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	<h1>MATHS OF PORTFOLIO THEORY</h1>

	<h2>1. What is an investment?</h2>

	<p>Imbalances between money and consumption desires will lead either to borrow or to save.</p>

	<p>When someone decides to save some of his income, he gives up immediate possession of the savings for a future, larger amount of money that will be available for future consumption.</p>

	<p>This trade-off of present consumption for a higher level amount of future consumption is the reason for savings. What is done with the savings to make them increase overtime is investment.</p>

	<p>Conversely, someone who consumes more than its current income must borrow the necessary funds and must be willing to pay back in the future more than he has borrowed. The rate of exchange between future and current consumption is the interest rate.</p>

	<p>An investor who exchanges €100 of certain income today with €105 of certain income in 1 year is to receive $\left(\frac{105}{100}\right)^{^{\frac{1}{n}}} - 1 $ of interest rate. This assumes however that the price level in the economy remains the same between today and next year. If the price level is expected to increase by 2%, the investor will ask for an additional 2% to preserve its purchasing power. The interest will be 5% (real) + 2% (exp. $ \displaystyle \infty $) = 7%. This interest rate is called the <strong>nominal interest rate</strong>. In addition, an incremental interest rate will be asked if the future payment of the investment is uncertain. This uncertainty is called the investment risk, the additional return asked to cover this risk is called the risk premium. In our example, if the investor demands for a future repayment of €110 the risk premium is 3%.</p>

	<p>Investors therefore want a rate of return that compensate them for the time, the expected rate of inflation, the uncertainty of return. This return is called the <mark>required rate of return.</mark></p>

	<h2>2. Measures of return and risk</h2>

	<p>Measures of historical return</p>

	<ul>
		<li>The Holding Period Return (HPR):</li>
	</ul>

	<p>$\displaystyle \frac{Ending\ value\ of\ investement}{Beginning\ value\ of\ investement} \ =\ \frac{\$220}{\$200} \ =\ 1.10$$ $</p>

	<ul>
		<li>The Holding Period Yield (HPY):</li>
	</ul>

	<p>$ HPR\ -\ 1\ =\ 0.10\ =\ 10\%\ per\ annum $</p>

	<p><mark>Example</mark>:</p>

	<p>Consider an investment that cost: $250 and is worth 350 after being held 2 years</p>

	<p>Annual HPR: $ \displaystyle \left(\frac{\$350}{\$250}\right)^{\frac{1}{2}} =\ 1.1832  $</p>

	<p>Annual HPY:  $\displaystyle 1.1832\ -\ 1\ =\ 0.1832\ =\ 18.32\%$ </p>

	<p>For a 6-month Investment:</p>

	<p>Annual HPR: $\displaystyle ( 1.40)^{\frac{1}{\frac{1}{2}}} \ =\ ( 1.40)^{2}$ = 1.96 </p>

	<p>Annual HPY: $ 1.96\ -\ 1\ =\ 0.96\ =\ 96\%\ p.a $</p>

	<h2>3. Computing mean historical returns</h2>

	<p>Used to indicate an investment’s rate of return expected to be received even on extended period of time.</p>

	<table>
		<thead>
			<tr>
				<th>
					Year
				</th>
				<th>
					Beginning Value
				</th>
				<th>
					Ending Value
				</th>
				<th>
					HPR
				</th>
				<th>
					HPV
				</th>
			</tr>
		</thead>
		<tbody>
			<tr>
				<td>
					1
				</td>
				<td>
					100
				</td>
				<td>
					115
				</td>
				<td>
					1.15
				</td>
				<td>
					15%
				</td>
			</tr>
			<tr>
				<td>
					2
				</td>
				<td>
					115
				</td>
				<td>
					138
				</td>
				<td>
					1.20
				</td>
				<td>
					20%
				</td>
			</tr>
			<tr>
				<td>
					3
				</td>
				<td>
					138
				</td>
				<td>
					110.4
				</td>
				<td>
					0.8
				</td>
				<td>
					-20%
				</td>
			</tr>
		</tbody>
	</table>

