 20$ Therefore,\n\n$bb(E)\\[P_2\\]= 80 + 20 bb(E)\\[B_2\\]$\n\n$bb(V)\\[P_2\\]= 0 + 20^2 bb(V)\\[B_2\\]$\n\nSo now let's try to understand the distribution of our random variable $B_2$.\n\nWe can actually leverage some common knowledge which tells us that the answer is based on the formulas.\n\nIf we have a general\n\n$X tilde.op N\\(mu\\,sigma^2\\)$\n\nand we want to know\n\n$bb(P)\\(X < a\\)$\n\nWe know this is the same thing as asking\n\n$phi.alt (frac(a - mu, sigma))$\n\nAnd then you can use the Excel formula\n\nNORMDIST((a - mu)/ sigma)\n\nRemember that in order to solve this, you can consult the table or Excel. There is a standardized reference.\n\n===== 5.\n
\nIf we have stock returns iid, with annual vol of 30% what is daily VaR at 99% confidence level.\n\nThis is tricky because we have to convert between annual and daily. If we have iid variables, than the VaR can be written as\n\n$V a R\\(X\\,alpha\\)= sigma Z_(1 - alpha)$\n\nIf we consider daily, we have\n\n$\"VaR\"_d = sigma_d dot.op Z_(1 - alpha)$\n\nand we are given that $alpha$ is 0.01.\n\nWe are not given information about $sigma_d$, but we know $sigma_A = 0.3$. To some extent, we are looking at the aggregation. If we have log returns, the annual returns of A is the summation of these returns from days until 252. So\n\n$sigma_A^2 = 252 dot.op sigma_d$\n\nThat means that actually\n\n$sigma_d = 1 / sqrt(252) dot.op sigma_A$\n\nWe can plug this value in to get $sigma_d$.\n\nThat's how we move between periods. For the $Z$, we just refer to the normal distribution table, which is the 99th percentile.\n\n===== 6.\n\nIn order to get the 10 day VaR, we use the formula\n\n$V a R_d = sqrt(d) dot.op V a R$.\n\n===== 7.\n\nWhat does value at risk mean?\n\nIn 1 out of however many days (represented by confidence level), the portfolio will decline by the value at risk #emph[or more].\n\ne.g.~for 95% confidence level, this is 1 out of 20 days.\n\n==== Regression Analysis\n\nThere's a portfolio on Canvas, an Excel spreadsheet in the FE\\_535\\_Session\\_01\\_Portfolio\\_CAPM.xlsx. The file looks at the composition of TSLA returns. That's what regression does.\n\n===== 8.\n\nWhat is the correlation between these two assets? What is $beta$, what does it stand for? It is the derivative of the regression, how will covariance change x?\n\n$beta = frac(\"Cov\"\\(X\\,Y\\), bb(V)\\[X\\])$\n\nIf $beta$ is positive, that tells us correlation is positive, so we can consider them related. This is also\n\n$sigma_(x y) / sigma_x^2$\n\nWe can play a little trick here to get\n\n$frac(sigma_(x y), sigma_x sigma_y) dot.op sigma_y / sigma_x$\n\nAnd that first expression is the correlation $rho_(x y)$\n\nSo we have calculated\n\n$beta = rho_(x y) sigma_y / sigma_x$\n\nLet's take the square here.\n\n$beta^2 = rho_(x y)^2 sigma_y^2 / sigma_x^2$\n\nI can write the volatility of this asset as systematic volatility.\n\n$sigma_y^2 = frac(beta^2 sigma_x^2, rho_(x y)^2)$\n\nThere's a proportion of total volatility that will be explained by this variable x.\n\nIf we have\n\n$R^2 = frac(beta^2 sigma_x^2, sigma_y^2)$\n\nthis is the meaning of this statistical property.\n\nWe can actually explain the total volatility as the systematic volatility ($beta$) as\n\n$sigma_y^2 = beta^2 sigma_x^2 + sigma_e^2$\n\nWhen you look at","dataUrl":"https://sharifhsn.dev/api/posts/probability-and-regression-review.json","date":"2024-10-17","datePublished":"2024-10-17","description":"We can actually do this from a portfolio perspective, it’s the same idea. We have some weight What’s the variance? 𝕍 [ 𝑍 ] = 𝑀 1 𝜎 1 2 + 𝑀 2 2 𝜎 2 2 + 2 𝑀 1 𝑀 2 𝜎 12 It’s the…","site":"https://sharifhsn.dev","slug":"probability-and-regression-review","source":"FE-535 | Risk Management","sourceUrl":null,"tags":["Risk Management","Probability","Regression"],"title":"Probability and Regression Review","url":"https://sharifhsn.dev/blog/probability-and-regression-review/","version":"1","wordCount":1350}