```{r} #| label: setup #| include: false set.seed(2026) library(ggplot2) theme_set(theme_minimal(base_size = 18) + theme(legend.text = element_text(size = 18), legend.title = element_text(size = 18))) ``` ## ๐ฌ The Idea in 3 Minutes ::: {style="text-align:center"} [Watch this short intuition video before (or after) the slides. Captions: CC button.]{style="font-size:22px"} ::: --- ## ๐ฏ Learning Objectives ::: {style="font-size: 32px"} By the end of this lecture, you will be able to: - **State** Definition 4.8 and read $\mu$ and $\sigma$ off a normal density (Theorem 4.7) - **Standardise** any normal variable with $Z = (Y - \mu)/\sigma$ and find probabilities from Table 4 and from `pnorm` - **Invert** the calculation with `qnorm` to find a quantile: a cut-off, a fill setting, a budget - **Compute** a one-day Value-at-Risk at the normal quantile - **Explain** why real daily returns have heavier tails than the normal model allows ::: --- ## ๐บ๏ธ Where We Are ::: {style="font-size: 30px"} **Wackerly ยง4.5** Saturday: **expected values for continuous random variables**, $E(Y) = \int y f(y)\,dy$, and the **uniform distribution**, where every probability is a length divided by a length. ::: {.fragment} The uniform made the integral easy. Today's density is the opposite: the one the whole course leans on, and the one whose integral **has no closed form at all**. ::: ::: {.fragment} So the skill today is not integration. It is **standardising**, and then reading a table or asking R. ::: ::: --- ## โ The Question This Lecture Answers ::: {style="font-size: 30px"} ::: {.callout-important} ## A risk desk in Baku A bank's trading book holds **2,000,000 AZN** in a broad equity index. The daily return has mean $0.04\%$ and standard deviation $1.2\%$. What loss will the book **not exceed on 99% of trading days**? ::: ::: {.fragment} That number is the one-day 99% **Value-at-Risk**. To compute it we need a model for the return, a way to find a tail area under that model, and a way to run the calculation backwards from an area to a cut-off. ::: ::: --- ## ๐ Definition 4.8 ::: {style="font-size: 30px"} ::: {.callout-note} ## Definition 4.8 A random variable $Y$ is said to have a **normal probability distribution** if and only if, for $\sigma > 0$ and $-\infty < \mu < \infty$, the density function of $Y$ is $$f(y) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-(y-\mu)^2/(2\sigma^2)}, \qquad -\infty < y < \infty.$$ ::: ::: {.fragment} Two parameters, $\mu$ and $\sigma$. The exponent is a squared distance from $\mu$, measured in units of $\sigma$: the further from $\mu$, the faster the density falls away. ::: ::: --- ## ๐ Theorem 4.7 ::: {style="font-size: 30px"} ::: {.callout-important} ## Theorem 4.7 If $Y$ is normally distributed with parameters $\mu$ and $\sigma$, then $$E(Y) = \mu \qquad \text{and} \qquad V(Y) = \sigma^2.$$ ::: The proof waits for ยง4.9, via the moment-generating function. ::: {.fragment} - $\mu$ **locates** the centre; the density is symmetric about it, so the median is $\mu$ too - $\sigma$ **measures the spread**: the peak is $1/(\sigma\sqrt{2\pi})$ at $y = \mu$, and the curve bends at $\mu \pm \sigma$ (Exercises 4.78, 4.79) ::: ::: --- ## ๐ $\mu$ Locates, $\sigma$ Spreads ```{r} #| label: three-normals #| echo: false #| fig-width: 10 #| fig-height: 4.6 y <- seq(-6, 6, by = 0.02) dd <- rbind( data.frame(y = y, f = dnorm(y, 0, 0.8), model = "a"), data.frame(y = y, f = dnorm(y, 0, 1.6), model = "b"), data.frame(y = y, f = dnorm(y, 2, 0.8), model = "c") ) ggplot(dd, aes(y, f, colour = model)) + geom_line(linewidth = 1.3) + scale_colour_manual(values = c("#14130f", "#3d6b9e", "#8b2635"), labels = c(expression(paste("calm index: ", mu == 0, ", ", sigma == 0.8)), expression(paste("volatile index: ", mu == 0, ", ", sigma == 1.6)), expression(paste("shifted: ", mu == 2, ", ", sigma == 0.8)))) + labs(x = "Daily return (%)", y = "f(y)", colour = NULL) + theme(legend.position = "top", legend.direction = "vertical", legend.text = element_text(size = 18, hjust = 0)) ``` ::: {style="font-size: 28px"} Doubling $\sigma$ halves the peak and doubles the width. Area stays 1. ::: --- ## ๐งฎ No Closed Form: Table 4 and R ::: {style="font-size: 30px"} $$P(a \le Y \le b) = \int_a^b \frac{1}{\sigma\sqrt{2\pi}}\, e^{-(y-\mu)^2/(2\sigma^2)}\,dy$$ has **no closed-form** antiderivative. It is evaluated numerically, and there are two ways to get the answer: - **Table 4, Appendix 3** gives $A(z) = P(Z > z)$, the area to the **right** of $z$, for the standard normal only - **R**: `pnorm(y0, mu, sigma)` gives $P(Y \le y_0)$; `qnorm(p, mu, sigma)` gives the $p$th quantile $\phi_p$ with $P(Y \le \phi_p) = p$ ::: {.fragment} The only advantage of software is **more decimal places**. Watch the direction: the table is a right tail, `pnorm` is a left tail. ::: ::: --- ## ๐ One Table for Every Normal ::: {style="font-size: 30px"} There are infinitely many normal distributions, but one table is enough, because $$Z = \frac{Y - \mu}{\sigma}$$ is a **standard normal** random variable: mean 0, standard deviation 1 (proved in Chapter 6). ::: {.fragment} $z$ is the distance from the mean **in standard deviations**. So $$P(Y > y_0) = P\!\left(Z > \frac{y_0 - \mu}{\sigma}\right) = A\!\left(\frac{y_0 - \mu}{\sigma}\right).$$ ::: ::: {.fragment} A return of $-2.36\%$ on a day with $\mu = 0.04\%$, $\sigma = 1.2\%$ is $z = -2$: two standard deviations below the mean, whatever the units. ::: ::: --- ## ๐ Example 4.8: Table vs R ::: {style="font-size: 28px"} $Z$ standard normal, Table 4: **(a)** $P(Z > 2) = A(2.0) = .0228$\ **(b)** $P(-2 \le Z \le 2) = 1 - 2(.0228) = .9544$\ **(c)** $P(0 \le Z \le 1.73) = .5 - A(1.73) = .5 - .0418 = .4582$ ::: ```{r} #| label: example-4-8 #| code-fold: false round(c(a = 1 - pnorm(2), b = pnorm(2) - pnorm(-2), c = pnorm(1.73) - 0.5), 5) ``` ::: {style="font-size: 28px"} Part (b) differs in the fourth decimal: $.9544$ from the rounded table, $.95450$ from R. Symmetry did the work in (b) and (c): the table only covers one side. ::: --- ## ๐ญ Worked Example: Filling Flour Sacks ::: {style="font-size: 28px"} A flour mill in Ganja packs **50 kg** sacks. The fill weight is normal with $\mu = 50.2$ kg and $\sigma = 0.4$ kg. What fraction of sacks is **underweight**? ::: {.fragment} $$z = \frac{50 - 50.2}{0.4} = -0.5, \qquad P(Y < 50) = P(Z < -0.5) = A(0.5) = .3085$$ Almost a third of the sacks break the label. Symmetry turned a left tail at $-0.5$ into the table's right tail at $+0.5$. ::: ::: {.fragment} Within $49.5$ to $51$ kg: $z_1 = -1.75$, $z_2 = 2.0$, so $$P = 1 - A(1.75) - A(2.0) = 1 - .0401 - .0228 = .9371$$ ::: ::: --- ## ๐ฏ Backwards: Setting the Mean ::: {style="font-size: 28px"} The regulator allows **1%** underweight. Keep $\sigma = 0.4$; where must the machine's mean be set? ::: {.fragment} We need $P(Y < 50) = .01$, so $50$ must sit $z_{.01}$ standard deviations below $\mu$. Table 4: $A(2.33) = .0099 \approx .01$. $$\mu = 50 + 2.33(0.4) = 50.932 \text{ kg}$$ ::: ```{r} #| label: fill-mean #| code-fold: false 50 + qnorm(0.99) * 0.4 ``` ::: {.fragment} The price of the rule: about **0.73 kg of extra flour per sack** over today's setting. The alternative is a better machine: a smaller $\sigma$ needs a smaller safety margin. ::: ::: --- ## ๐ณ Standardising Credit Scores ::: {style="font-size: 28px"} A Baku bank scores applicants on a scale that is normal with $\mu = 640$, $\sigma = 80$, and approves at **700** or above. ::: {.fragment} $$z = \frac{700 - 640}{80} = 0.75, \qquad P(Y \ge 700) = A(0.75) = .2266$$ ::: ::: {.fragment} It buys a second bureau's scores, normal with $\mu = 520$, $\sigma = 60$. The **comparable** cut-off is the one with the same $z$: $$520 + 0.75(60) = 565$$ ::: ::: {.fragment} Between 600 and 720 on the first scale: $1 - A(0.5) - A(1.0) = 1 - .3085 - .1587 = .5328$. ::: ::: --- ## ๐ Value-at-Risk at the Normal Quantile ::: {style="font-size: 28px"} Back to the trading book: $R \sim$ normal, $\mu = 0.04\%$, $\sigma = 1.2\%$, position $2{,}000{,}000$ AZN. ::: {.fragment} The 1% quantile of the return is $\mu - z_{.01}\,\sigma$. With the table's $2.33$: $$\phi_{.01} = 0.04 - 2.33(1.2) = -2.756\%$$ ::: ::: {.fragment} $$\text{VaR}_{99\%} = 0.02756 \times 2{,}000{,}000 \approx 55{,}120 \text{ AZN}$$ On 99 days in 100 the book loses less than this. On the other one day it loses **more**, and the model says nothing about how much more. ::: ::: --- ## ๐ป VaR in R ```{r} #| label: var V <- 2e6; mu <- 0.04; sigma <- 1.2 q01 <- qnorm(0.01, mu, sigma) # 1% quantile of the return, in % data.frame(quantile_pct = round(q01, 4), VaR_AZN = round(-q01 / 100 * V)) ``` ```{r} #| label: var-figure #| echo: false #| fig-width: 10 #| fig-height: 3.3 r <- seq(-5, 5, by = 0.01) tail_df <- data.frame(r = r[r <= q01], f = dnorm(r[r <= q01], mu, sigma)) ggplot(data.frame(r = r, f = dnorm(r, mu, sigma)), aes(r, f)) + geom_area(data = tail_df, fill = "#8b2635", alpha = 0.7) + geom_line(linewidth = 1.2) + geom_vline(xintercept = q01, linetype = "dashed") + annotate("text", x = q01 - 0.1, y = 0.25, hjust = 1, size = 6, label = "1% of days") + labs(x = "Daily return (%)", y = "f(r)") ``` ::: {style="font-size: 28px"} `qnorm` gives $55{,}032$ AZN; the table's rounded $2.33$ overstates by about 90 AZN. ::: --- ## โ ๏ธ Why Real Tails Are Heavier ::: {style="font-size: 28px"} Under the normal model a move beyond $4\sigma$ has probability $2A(4) \approx 0.000063$: **once in about 63 years** of 250 trading days. Equity indices produce such days far more often. A heavier-tailed model with the **same** $\sigma$ (Student's $t$ with 3 df, rescaled; met properly in Chapter 7), simulated for ten years: ::: ```{r} #| label: heavy-tails #| code-fold: true n_days <- 2500 r_t <- 0.04 + 1.2 * rt(n_days, df = 3) / sqrt(3) # sd = 1.2, fat tails data.frame(beyond = c("3 sigma", "4 sigma"), simulated_days = c(sum(abs(r_t - 0.04) > 3 * 1.2), sum(abs(r_t - 0.04) > 4 * 1.2)), normal_expected = round(n_days * 2 * pnorm(-c(3, 4)), 2)) ``` --- ## ๐ The Tail, on a Log Scale ```{r} #| label: tail-figure #| echo: false #| fig-width: 10 #| fig-height: 4.4 k <- seq(1, 5, by = 0.05) tl <- rbind(data.frame(k = k, p = 2 * pnorm(-k), model = "normal"), data.frame(k = k, p = 2 * pt(-k * sqrt(3), 3), model = "heavy-tailed (t, 3 df)")) ggplot(tl, aes(k, p, colour = model)) + geom_line(linewidth = 1.3) + scale_y_log10(breaks = 10^(-6:0), labels = c("0.000001", "0.00001", "0.0001", "0.001", "0.01", "0.1", "1")) + scale_colour_manual(values = c("#8b2635", "#14130f")) + labs(x = "k (standard deviations from the mean)", y = expression(P(abs(R - mu) > k * sigma)), colour = NULL) + theme(legend.position = "top") ``` ::: {style="font-size: 28px"} Same mean, same $\sigma$; the curves cross near $2\sigma$. At $4\sigma$ the heavy tail is about **100 times** as likely, so normal VaR **understates** extreme losses. ::: --- ## ๐ง Think-Pair-Share ```{r} #| label: tps-timer #| echo: false # The timer is the only thing in this deck that needs a package beyond base R. # Guarded so a machine without it renders the deck anyway, with a static # figure in the same corner, rather than halting the whole build. if (requireNamespace("countdown", quietly = TRUE)) { countdown::countdown(minutes = 4, seconds = 0, top = 0, right = 0, font_size = "2em", warn_when = 30) } else { htmltools::HTML(paste0( '4:00
')) } ``` ::: {style="font-size: 30px"} A mobile operator models a subscriber's monthly data use as normal with $\mu = 12$ GB and $\sigma = 4$ GB. **Four minutes, in pairs:** 1. What fraction of subscribers use more than 20 GB? 2. A "heavy user" tariff targets the top 5%. Where is its threshold? 3. What does the model say about $P(Y < 0)$, and what does that tell you about the model? ::: --- ## โ
