Chapter 11, Problem 22QP

Portfolio Returns and Deviations Consider the following information about throe stocks:

1. a. If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the portfolio expected return? The variance? The standard deviation?
2. b. If the expected T-bill rate is 3.80 percent, what is the expected risk premium on the portfolio?
3. c. If the expected inflation rate is 3.50 percent, what are the approximate and exact expected real returns on the portfolio? What are the approximate and exact expected real risk premiums on the portfolio?

Summary Introduction

To determine: The Expected Return, Variance and Standard Deviation.

Introduction: Expected Return is a process of estimating the profits and losses an investor earns through the expected rate of returns. Standard deviation is apportioned of distribution of a collection of figures from its mean.

Answer to Problem 22QP

The Expected Return is 10.65%, Variance is 0.02317 and Standard Deviation is 15.22%.

Explanation of Solution

Determine the Portfolio Return for each Stock

$\begin{array}{c}Portfolio\mathrm{Re}turn{\left(R_P\right)}_{Boom}=\left[\begin{array}{l}\left(Weigh{t}_{StockA}\times \mathrm{Re}tur{n}_{StockA}\right)+\left(Weigh{t}_{StockB}\times \mathrm{Re}tur{n}_{StockB}\right)+\\ \left(Weigh{t}_{StockC}\times \mathrm{Re}tur{n}_{StockC}\right)\end{array}\right]\\ =\left[\left(0.40\times 0.20\right)+\left(0.40\times 0.15\right)+\left(0.20\times 0.01\right)\right]\\ =\left[0.08+0.1+0.12\right]\\ =0.30or30\%\end{array}$

$\begin{array}{c}Portfolio\mathrm{Re}turn{\left(R_P\right)}_{Normal}=\left[\begin{array}{l}\left(Weigh{t}_{StockA}\times \mathrm{Re}tur{n}_{StockA}\right)+\left(Weigh{t}_{StockB}\times \mathrm{Re}tur{n}_{StockB}\right)+\\ \left(Weigh{t}_{StockC}\times \mathrm{Re}tur{n}_{StockC}\right)\end{array}\right]\\ =\left[\left(0.40\times 0.15\right)+\left(0.40\times 0.11\right)+\left(0.20\times 0.05\right)\right]\\ =\left[0.06+0.044+0.01\right]\\ =0.114or11.40\%\end{array}$

$\begin{array}{c}Portfolio\mathrm{Re}turn{\left(R_P\right)}_{Bust}=\left[\begin{array}{l}\left(Weigh{t}_{StockA}\times \mathrm{Re}tur{n}_{StockA}\right)+\left(Weigh{t}_{StockB}\times \mathrm{Re}tur{n}_{StockB}\right)+\\ \left(Weigh{t}_{StockC}\times \mathrm{Re}tur{n}_{StockC}\right)\end{array}\right]\\ =\left[\left(0.40\times 0.01\right)+\left(0.40\times \left(-0.15\right)\right)+\left(0.20\times \left(-0.05\right)\right)\right]\\ =\left[0.004+\left(-0.06\right)+\left(-0.1\right)\right]\\ =-0.156or15.60\%\end{array}$

Therefore the Portfolio Return for Boom is 30%, Normal is 11.40%, and Bust is -15.60%.

Determine the Expected Return

$\begin{array}{c}Expected\mathrm{Re}turn\left(E_{Rp}\right)=\left[\begin{array}{l}\left(\mathrm{Pr}obabilit{y}_{Boom}\times \mathrm{Re}tur{n}_{Boom}\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Normal}\times \mathrm{Re}tur{n}_{Normal}\right)+\\ +\left(\mathrm{Pr}obabilit{y}_{Bust}\times \mathrm{Re}tur{n}_{Bust}\right)\end{array}\right]\\ =\left[\left(25\%\times 30\%\right)+\left(55\%\times 11.40\%\right)+\left(20\%\times \left(-15.60\%\right)\right)\right]\\ =\left[0.075+0.0627+\left(-0.0312\right)\right]\\ =0.1065or10.65\%\end{array}$

Therefore the Expected Return on Portfolio is 11.17%.

Determine the Variance

$\begin{array}{c}Variance\left({\sigma }_P{}^2\right)=\left[\begin{array}{l}\left(\mathrm{Pr}obabilit{y}_{Boom}\times {\left(\mathrm{Re}tur{n}_{Boom}-Expected\mathrm{Re}turn\left(E_{Rp}\right)\right)}^2\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Normal}\times {\left(\mathrm{Re}tur{n}_{Normal}-Expected\mathrm{Re}turn\left(E_{Rp}\right)\right)}^2\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Bust}\times {\left(\mathrm{Re}tur{n}_{Bust}-Expected\mathrm{Re}turn\left(E_{Rp}\right)\right)}^2\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(25\%\times {\left(30\%-10.65\%\right)}^2\right)+\left(55\%\times {\left(11.40\%-10.65\%\right)}^2\right)+\\ \left(20\%\times {\left(-15.60\%-10.65\%\right)}^2\right)\end{array}\right]\\ =\left[\left(25\%\times 0.037442\right)+\left(55\%\times 0.000056\right)+\left(20\%\times 0.068906\right)\right]\\ =\left[0.009361+0.000031+0.013781\right]\\ =0.023173or0.02317\end{array}$

Therefore the Variance on Portfolio is 0.02317.

