---
title: "Monte Carlo: Collective Risk Model"
author: "Milica Cudina"
date: "`r Sys.Date()`"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
## The model
We are redoing problem \#32 from Exam 3, November 2000, but using Monte Carlo and without the normal approximation. We have that the number of losses and loss amounts are independent with 
$$ N \sim Poisson(\lambda=25) $$
and 
$$ X \sim U(5, 95) $$.
```{r}
lambda=25
min=5
max=95
```


## Simulating a single realized aggregate loss
```{r}
n=rpois(1, lambda)
n
x.s=runif(n, min, max)
x.s
s=sum(x.s)
s
```

## Creating a function which draws a single aggregate loss
```{r}
one.loss<-function(){
n=rpois(1, lambda)
x.s=runif(n, min, max)
s=sum(x.s)
return(s)  
}
```

## Running this function many times
```{r}
nsim=10000
all.losses=replicate(nsim, one.loss())
all.losses
hist(all.losses, breaks=50, prob=TRUE)
```

## The theoretical values of the mean and the standard deviation
```{r}
mu.s=lambda*0.5*(min+max)
mu.s
sd.s=sqrt(lambda*((1/12)*(max-min)^2+(0.5*(min+max))^2))
sd.s
```

## Let's see what the MC estimates of the mean and the standard deviation are
```{r}
mean(all.losses)
sd(all.losses)
```

## What if we superimpose the normal density onto the histogram of simulated aggregate losses
```{r}
hist(all.losses, breaks=50, prob=TRUE)
curve(dnorm(x,mean=mu.s, sd=sd.s), add=TRUE)
```

## What about the normal cdf and the "empirical" cdf?
```{r}
plot(ecdf(all.losses))
curve(pnorm(x,mean=mu.s,sd=sd.s), add=TRUE, col="green")
```

## Here is the probability from the problem
```{r}
#the simulated
sim.prob=mean(all.losses>2000)
sim.prob
#the normal approximation
1-pnorm(2000, mean=mu.s, sd=sd.s)
```

## What about the normal cdf and the "empirical" cdf?
```{r}
plot(ecdf(all.losses), xlim=c(1980,2020), ylim=c(0.989,1))
curve(pnorm(x,mean=mu.s,sd=sd.s), add=TRUE, col="green")
```