	<ul>
		<li>$\displaystyle  \begin{array}{{>{\displaystyle}l}}
			Arithmetic\ mean:\\
			\frac{15\%\ +\ 20\%\ +\ ( -20\%)}{3} \ =\ 5\%
			\end{array}$</li>
		<li>$\displaystyle  \begin{array}{{>{\displaystyle}l}}
			Geometric\ mean:\\
			1. [( 1.15) \ *\ ( 1.20) \ *\ ( 0.80)]^{\frac{1}{3}} \ -\ 1\ =\ 3.35\%
			\end{array}$
			<ol start="2">
				<li> $\displaystyle \left(\frac{110.4}{100}\right)^{\frac{1}{3}} -1\ =\ 3.35\%$</li>
				<li> $\displaystyle 100( 1+x)^{3} =\ 110.4$ </li>
			</ol></li>
	</ul>

	\begin{array}{l}
	( 1+x)^{3} \ =\ \frac{110.4}{100}\\
	( 1+x)^{3*\frac{1}{3}} \ =\ \left(\frac{110.4}{100}\right)^{\frac{1}{3}}\\
	1+x\ =\ \left(\frac{110.4}{100}\right)^{\frac{1}{3}}\\
	x\ =\ \left(\frac{110.4}{100}\right)^{\frac{1}{3}} -1\\
	x\ =\ HPY\ =\ 3.35\%
	\end{array}
	<p>The Arithmetic Mean(AM) is biased upward if you are attempting to measure an asset long-term performance, especially if it is a volatile security.</p>

	<h2>A portfolio of investment</h2>

	<table>
		<caption>*Based on Buy Market Value</caption>
		<thead>
			<tr>
				<th>
					Invest
				</th>
				<th>
					# of shares
				</th>
				<th>
					Beg. Value
				</th>
				<th>
					Beg. Market Value
				</th>
				<th>
					Ending Value
				</th>
				<th>
					Ending Market Value
				</th>
				<th>
					HPR
				</th>
				<th>
					HPY
				</th>
				<th>
					Market* Weight
				</th>
				<th>
					Weighted HPY
				</th>
			</tr>
		</thead>
		<tbody>
			<tr>
				<td>
					A
				</td>
				<td>
					100,000
				</td>
				<td>
					$10
				</td>
				<td>
					$1,000,000
				</td>
				<td>
					$12
				</td>
				<td>
					$1,200,000
				</td>
				<td>
					1.2
				</td>
				<td>
					20%
				</td>
				<td>
					5%
				</td>
				<td>
					0.01(1M/20M)
				</td>
			</tr>
			<tr>
				<td>
					B
				</td>
				<td>
					200,000
				</td>
				<td>
					$20
				</td>
				<td>
					$4,000,000
				</td>
				<td>
					$21
				</td>
				<td>
					$4,200,000
				</td>
				<td>
					1.05
				</td>
				<td>
					5%
				</td>
				<td>
					20%
				</td>
				<td>
					0.01
				</td>
			</tr>
			<tr>
				<td>
					C
				</td>
				<td>
					500,000
				</td>
				<td>
					$30
				</td>
				<td>
					$15,000,000
				</td>
				<td>
					$33
				</td>
				<td>
					$16,500,000
				</td>
				<td>
					1.1
				</td>
				<td>
					10%
				</td>
				<td>
					75%
				</td>
				<td>
					0.075
				</td>
			</tr>
			<tr>
				<td>
					TOTAL
				</td>
				<td>
					X
				</td>
				<td>
					X
				</td>
				<td>
					$20,000,000
				</td>
				<td>
					X
				</td>
				<td>
					$21,900,000
				</td>
				<td>
					X
				</td>
				<td>
					X
				</td>
				<td>
					100
				</td>
				<td>
					0.095
				</td>
			</tr>
		</tbody>
	</table>

	\begin{array}{l}
	HPR\ =\ \frac{21,900,000}{20,000,000} =1.095\\
	HPY=1.095-1\ =\ 9.5\%
	\end{array}
	<p>Example:</p>

	<p>Price / HPY</p>

	<ol>
		<li>20 / -</li>
		<li>23 / +15%</li>
		<li>18 / -21%</li>
		<li>27 / 50%</li>
		<li>35 / 29%</li>
	</ol>

	<p>AM = 18.25% ⇨  $\displaystyle \frac{15-21+50+29}{4} =18.25$</p>

	<p>GM = 17.9% ⇨  $\displaystyle \ \left(\frac{35}{20}\right)^{\frac{1}{4}} - 1 \ =\ 15.01 \% $</p>

	<p> $\displaystyle \ [( 1.15) *( 0.79) *( 1.5) *( 1.29)]^{\frac{1}{4}} \ =\ 15.14\ \%$</p>

	<h2>Calculating expected return</h2>

	<p>Expected return = $\displaystyle \sum _{i\ =\ 1}^{n}( Probability\ of\ return)( Rate \ of \ return)$</p>

	<p> $\displaystyle E( R) \ =\ ( P_{1})( R_{1}) +( P_{2})( R_{2}) +( P_{3})( R_{3}) +...+( P_{n})( R_{n})$</p>

