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2019 | Book

1 Introduction Read chapter Read first chapter

Solutions to Financial Economics

Exercises on Classical and Behavioral Finance

Authors: Prof. Dr. Thorsten Hens, Prof. Dr. Marc Oliver Rieger

Publisher: Springer Berlin Heidelberg

Book Series : Springer Texts in Business and Economics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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About this book

This book offers a concise introduction to the field of financial economics and presents, for the first time, recentbehavioral finance research findings that help us to understand many puzzles in traditional finance. Tailor-made formaster’s and PhD students, it includes tests and exercises that enable students to keep track of their progress. Parts of the book can also be used at the bachelor level.

Table of Contents

Frontmatter

Exercises

Frontmatter
1 Introduction
Abstract
In this part, we present a large number of exercises that can accompany our book Financial Economics. They are sorted by the chapters of the book. Within each chapter, the exercises are roughly sorted by topics such that topics covered earlier in the chapter come first. A second criterion is by the difficulty (the easier exercises first). Many exercises are far from being “routine”. We think that exercises that just plug in numbers into formulas being learned by heart do not help students much to comprehend a topic. They also don’t help the lecturer when designing exams or homework assignments: such simple “plug-and-play” exercises are easier designed from scratch then copied from a textbook. Instead, we tried to design exercises that inspire thinking and encourage deeper understanding of the subject. Often there is not only one solution and sometimes students who do not find the optimal solution can at least try to get partial or approximative results. Teasing out the creativity of students in solving problems is very helpful for guiding them into making their own research and this is what we aim to achieve with our exercises.
Thorsten Hens, Marc Oliver Rieger
2 Decision Theory
Abstract
Consider the following game: you roll a dice, if you roll a 6, you win 6 million € otherwise you win nothing. You can play only once. Let us assume your expected utility function is given by u(x) =log10 x (base 10 logarithm, i.e., log10(10n) = n) and your initial wealth is 10,000 €.
Thorsten Hens, Marc Oliver Rieger
3 Two-Period Model: Mean-Variance Approach
Abstract
There are two risky assets, k = 1, 2 and one risk-free asset with return of 2%. Risky assets cannot be short sold. The expected returns of the risky assets are μ 1 := 5% and μ 2 := 7.5%. The covariance matrix is:
$$\displaystyle COV := \begin {pmatrix}2\% & -1\%\\ -1\% & 4\% \end {pmatrix}.$$
Thorsten Hens, Marc Oliver Rieger
4 Two-Period Model: State-Preference Approach
Abstract
Consider a two-period economy with uncertainty in the second period. Consumption is in terms of a single consumer good. In the second period there are S many possible states and every consumer aims to maximize the consumption across states. There are I many consumers with utility functions U i(strictly increasing, concave and continuous). The consumption good has a price π s in each state.
Thorsten Hens, Marc Oliver Rieger
5 Multiple-Periods Model
Abstract
Let us consider the following three-period model where the returns of two assets are marked at each node
Thorsten Hens, Marc Oliver Rieger
6 Theory of the Firm
Abstract
Suppose S = 1 and wealth in period one is produced from wealth in period zero by the production function \(\sqrt {\quad}\). Define the production technology set \(Y \subset \mathbb {R}^2\) and check whether the properties (i)–(v) given in Assumption 6.1 in the book are satisfied.
Thorsten Hens, Marc Oliver Rieger
7 Information Asymmetries on Financial Markets
Abstract
There are two time periods t = 0, 1 and two states in the second period s = 1, 2. There are two consumers i = 1, 2. The first consumer is rich today and poor tomorrow, w 1 = (1, 0, 0). The second is rich tomorrow and poor today, w 2 = (0, 1, 1). There are two Arrow securities, i.e. \(A = \begin {pmatrix}1 & 0 \\ 0 & 1\end {pmatrix}\). The first consumer does not know which state occurs and a priori assigns equal probabilities to them. Before trading the asset the second consumer gets a signal revealing the state of the world. Both consumers have ln-utility of wealth, and no time discount rate. All this information is common knowledge. No consumer acts strategically (as if there were two types of infinitely many consumers).
Thorsten Hens, Marc Oliver Rieger
8 Time-Continuous Model
Abstract
Let W be a Wiener process. Find the expressions for
$$\displaystyle d(W^{2}) \; \text{and } (dW)^{2}.$$
Thorsten Hens, Marc Oliver Rieger