Think-Pair-Share: Solution ::: {style="font-size: 30px"} 1. $z = (20 - 12)/4 = 2$, so $P(Y > 20) = A(2.0) = .0228$: about 2.3% of subscribers. 2. The top 5% starts at $z_{.05} = 1.645$: $\;12 + 1.645(4) = 18.58$ GB. In R, `qnorm(0.95, 12, 4)`. ::: {.fragment} 3. $P(Y < 0) = P(Z < -3) = A(3.0) = .00135$. Small, but **impossible** usage has positive probability. Data use is bounded below by zero and skewed to the right; the normal is a working approximation near the centre, not a law of nature. Check the support before you trust the tails. ::: ::: --- ## ๐ Quiz #1: Reading the Table {.quiz-question} $Z$ is standard normal. What is $P(Z > 1.5)$? - [$0.0668$]{.correct data-explanation="โ
Table 4 tabulates the right tail directly: A(1.5) = .0668. In R, 1 - pnorm(1.5)."} - $0.9332$ - $0.4332$ - $0.1336$ --- ## ๐ Quiz #2: Setting a Budget {.quiz-question} A power plant's weekly fuel bill is normal with $\mu = 4000$ AZN and $\sigma = 250$ AZN. What budget is exceeded in only 10% of weeks? - [About $4320$ AZN]{.correct data-explanation="โ
The 90th percentile: 4000 + 1.28(250) = 4320. In R, qnorm(0.90, 4000, 250) = 4320.4."} - About $4411$ AZN - About $4490$ AZN - About $4250$ AZN --- ## ๐ Quiz #3: What the Normal Misses {.quiz-question} Real daily index returns show many more moves beyond $4\sigma$ than the normal model predicts. What does this do to a 99.9% normal VaR? - [It understates the loss the book can actually suffer]{.correct data-explanation="โ
The normal quantile sits too close to the mean when the true tail is heavier, so extreme losses are more likely than the VaR suggests."} - It overstates the loss, so the bank holds too much capital - Nothing: VaR depends only on $\mu$ and $\sigma$, which are unchanged - It makes VaR undefined --- ## ๐ Key Formulas ::: {style="font-size: 30px"} | | Statement | |---|---| | Definition 4.8 | $f(y) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-(y-\mu)^2/(2\sigma^2)}$ | | Theorem 4.7 | $E(Y) = \mu, \; V(Y) = \sigma^2$ | | standardising | $Z = (Y - \mu)/\sigma$ | | Table 4 | $A(z) = P(Z > z)$ | | R, forwards | `pnorm(y0, mu, sigma)` $= P(Y \le y_0)$ | | R, backwards | `qnorm(p, mu, sigma)` $= \phi_p$ | | quantiles | $z_{.10} = 1.28,\ z_{.05} = 1.645,\ z_{.025} = 1.96,\ z_{.01} = 2.33$ | ::: --- ## ๐ Summary ::: {style="font-size: 30px"} - The normal density has two parameters: $\mu$ is the mean and centre, $\sigma$ the standard deviation and spread - Its integral has no closed form, so every probability goes through $Z = (Y - \mu)/\sigma$ and then Table 4 or `pnorm` - Quantiles run the same step backwards: $y = \mu + z\sigma$, or `qnorm` - A fill setting, an equivalent cut-off and a VaR are all one quantile - Real returns have heavier tails than the normal, so normal VaR is optimistic about the worst days ::: --- ## ๐ Practice Problems ::: {style="font-size: 28px"} **Wackerly, 7th edition, ยง4.5** - Table practice: 4.58, 4.59; then 4.63 โ 4.65 and 4.68 โ 4.71 - Quantiles and settings: 4.73 โ 4.77; for the brave, 4.78 and 4.79 - Do each one twice: once with Table 4, once with `pnorm`/`qnorm` **Week 10, Problem Set 1** is open now and closes **Sunday 22 November at 23:59** on WeBWorK, covering ยง4.5. **Next class:** 14 November, the gamma, exponential and chi-square distributions (Wackerly ยง4.6). ::: --- ## ๐ Thank You ::: {style="font-size: 34px"} **Dr. Samir Orujov** ๐ง sorujov@ada.edu.az\ ๐ข Building D, Room D325\ ๐ Office hours: Wednesday, 16:00 โ 18:00 Slides and readings: **sorujov.net/teaching** ::: --- ## โ Questions ::: {style="font-size: 32px"} - Why does one table suffice for infinitely many normal distributions? - If $\sigma$ halves, what happens to the 99% VaR, and to the fill setting in the flour example? - Where else in finance would a normal model put positive probability on an impossible value? ::: 