Chapter 10, Problem 28QP

a

Summary Introduction

To determine: The probability of long-term bonds and t-bill

Introduction:

The probability distribution refers to a function that provides the possibilities (probabilities) of occurrence of various possible outcomes in an investment.

Explanation of Solution

Given information:

It is given that the return on long-term bonds and treasury-bills are normally distributed.

The formula of Z-score equation:

$\text{Z}=\frac{\text{X}-\mu }{\sigma }$

Where,

Z refers to z-scores

X refers to return

µ refers to mean return

σ refers to standard deviation

When the return of long-term bonds is greater than 10%, the probability would be:

$\begin{array}{c}\text{Z}=\frac{\text{X}-\mu }{\sigma }\\ =\frac{10\%-6.4\%}{8.4\%}\\ =0.4286\end{array}$

Considering, the Z value 0.4286 is approximately equal to 0.43.

$\text{Pr}(\text{R}\ge \text{1}0\%)=\text{ 1}-\text{ Pr}(\text{R}\le \text{1}0\%)$

$\begin{array}{c}=\text{ 1}-\text{ Pr}(\text{R}\le \text{1}0\%)\\ =\text{ 1}-0.\text{666}\\ =\text{33}.\text{4}\end{array}$

Hence, the probability if the return is greater than 10% is 33.4%.

When the return of long-term bonds is less than 0%, the probability would be:

$\begin{array}{c}\text{Z}=\frac{\text{X}-\mu }{\sigma }\\ =\frac{0\%-6.4\%}{8.4\%}\\ =-0.7619\end{array}$

Considering, the Z value $-0.7619$ is approximately equal to $-0.76$.

$\begin{array}{c}\text{Z =Pr}(\text{R}<0\%)\\ =\text{Pr}(\text{R}\text{>}-0.76)\\ =0.223\end{array}$

Hence, the probability if the return is less than 0% is 22.3%.

The probability is 33.4%, when the return on long-term corporate bonds is greater than 10%. The probability is 22.3%, when the return on long-term corporate bonds is less than 0%.

(b)

Summary Introduction

To determine: The probability of long-term bonds and t-bill

Explanation of Solution

When the returns of t-bills are greater than 10%, then the probability would be:

$\begin{array}{c}\text{Z}=\frac{\text{X}-\mu }{\sigma }\\ =\frac{10\%-3.5\%}{3.1\%}\\ =2.0968\end{array}$

$\begin{array}{c}\text{Pr}(\text{R}\ge \text{1}0\%)=\text{ 1}-\text{ Pr}(\text{R}\le \text{1}0\%)\\ =\text{ 1}-\text{ Pr}(\text{R}\le \text{1}0\%)\\ =\text{ 1}-0.\text{9802}\\ =1.80\%\end{array}$

Hence, the probability if the return on t-bills is greater than 10% is 1.80%.

When the returns of t-bills are less than 0%, then the probability would be:

$\begin{array}{c}\text{Z}=\frac{\text{X}-\mu }{\sigma }\\ =\frac{0\%-3.5\%}{3.1\%}\\ =-1.129\end{array}$

Considering, the Z value $-1.129$ is approximately equal to $-1.13$.

$\begin{array}{c}\text{Z =Pr}(\text{R}<0\%)\\ =\text{Pr}(\text{R}\text{>}-1.13)\\ =0.129\end{array}$

Hence, the probability if the return on t-bills is less than 0% is 12.9%.

The probability is 1.80%, when the return on t-bill is greater than 10%. The probability is 12.9%, when the return on long-term corporate bonds is less than 0%.

c)

Summary Introduction

To determine: The probability of long-term bonds and t-bill

Explanation of Solution

When the return of long-term bonds is less than -4.18%, then the probability would be:

$\begin{array}{c}\text{Z}=\frac{\text{X}-\mu }{\sigma }\\ =\frac{-4.18\%-6.4\%}{8.4\%}\\ =-1.2595\end{array}$

Considering, the Z value $-1.259$ is approximately equal to $-1.26$.

$\begin{array}{c}\text{Z =Pr}(\text{R}\le -4.18\%)\\ =\text{Pr}(\text{R}\text{>}-1.26)\\ =0.1038\end{array}$

Hence, the probability if the return on t-bills are less than – 4.18% is 10.38%.

When the return of T-bills is greater than 10.56%, the probability would be:

$\begin{array}{c}\text{Z =}\frac{\text{X}-\mu }{\sigma }\\ =\frac{10.56\%-3.6\%}{3.1\%}\\ =2.2774\end{array}$

Considering, the Z value 2.2774 is approximately equal to 2.28.

$\text{Pr}(\text{R}\ge \text{1}0.56\%)=\text{ 1}-\text{ Pr}(\text{R}\le 10.56\%)$

$\begin{array}{c}=\text{ 1}-\text{ Pr}(\text{R}\le 10.56\%)\\ =\text{ 1}-0.\text{9887}\\ =0.0113\end{array}$

Hence, the probability if the return on t-bills is greater than 10.56% is 1.13%. The probability is 10.38%, when the long-term bonds are less than -4.18. The probability is 1.13%, when the long-term bonds are greater than 10.56%.