	<p>⇨  $\displaystyle E( R_{C}) =\sum _{i=1}^{n} P_{i} R_{i}$</p>

	<p><mark>Example</mark>:</p>

	<table>
		<thead>
			<tr>
				<th>
					Economic condition
				</th>
				<th>
					Probability
				</th>
				<th>
					Rate of return
				</th>
			</tr>
		</thead>
		<tbody>
			<tr>
				<td>
					Strong economy
				</td>
				<td>
					0.15
				</td>
				<td>
					0.20
				</td>
			</tr>
			<tr>
				<td>
					Weak economy
				</td>
				<td>
					0.15
				</td>
				<td>
					-0.20
				</td>
			</tr>
			<tr>
				<td>
					No major change in economy
				</td>
				<td>
					0.7
				</td>
				<td>
					0.10
				</td>
			</tr>
		</tbody>
	</table>

	<p>$\displaystyle E( R_{i}) \ =\ [( 0.15)( 0.20) +( 0.15)( -0.20) +( 0.70)( 0.10)]$</p>

	<p> $\displaystyle E( R_{i}) \ =\ 0.07\ =\ 7\%$</p>

	<h2>Measuring the risk of expected return</h2>

	<p>2 possible measures of risk (uncertainty): The variance and the standard deviation of the expected distribution of expected return.</p>

	<h3>The variance:</h3>

	\begin{array}{l}
	Variance\left( \sigma ^{2}\right) =\sum\limits _{i\ =1}^{n}( Probability) *( Possible\ return\ -\ Expected\ return)^{2}\\
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\ \sum\limits _{i\ =1}^{n} P_{i}[( R_{i} -E( R_{i})]^{2}\\
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sigma ^{2} \ =\ ( 0.15)( 0.20-0.07)^{2} +( 0.15)( -0.20-0.07)^{2} +( 0.70)( 0.10-0.07)^{2}\\
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\ 0.0141
	\end{array}
	<p>The larger the variance for an expected rate of return, the greater the dispersion of expected return and the greater the uncertainty on risk of the investment.</p>

	<h3>Standard deviation:</h3>

	<p>The square root of the variance.</p>

	<p> $\displaystyle \sigma =\sqrt{\sum\limits _{i\ =1}^{n} P_{i}[( R_{i} -E( R_{i})]^{2}}$</p>

	<p><mark>In our example</mark>:  $\displaystyle \sigma =\sqrt{0.0141} =0.11874=11.874\%$</p>

	<h3>A relative measure of risk</h3>

	<p>The coefficient of variation =  $\displaystyle \frac{Standard\ deviation}{Expected\ rate\ of\ return} =\frac{\sigma _{i}}{E( R_{i})} =\frac{0.11874}{0.07} =1.6963$</p>

	<p>It is used to compare alternative investments with widely different rates of return and standard deviation of returns.</p>

	<p><mark>Example</mark>:</p>

	<table>
		<thead>
			<tr>
				<th>
					
				</th>
				<th>
					Investissement A
				</th>
				<th>
					Investissement B
				</th>
			</tr>
		</thead>
		<tbody>
			<tr>
				<td>
					𝐸(𝑅)
				</td>
				<td>
					0.07
				</td>
				<td>
					0.12
				</td>
			</tr>
			<tr>
				<td>
					𝜎
				</td>
				<td>
					0.05
				</td>
				<td>
					0.07
				</td>
			</tr>
		</tbody>
	</table>

	<p>B seems riskier. In contrast, the CV figures show that B has less relative variability or lower risk per unit of expected return.</p>

	$\displaystyle  \begin{array}{{>{\displaystyle}l}}
	CV_{A} =\frac{0.05}{0.07} =0.7142\\
	CV_{B} =\frac{0.07}{0.12} =0.5833
	\end{array}$
	<h2>Determinants of required rate of return</h2>

	<p>We recall that the required rate of return compensate:</p>

	<ol>
		<li>The time value of money during the period of investment</li>
		<li>The expected rate of inflation during the period</li>
		<li>The risk involved</li>
	</ol>

	<h3>The Real Risk-free Rate</h3>

	<p>It is the basic interest rate, assuming no inflation and no uncertainty about future flows.</p>

	<p>2 factors, one subjective and one objective, influence the exchange price between current goods and future goods.</p>

	<ul>
		<li>The subjective one is the time preference of individuals for the consumption of income. The strength of the human desire for current consumption influences the rate of compensation required.</li>
		<li>The objective one is the set of investment opportunities available in the economy, themselves determined in turn by the long-term real growth rate of the economy.</li>
	</ul>