Solutions

Frontmatter
1 Introduction
Abstract
In this part, we provide the solutions to the exercises from Part I. Sometimes we will be very brief (if we think that this is sufficient to understand the solution), sometimes we will be a bit more wordy (if the solution requires some careful argument).
Thorsten Hens, Marc Oliver Rieger
2 Decision Theory
Abstract
The efficiency of the measure can be determined through computing the expected utility of playing the lottery of illegal parking under that measure. The measure with the lowest expected utility of illegal parking should be taken.
Thorsten Hens, Marc Oliver Rieger
3 Two-Period Model: Mean-Variance Approach
Abstract
The minimum–variance portfolio is the portfolio of risky assets with the minimal portfolio variance:
$$\displaystyle(\lambda _1^{MV}, \lambda _2^{MV}) = \arg \min _{\lambda } \;\sigma _\lambda ^2, \quad \text{ s.t. } 0 \le \lambda _1, \, 0 \le \lambda _2, \, \lambda _1+\lambda _2=1,$$
where the variance of portfolio λ is
$$\displaystyle\sigma _\lambda ^2 = \sigma ^2(\lambda _1 R_1+ \lambda _2 R_2) = \lambda ^T COV \lambda .$$
Thorsten Hens, Marc Oliver Rieger
4 Two-Period Model: State-Preference Approach
Abstract
(a) The budget sets are:
Thorsten Hens, Marc Oliver Rieger
5 Multiple-Periods Model
Abstract
Financial market is complete if any consumption stream can be attained with at least one initial wealth. The necessary and sufficient condition for a financial market to be complete is that each two-period submarket is complete.
Thorsten Hens, Marc Oliver Rieger
6 Theory of the Firm
Abstract
\(f = \left \{(y_0, y_1) \mid y_1 \leq \sqrt {-y_0}, \; y_0 \leq 0 \right \}\)
(i)
f is closed: defined by ≤ 0
 
(ii)
f convex: since https://static-content.springer.com/image/chp%3A10.1007%2F978-3-662-59889-4_14/328309_1_En_14_IEq2_HTML.gif is concave
 
(iii)
y 0 < 0, y 1 > 0
 
(iv)
y 0 = 0 ⇒ y 1 = 0
 
(v)
max production is \(\sqrt {w_0}\)
 
Thorsten Hens, Marc Oliver Rieger
7 Information Asymmetries on Financial Markets
Abstract
Demand of consumer 1
$$\displaystyle \begin{aligned} \begin {array}{c} \max _{\theta _1^1,\theta _2^1} \; \ln (x_0^1) + \frac {1}{2} \ln (x_1^1) + \frac {1}{2} \ln (x1_2^1) \\ x_0^1 + q_1 \theta _1^1 + q_2 \theta _2^1 = 1 \\ x_1^1 = \theta _1^1, \; x_2^1 \\ \implies x_0^1 = \frac {1}{2}, \; x_1^1 = \frac {1}{4 q_1}, \; x_2^1 = \frac {1}{4q_2} \end {array} \end{aligned} $$
Thorsten Hens, Marc Oliver Rieger
8 Time-Continuous Model
Abstract
$$\displaystyle d(W^{2}) = 2WdW + \frac {1}{2}^{2}(dW)^{2} = 2WdW + dt.$$
Thorsten Hens, Marc Oliver Rieger
Backmatter
Metadata
Title
Solutions to Financial Economics
Authors
Prof. Dr. Thorsten Hens
Prof. Dr. Marc Oliver Rieger
Copyright Year
2019
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-59889-4
Print ISBN
978-3-662-59887-0
DOI
https://doi.org/10.1007/978-3-662-59889-4