Compute and plot all pair$-$wise scatterplots between these $5$ funds. Briefly comment on any relationships you see. $(5$p$)$

Compute the sample covariance matrix of the returns on these $5$ funds and comment on the direction of linear association between the asset returns. $(5$p$)$

Compute the sample correlation matrix of the returns on these $5$ funds. $(5$p$)$

a$.$ Which funda are most highly correlated? $(5$p$)$

b$.$ Which are least correlated? $(5$p$)$

c$.$ Based on the estimated correlation values do you think diversification will reduce risk with these assets? $(5$p$)$

Part $3.$ Estimating expected returns

In this section, you need to use information from Part $1$ and Part $2.$

Use the Eama$-$Erench, Three$-$Factor model augmented by "Momentum"$($MOM$)$ to estimate the expected returns of each of the funds from part $2$ :

European stock fund: Fidelity Europe $($FIEUX$)$

Latin America Fund: Fidelity Latin America $($FLATX$)$

Long$-$term bond fund: Fidelity Corporate Bond $($FCBFX$)$

Real Estate fund: T$.$ Rowe Price Real Estate $($TRREX$)$

Small Cap Stock Fund: Fidelity Small Cap Stock $($FSLCX$)$

$R_{i,t}^e={}_i+{}_{i,M}{R}_{MKT}^e+{}_{i,SMB}SM{B}_t+{}_{i,HML}HM{L}_t+{}_{i,\text{M}\text{O}\text{M}}MO{M}_t+e_{i,t}$

To simplify notation in the regression notice that $R_{i,t}^e=R_{i,t}-R_{F,t}=$ is stock or portfolio $i^{th}$ excess return and $R_{MKT}^e=R_{Mt}-R_{F,t}=$ is the excess return on a "stock market portfolio"

In order to do this follow $3$ simple steps:

Step $1.$ Estimate the risk premia for each factor $(5$p$)$

${}_{MKT}=\frac{1}{T}{\displaystyle \underset{t=1}{\overset{T}{}}}(R_{Mt}-R_{F,t}),{}_{SMB}=\frac{1}{T}{\displaystyle \underset{t=1}{\overset{T}{}}}SM{B}_t,{}_{HML}=\frac{1}{T}{\displaystyle \underset{t=1}{\overset{T}{}}}HM{L}_t,{}_{\text{M}\text{O}\text{M}}=\frac{1}{T}{\displaystyle \underset{t=1}{\overset{T}{}}}MO{M}_t$

Step $2.$ Estimate the sensitivities of the $i^{th}$ stock to each of those factors. $(5$p$)$

$R_{i,t}^e={}_i+{}_{iM}{R}_{MKT}^e+{}_{\text{i}\text{S}\text{M}\text{B}}SM{B}_t+{}_{\text{i}\text{H}\text{M}\text{L}}HM{L}_t+{}_{i,\text{M}\text{O}\text{M}}MO{M}_t+e_{it}$

Step $3.$ The expected returns can be calculated by combining the results of the previous steps. $(5$p$)$

$E(R_i^e)=$hat$({)}_{i,M}{}_{MKT}+$hat$({)}_{i,SMB}{}_{SMB}+$hat$({)}_{i,HML}{}_{HML}+$hat$({)}_{i,\text{M}\text{O}\text{M}}{}_{\text{M}\text{O}\text{M}}$

Which fund has the highest and lowest expected return? $(5$p$)$