	<h3>The Nominal Risk-free Rate</h3>

	<p>Factors influencing this rate:</p>

	<ul>
		<li>The expected rate of inflation</li>
		<li>The relative ease on tightness in the capital markets</li>
		<li>Risk premium:

			<p>Major sources of uncertainty</p>

			<ul>
				<li>Business risk</li>
				<li>Financial risk</li>
				<li>Liquidity risk</li>
				<li>FX risk.</li>
			</ul></li>
	</ul>

	<h3>Risk premium and portfolio theory</h3>

	<p>Investors want to hold a completely diversified market portfolio of risky assets to reach a risk that is consistant with their risk preference.</p>

	<p>The relevant risk measure for an individual asset is its co-movement with the market portfolio (the asset’s systematic risk); the portion of an individual asset’s total variance attributable to the variability of the total market portfolio.</p>

	<h3>Variance (Std Dev.) of returns of a portfolio.</h3>

	<p>Covariance of returns: measure of the degree to which 2 variables move together relative to their individual mean values overtime.</p>

	<p>The magnitude of the covariance depends on the variances of the individual return series, as well as on the relationship between the series.</p>

	<p>The covariance provides an absolute measure of how the rate of returns for stocks and bonds move together overtime.</p>

	<p> $\displaystyle Cov_{i,j} =E[ R_{i} -E( R_{i})][ R_{j} -E( R_{j})]$ </p>

	<p> $\displaystyle \frac{1}{*11}\sum _{i=1}^{12}\left[ R_{i} -\overline{R_{i}}\right]\left[ R_{j} -\overline{R_{j}}\right]$ </p>

	<p>*Here we use sample means  $\displaystyle \overline{R}$ as an estimate of the return and divide by (n-1) rather than n to avoid statistical bias.</p>

	<p><mark>Example</mark>:</p>

	<table>
		<thead>
			<tr>
				<th>
					Date
				</th>
				<th>
					Wilshire 5000<br/>Index Monthly rate of return (%)
				</th>
				<th>
					Lehman Brothers.<br/>TreasuryBonds Monthly returns (%)
				</th>
				<th>
					Covariances<br/>(Rate(Wilshire)-Mean) x (Rate(LB)-Mean)
				</th>
			</tr>
		</thead>
		<tbody>
			<tr>
				<td>
					1/04
				</td>
				<td>
					2.23
				</td>
				<td>
					1.77
				</td>
				<td>
					1.21 x 1.10 = 1.23
				</td>
			</tr>
			<tr>
				<td>
					2
				</td>
				<td>
					1.46
				</td>
				<td>
					2.0
				</td>
				<td>
					0.44 x 1.33 = 0,59
				</td>
			</tr>
			<tr>
				<td>
					3
				</td>
				<td>
					-1.07
				</td>
				<td>
					1.50
				</td>
				<td>
					-2.09 x 0.83 = -1.74
				</td>
			</tr>
			<tr>
				<td>
					4
				</td>
				<td>
					-2.13
				</td>
				<td>
					-5.59
				</td>
				<td>
					-3.15 x -6.26 = 19.73
				</td>
			</tr>
			<tr>
				<td>
					5
				</td>
				<td>
					1.38
				</td>
				<td>
					-0.54
				</td>
				<td>
					0.36 x -1.21 = -0.43
				</td>
			</tr>
			<tr>
				<td>
					6
				</td>
				<td>
					2.08
				</td>
				<td>
					0.95
				</td>
				<td>
					1.06 x 0.28 = 0.30
				</td>
			</tr>
			<tr>
				<td>
					7
				</td>
				<td>
					-3.82
				</td>
				<td>
					1.73
				</td>
				<td>
					-4.84 x 1.06 = -5.16
				</td>
			</tr>
			<tr>
				<td>
					8
				</td>
				<td>
					0.33
				</td>
				<td>
					3.74
				</td>
				<td>
					-0.69 x 3.07 = -2.12
				</td>
			</tr>
			<tr>
				<td>
					9
				</td>
				<td>
					1.78
				</td>
				<td>
					0.84
				</td>
				<td>
					0.76 x 0.17= 0.13
				</td>
			</tr>
			<tr>
				<td>
					10
				</td>
				<td>
					1.71
				</td>
				<td>
					1.51
				</td>
				<td>
					0.69 x 0.84= 0.58
				</td>
			</tr>
			<tr>
				<td>
					11
				</td>
				<td>
					4.68
				</td>
				<td>
					-2.19
				</td>
				<td>
					3.66 x 2.86= -10.46
				</td>
			</tr>
			<tr>
				<td>
					12
				</td>
				<td>
					3.63
				</td>
				<td>
					2.31
				</td>
				<td>
					2.61 x 1.64 = 4.28
				</td>
			</tr>
		</tbody>
		<thead>
			<tr>
				<th>
					Arithmetic Mean
				</th>
				<th>
					1.02
				</th>
				<th>
					0.67
				</th>
				<th>
					$\sum $ = 7
				</th>
			</tr>
		</thead>
	</table>

