<commit_before>---
layout: post
title: "Investment and Financial Market (IFM)"
modified:
categories: blog
excerpt: "SOA exams preparation"
tags: [IFM, SOA, ASA, FSA, Investment, Finance, Financial Market]
image:
  feature:
comments: true
share: true
date: 2020-10-23T08:08:50-04:00
---

This article provides an overview of actuarial and financial subjects related to Investment and Financial Market. Following the syllabus of IFM exam given by the SOA, it can serve as materials for students to prepare the exam, or for actuaries & other financial professions to consult for their work.  

* Table of Contents
{:toc}

 Topics
## 1. Mean-Variance Portfolio Theory

### 1.1. Risk and Return of a Single Asset

Consider a single asset with return R (a random variable following a discrete distribution : \\( \forall i = 1, ..., n: P(R=R_i) = p_i  \\). We are interested in calculating the mean and variance of its return (also called risk in this case) : 
\\[ E[R]=\sum_{i=1}^{n}{p_i*R_i} \\] 
\\[ Var(R)= E[(R_i-E[R])^2] =\sum_{i=1}^{n}{p_i(R_i-E[R])^2} = E[R^2]-(E[R])^2 \\]
The standard deviation of R is : \\( SD(R)=\sqrt{Var(R)} \\)

Now the question is, how do we define the (rate of) return R for an asset ? 
Let's consider an asset (an equity for example) with price \\( P_t \\) at time \\( t \\) and \\( P_{t+1} \\) at time \\( t+1 \\). In addition, this asset pays a dividend \\( D_{t+1} \\) between \\( t \\) and \\( t+1 \\). Its realiaed return in the interval \\( [t, t+1] \\) is then the sum of capital gain and the dividend yield :
\\[ R_{t+1} = \frac{P_{t+1}-P_t}{P_t} + \frac{D_{t+1}}{P_t} \\].

In practice, the rate of return can be calculated on a daily, monthly, quarterly or annual basis, depending on the type of asset. one can easily convert the rate of return between these different time intervals (or basis). For example, an annual realized return can be deducted from the four quarterly returns :
\\[ 1 + R_{t+1} = (1+R_{Q1})...(1+R_{Q4}) \\].

The return of an asset may vary from a given year to another, depending on its performance. One can be interested in the mean of all annual return, called  average annual return, for a period of T years :
\\[ \overline{R} = \frac{1}{T} sum_{t=1}^{T} R_t \\]
Once again, the "risk" can be measured by the variance :
\\[ Var(R) = \frac{1}{T-1} \sum_{t=1}^{T} (R_t- \overline{R})^2 \\]
<commit_msg>Update 2020-10-23-IFM.md<commit_after>---
layout: post
title: "Investment and Financial Market (IFM)"
modified:
categories: blog
excerpt: "SOA exams preparation"
tags: [IFM, SOA, ASA, FSA, Investment, Finance, Financial Market]
image:
  feature:
comments: true
share: true
date: 2020-10-23T08:08:50-04:00
---

This article provides an overview of actuarial and financial subjects related to Investment and Financial Market. Following the syllabus of IFM exam given by the SOA, it can serve as materials for students to prepare the exam, or for actuaries & other financial professions to consult for their work.  

* Table of Contents
{:toc}

 Topics
## 1. Mean-Variance Portfolio Theory

### 1.1. Risk and Return of a Single Asset

Consider a single asset with return R (a random variable following a discrete distribution : \\( \forall i = 1, ..., n: P(R=R_i) = p_i  \\). We are interested in calculating the mean and variance of its return (also called risk in this case) : 
\\[ E[R]=\sum_{i=1}^{n}{p_i*R_i} \\] 
\\[ Var(R)= E[(R_i-E[R])^2] =\sum_{i=1}^{n}{p_i(R_i-E[R])^2} = E[R^2]-(E[R])^2 \\]
The standard deviation of R is : \\( SD(R)=\sqrt{Var(R)} \\)

Now the question is, how do we define the (rate of) return R for an asset ? 
Let's consider an asset (an equity for example) with price \\( P_t \\) at time \\( t \\) and \\( P_{t+1} \\) at time \\( t+1 \\). In addition, this asset pays a dividend \\( D_{t+1} \\) between \\( t \\) and \\( t+1 \\). Its realiaed return in the interval \\( [t, t+1] \\) is then the sum of capital gain and the dividend yield :
\\[ R_{t+1} = \frac{P_{t+1}-P_t}{P_t} + \frac{D_{t+1}}{P_t} \\].

In practice, the rate of return can be calculated on a daily, monthly, quarterly or annual basis, depending on the type of asset. one can easily convert the rate of return between these different time intervals (or basis). For example, an annual realized return can be deducted from the four quarterly returns :
\\[ 1 + R_{t+1} = (1+R_{Q1})...(1+R_{Q4}) \\].

The return of an asset may vary from a given year to another, depending on its performance. One can be interested in the mean of all annual return, called  average annual return, for a period of T years :
\\[ \overline{R} = \frac{1}{T} \sum_{t=1}^{T} R_t \\]
Once again, the "risk" can be measured by the variance :
\\[ Var(R) = \frac{1}{T-1} \sum_{t=1}^{T} (R_t- \overline{R})^2 \\]

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