Determine the Standard Deviation

$\begin{array}{c}S\tan dardDeviation\left(\sigma \right)=\sqrt{Variance\left({\sigma }_P{}^2\right)}\\ =\sqrt{0.02317}\\ =0.152226or15.22\%\end{array}$

Therefore the Standard Deviation on Portfolio is 15.22%.

Summary Introduction

To determine: The Expected Risk Premium on Portfolio.

Introduction: Expected Return is a process of estimating the profits and losses an investor earns through the expected rate of returns. Standard deviation is apportioned of distribution of a collection of figures from its mean.

Answer to Problem 22QP

The Expected Risk Premium on Portfolio is 6.85%.

Explanation of Solution

Determine the Expected Risk Premium on Portfolio

$\begin{array}{c}ExpectedRisk\mathrm{Pr}emium=\left[Expected\mathrm{Re}turn\left(E_r\right)-RiskfreeRate\left(R_f\right)\right]\\ =\left[10.65\%-3.8\%\right]\\ =6.85\%\end{array}$

Therefore the Expected Risk Premium on Portfolio is 6.85%.

Summary Introduction

To determine: The Approximate and Exact Expected Real Returns and Approximate and Exact Expected Real Risk Premium on Portfolio.

Answer to Problem 22QP

The Approximate Real Returns is 7.15% and Exact Expected Real Returns is 6.91% and Approximate Real Risk Premium is 6.85% and Exact Expected Real Risk Premium is 6.62%.

Explanation of Solution

Determine the Approximate Expected Real Returns

$\begin{array}{c}ApproximateExpected\mathrm{Re}al\mathrm{Re}turn=\left[Expected\mathrm{Re}turn\left(E_{Rp}\right)-ExpectedInflationRate\right]\\ =\left[10.65\%-3.5\%\right]\\ =7.15\%\end{array}$

Therefore the Approximate Expected Real Returns is 7.15%.

Determine the Exact Expected Real Returns

Using Fisher equation we calculate the exact expected real returns as,

$\begin{array}{c}1+Expected\mathrm{Re}turn\left(E_{Rp}\right)=\left(1+InflationRate\times \left(1+Exact\mathrm{Re}turn\right)\right)\\ 1+10.65\%=\left(1+3.50\%\times \left(1+Exact\mathrm{Re}turn\right)\right)\\ Exact\mathrm{Re}turn=\left[\left(\frac{1.1065}{1.0350}\right)-1\right]\\ =\left[1.069082-1\right]\\ =0.069082or6.91\%\end{array}$

Therefore the Exact Expected Real Returns is 6.91%.

Determine the Approximate Real Risk-free Rate

$\begin{array}{c}Approximate\mathrm{Re}alRiskfreeRate=\left[ExpectedTBillRate-InflationRate\right]\\ =\left[3.80\%-3.50\%\right]\\ =0.30\%\end{array}$

Therefore the Exact Expected Real Returns is 0.30%

Determine the Exact Real Risk-free Rate

Using Fisher equation we calculate the exact expected real returns as,

$\begin{array}{c}1+TBillRate=\left(1+InflationRate\times \left(1+Exact\mathrm{Re}turn\right)\right)\\ 1+3.80\%=\left(1+3.50\%\times \left(1+Exact\mathrm{Re}turn\right)\right)\\ Exact\mathrm{Re}turn=\left[\left(\frac{1.0380}{1.0350}\right)-1\right]\\ =\left[1.002899-1\right]\\ =0.002899or0.29\%\end{array}$

Therefore the Exact Real Risk-free Rate is 0.29%

Determine the Approximate Real Risk Premium

$\begin{array}{c}Approximated\mathrm{Re}alRisk\mathrm{Pr}emium=\left[\begin{array}{l}ApproximateExpected\mathrm{Re}al\mathrm{Re}turn-\\ Approximate\mathrm{Re}alRiskfreeRate\end{array}\right]\\ =\left[7.15\%-0.30\%\right]\\ =6.85\%\end{array}$

Therefore the Approximate Real Risk Premium is 6.85%

Determine the Exact Real Risk Premium

$\begin{array}{c}Exact\mathrm{Re}alRisk\mathrm{Pr}emium=\left[\begin{array}{l}ExactExpected\mathrm{Re}al\mathrm{Re}turn-\\ Exact\mathrm{Re}turn\mathrm{Re}alRiskfreeRate\end{array}\right]\\ =\left[6.91\%-0.29\%\right]\\ =6.62\%\end{array}$

Therefore the Exact Real Risk Premium is 6.62%.