	<p>Interpretation is not easy. All we know is that the relationship between the 2 assets is generally positive, but it is not possible to be more specific. (During 5 months, the 2 assets moved counter to each other; therefore the number is small but positive. </p>

	<p>Standardising the covariance by the product of the individual variances yields the correlation coefficient  $\displaystyle r_{i,j}$ which can vary in the range of -1 to +1.</p>

	<p>A coefficient of +1 indicate a perfect linear relationship between  $\displaystyle R_{i}$ and  $\displaystyle R_{j}$.</p>

	<p> $\displaystyle r_{i,j} =\frac{cov_{i,j}}{\sigma _{i} \sigma _{j}} \ also\ cov_{i,j} =\sigma _{i,j} *\sigma _{i} *\sigma _{j}$</p>

	<h2>Portfolio Standard deviation</h2>

	\begin{array}{l}
	\sigma _{port} =\sqrt{\underbrace{\sum\limits _{i\ =\ 1}^{n} W_{i}^{2} \sigma _{i}^{2}}_{1} \ +\ \underbrace{\sum\limits _{i\ =\ 1}^{n}\sum\limits _{j\ =\ 1}^{n} W_{i} \ W_{j} \ Cov_{i}{}_{,}{}_{j}}_{2}}\\
	W\ =\ Weight
	\end{array}
	<p>We see 2 effects:</p>

	<ol>
		<li>The asset’s own variance of returns.</li>
		<li>The covariance between the returns of this new asset and the returns of every single other asset that is already in the portfolio.</li>
	</ol>

	<p>The relative weight of these numerous covariances is substantially greater than the asset’s unique variance. This means that the important factor to consider when adding an investment to a portfolio that contains several other investments is not the new security’s own variance(1) but it’s average covariance with all the other investments in the portfolio(2).</p>

	<table>
		<thead>
			<tr>
				<th>
					SD
				</th>
				<th>
					1
				</th>
				<th>
					2
				</th>
				<th>
					3
				</th>
				<th>
					…
				</th>
				<th>
					n
				</th>
			</tr>
		</thead>
		<tbody>
			<tr>
				<td>
					1
				</td>
				<td>
					$\sigma _{1}^{2} $
				</td>
				<td>
					$\sigma _{1,2} $
				</td>
				<td>
					$\sigma _{1,3} $
				</td>
				<td>
					
				</td>
				<td>
					$\sigma _{1,n} $
				</td>
			</tr>
			<tr>
				<td>
					2
				</td>
				<td>
					$\sigma _{2,1} $
				</td>
				<td>
					$\sigma _{2}^{2} $
				</td>
				<td>
					$\sigma _{2,3} $
				</td>
				<td>
					
				</td>
				<td>
					$\sigma _{2,n} $
				</td>
			</tr>
			<tr>
				<td>
					3
				</td>
				<td>
					$\sigma _{3,1} $
				</td>
				<td>
					$\sigma _{3,2} $
				</td>
				<td>
					$\sigma _{3}^{2} $
				</td>
				<td>
					
				</td>
				<td>
					$\sigma _{3,n} $
				</td>
			</tr>
			<tr>
				<td>
					…
				</td>
				<td>
					
				</td>
				<td>
					
				</td>
				<td>
					
				</td>
				<td>
					
				</td>
				<td>
					
				</td>
			</tr>
			<tr>
				<td>
					n
				</td>
				<td>
					$\sigma _{n,1} $
				</td>
				<td>
					$\sigma _{n,2} $
				</td>
				<td>
					$\sigma _{n,3} $
				</td>
				<td>
					
				</td>
				<td>
					$\sigma _{n}^{2} $
				</td>
			</tr>
		</tbody>
	</table>

	<h3>The benefits of diversification</h3>

	<p>Let’s take 2 assets with equal expected return and Standard Deviation of return and same equal weight.</p>

	\begin{gather*}
	E( R_{1}) =0.20\ \ \ E( \sigma _{1}) =0.10\ \ \ W_{1} =0.50\\
	E( R_{2}) =0.20\ \ \ E( \sigma _{2}) =0.10\ \ \ W_{2} =0.50
	\end{gather*}
	<p>Since $Cov_{i,j} =\sigma _{i,j}. \sigma _{i}. \sigma _{j} $ we can calculate different covariances if the $Cov$ coefficient are different.</p>

	$\displaystyle  \begin{array}{{>{\displaystyle}l}}
	a.\sigma _{1,2} =1.0\\
	b.\sigma _{1,2} =0.5\\
	c.\sigma _{1,2} =-1.0\\
	\\
	a.Cov_{1,2} =( 1.0)( 0.10)( 0.10) =0.01\\
	b.Cov_{1,2} =( 0.5)( 0.10)( 0.10) =0.05\\
	c.Cov_{1,2} =( -1.0)( 0.10)( 0.10) =-0.01
	\end{array}$
	<p>When we apply the general portfolio to a 2-asset portfolio, we get for:</p>

	$\displaystyle \begin{array}{{>{\displaystyle}l}}
	a.\sigma _{port_{a}} =\sqrt{\underbrace{W_{1}^{2} \sigma _{1}^{2} +W_{2}^{2} \sigma _{2}^{2}}_{1} +2W_{1} W_{2}\underbrace{\sigma _{1,2} \sigma _{1} \sigma _{2}}_{Cov_{1,2}}}\\
	\ \ \ \ \ \ \ \ \ \ =\sqrt{( 0.5)^{2}( 0.1)^{2} +( 0.5)^{2}( 0.1)^{2} +2( 0.5)( 0.5)( 0.01)}\\
	\ \ \ \ \ \ \ \ \ \ =\sqrt{0.01} \ =\ 0.10\\
	\\
	b.\sigma _{port_{b}} =0.0866\\
	c.\sigma _{port_{c}} =0
	\end{array}$
	<p>In case C, the negative covariance term exactly offsets the individual variance terms, leaving an overall standard deviation of zero</p>

	<h2>Combining stocks and different Returns and Risks</h2>

	<table>
		<thead>
			<tr>
				<th>
					Asset
				</th>
				<th>
					$E( R_{i})$
				</th>
				<th>
					$ W_{i} $
				</th>
				<th>
					$\sigma _{i}^{2} $
				</th>
				<th>
					$\sigma _{i} $
				</th>
			</tr>
		</thead>
		<tbody>
			<tr>
				<td>
					1
				</td>
				<td>
					0.10
				</td>
				<td>
					0.50
				</td>
				<td>
					0.0049
				</td>
				<td>
					0.07
				</td>
			</tr>
			<tr>
				<td>
					2
				</td>
				<td>
					0.20
				</td>
				<td>
					0.50
				</td>
				<td>
					0.010
				</td>
				<td>
					0.10
				</td>
			</tr>
		</tbody>
	</table>

	<p>We use previous set of correlation coefficient</p>

	<table>
		<thead>
			<tr>
				<th>
					Case
				</th>
				<th>
					Correlation coefficient
				</th>
				<th>
					$Cov_{1,2} =\sigma _{1,2} \sigma _{1} \sigma _{2} $
				</th>
			</tr>
		</thead>
		<tbody>
			<tr>
				<td>
					a
				</td>
				<td>
					+1.0
				</td>
				<td>
					0.007
				</td>
			</tr>
			<tr>
				<td>
					b
				</td>
				<td>
					+0.5
				</td>
				<td>
					0.0035
				</td>
			</tr>
			<tr>
				<td>
					c
				</td>
				<td>
					-1.0
				</td>
				<td>
					-0.007
				</td>
			</tr>
		</tbody>
	</table>

	<p>$E( R_{port}) =( 0.5)( 0.10) +( 0.5)( 0.20) =0.15 $</p>

	$\displaystyle \begin{array}{{>{\displaystyle}l}}
	\sigma _{port_{a}} =\sqrt{( 0.5)^{2}( 0.0049) +( 0.5)^{2}( 0.01) +2( 0.5)^{2}( 0.007)} \ =0.085\\
	\sigma _{port_{b}} =0.07399\\
	\sigma _{port_{c}} =0.015
	\end{array}$
	<p>It is possible for a conservative investor to experience both lower risk and higher return by diversifying into a higher risk - higher return asset, assuming that the correlation coefficient between the 2 assets is fairly low.</p>

	<figure><img src="diagram-20221104.svg"/></figure>

	<p>$For\ \sigma _{i,j} =0 $</p>

	<table>
		<thead>
			<tr>
				<th>
					case
				</th>
				<th style="text-align: left;">
					$ W _{1}$
				</th>
				<th style="text-align: left;">
					$ W _{2}$
				</th>
				<th style="text-align: left;">
					$E( R_{port})$
				</th>
				<th style="text-align: left;">
					$E( \sigma _{port}) $
				</th>
			</tr>
		</thead>
		<tbody>
			<tr>
				<td>
					
				</td>
				<td style="text-align: left;">
					0
				</td>
				<td style="text-align: left;">
					1.0
				</td>
				<td style="text-align: left;">
					0.20
				</td>
				<td style="text-align: left;">
					0.10
				</td>
			</tr>
			<tr>
				<td>
					
				</td>
				<td style="text-align: left;">
					0.20
				</td>
				<td style="text-align: left;">
					0.80
				</td>
				<td style="text-align: left;">
					0.18
				</td>
				<td style="text-align: left;">
					0.812
				</td>
			</tr>
			<tr>
				<td>
					
				</td>
				<td style="text-align: left;">
					0,40
				</td>
				<td style="text-align: left;">
					0.60
				</td>
				<td style="text-align: left;">
					0.16
				</td>
				<td style="text-align: left;">
					0.662
				</td>
			</tr>
			<tr>
				<td>
					
				</td>
				<td style="text-align: left;">
					0,50
				</td>
				<td style="text-align: left;">
					0.50
				</td>
				<td style="text-align: left;">
					0.15
				</td>
				<td style="text-align: left;">
					0.061
				</td>
			</tr>
			<tr>
				<td>
					Minimum Risk Test
				</td>
				<td style="text-align: left;">
					0,60
				</td>
				<td style="text-align: left;">
					0.40
				</td>
				<td style="text-align: left;">
					0.14
				</td>
				<td style="text-align: left;">
					0.058
				</td>
			</tr>
			<tr>
				<td>
					
				</td>
				<td style="text-align: left;">
					0,80
				</td>
				<td style="text-align: left;">
					0.20
				</td>
				<td style="text-align: left;">
					0.12
				</td>
				<td style="text-align: left;">
					0.595
				</td>
			</tr>
			<tr>
				<td>
					
				</td>
				<td style="text-align: left;">
					1.0
				</td>
				<td style="text-align: left;">
					0
				</td>
				<td style="text-align: left;">
					0.10
				</td>
				<td style="text-align: left;">
					0.07
				</td>
			</tr>
		</tbody>
	</table>

	<p>$\displaystyle \sigma _{port} =\sqrt{( 0.20)^{2}( 0.07)^{2} +( 0.80)^{2}( 0.10)^{2} +2( 0.20)( 0.80) Cov_{1,2}}$ </p>

	<h3>3 asset portfolio using matrix notations</h3>

	<p>The expected return:</p>

	\begin{array}{l}
	R_{P,W} =\mathit{(} W_{A} W_{B} W_{C}) \ \begin{pmatrix}
	R_{A}\\
	R_{B}\\
	R_{C}
	\end{pmatrix}\\
	=W_{A} R_{A} +W_{B} R_{B} +W_{C} R_{C}
	\end{array}
	<p>The variance of the portfolio :</p>

	$\displaystyle  \begin{array}{{>{\displaystyle}l}}
	\sigma _{P,W}^{2} -V_{A}{}_{R}( R_{P}{}_{,}{}_{W}) \ =\ V_{A}{}_{R}( W'R) =W'\sum W\\
	( W_{A} W_{B} W_{C})\begin{pmatrix}
	\sigma _{A}^{2} & \sigma _{AB} & \sigma _{AC}\\
	\sigma _{BA} & \sigma _{B}^{2} & \sigma _{BC}\\
	\sigma _{CA} & \sigma _{CB} & \sigma _{C}^{2}
	\end{pmatrix}\begin{pmatrix}
	W_{A}\\
	W_{B}\\
	W_{C}
	\end{pmatrix}\\
	=W_{A}^{2} \sigma _{A}^{2} +W_{B}^{2} \sigma _{B}^{2} +W_{C}^{2} \sigma _{C}^{2} +2W_{A} W_{B} \sigma _{A}{}_{B} +2W_{A} W_{C} \sigma _{A}{}_{C} +2W_{B} W_{C} \sigma _{B}{}_{C}\\
	\end{array}$
	<h3>Example of hedging</h3>

	<p>Today the 1st of march we will receive a payment A/R of 50 Million Yens at the end of july</p>

	<p>Yen contracts on the CME delivery months of March, June, September and December.</p>

	<p>One contract is for the delivery of 12.5 M Yen</p>

	<p>The company shorts 4 september-yen futures contracts on 3/1</p>

	<p>When the yens are received in July the company closes its position.</p>

	<p>We have F1 : 0.78 $cent/Yen (Initial future price)</p>

	<p>Spot in the future  S2: 0.72 $cent/Yen</p>

	<p>Fonction du spot  F2: 0.725 $cent/Yen</p>

	<p>The Gain on future contracts is 0.78(F1) - 0.725(F2) = 0.055 $Cents/Yen</p>

	<p>The final price obtained in cents/yen is final spot S2 plus the gain on the futures</p>

	<p>0.7200+0.0550 = 0.7750 cents/yen</p>

	<p>Total amount received by the company is:</p>

	<p>50M ¥ * 0.0077 $/¥ = 387.5$</p>

	<h3>Cross hedging</h3>

	<p>Occurs when the 2 assets are different</p>

	<p>Example: </p>

	<p>Jet fuel(assets) and heating oil futures(underlying asset)</p>

	<p>The hedge ratio is the ratio of the size of the position futures contracts to the size of the exposure</p>

	<h4>Calculating the minimum variance hedge ratio</h4>

	<p>The MVHR depends on the relationships between changes in the spot price(Δ S) and the futures price(Δ F)</p>

	$\displaystyle h^{*} =\varphi \frac{\sigma _{S}}{\sigma _{F}} \ \varphi \ is\ the\ coeff\ of\ correlation\ s\ betwwen\ \Delta _{S} \ et\ \Delta _{F}$ 
	<p><mark>Optimum number of contracts</mark></p>

	$\displaystyle N^{*} \ =\ \frac{h^{*} \ Q_{A} \ \leftarrow \ Size\ of\ position\ being\ hedged( units)}{Q_{F} \ \leftarrow \ Size\ of\ 1\ futures\ contracts( units)}$
	<p><mark>Example continued</mark></p>

	<p>An airline expects to purchase 2 million gallons of jet fuel in 1 month and decides to use hedging oil futures contracts for hedging (Each heating oil contracts on NYMEX is for 42000 galons)</p>

	<p>Datas are collected over the last 15 months</p>


	\begin{gather*}
	\sigma _{S} =0.0263\\
	\sigma _{F} =0.0313\\
	\varphi =0.928\\
	h^{*} =0.928*\frac{0.0263}{0.0313} =0.779757\\
	N^{*} =0.7797*\frac{2\ 000\ 000}{42\ 000} =37.13\ contracts
	\end{gather*}
	<h3>Hedging an equity portfolio</h3>

	<p>VA = current value of the portfolio</p>

	<p>VF = Current value of one futures contract</p>

	<p><mark>IF the portfolio mirrors the index</mark></p>

	<p>$h^{*} =1\ therefore\ N^{*} =1*\frac{V_{A}}{V_{F}} =\frac{V_{A}}{V_{F}} $</p>

	<p>If a portfolio with 5,050,000dollar mirrors the index (S&amp;P 500) and if the index futures price is 1,010 and each futures contracts is on 250 times the index, $\frac{V_{A}}{V_{F}} =\frac{5,050,000}{252,500} =20\ contracts( short) $</p>

	<p>If the portfolio doesn’t mirror the index we use the CAPM(MEDAF)</p>

	<p>$\displaystyle N^{*} =\beta \frac{V_{A}}{V_{F}}$</p>

	<p>Example:</p>

	<p> Value of S&amp;P500 index: 1000</p>

	<p>S&amp;P500 futures price: 1010</p>

	<p>Value of portfolio: $5,050,000</p>

	<p>r: 4% p.a</p>

	<p>Dividend yield: 1% p.a</p>

	<p>$\displaystyle \beta_{port}: 1,5$</p>

	<p>One futures contract is for the delivery of 250 times the index.</p>

	<p>$N^{*} =1.5\frac{\$5,0,50,000}{\$252,250/contract} =30\ contracts $</p>

	<p>If the index turns out to be 900 in 3 months and the futures price is 902</p>

	<p>The gain from the short futures position is:</p>

	<p>30 x (1,010 - 902) x 250 = $810,000</p>

	<p>The loss on the index is -10% or -9,75% when the dividends are taken into account (1% p.a/4)</p>

	<p>Expected return on the portfolio = 1% + 1.5 (-9.75%-1%) = 15.125%</p>

	<p>The expected value of the portfolio at the end of 3 months is:</p>

	<p>$5,050,000 x (1 - 0.15125) = $4,286,187</p>

	<p>Expected value of the hedger’s position is : $4,286,187 + $810,000 = $5,096,187</p>

	<h3>Changing the $\beta$eta of a portfolio</h3>

	<p>Examples:</p>

	<p>S&amp;P500 index = 1000</p>

	<p>S&amp;P500 futures price = 1,010</p>

	<p>Value of portfolio: $5,050,000</p>

	<p>$\beta eta = 1.5$</p>

	<p>As before $N^{*} = 30$ contracts to be shorted</p>

	<figure><img src="IMG_0381.jpg"/></figure>
</article>