Problem set 2 Q&A this morning

Q (from a 4366 student earlier this morning): In problem set #2 (question 1-E) to calculate the standard deviation of the portfolio, do I need to omit the correlation term in the standard deviation formula?

A (from Dr. Garven): For a two asset portfolio, the standard deviation formula is:

{\sigma _p} = \sqrt {w_1^2\sigma _1^2 + w_2^2\sigma _2^2 + {w_1}{w_2}{\sigma _{12}}},

where {{w_1}} and {{w_2}} correspond to the percentage portfolio allocations for assets 1 and 2, {w_1} + {w_2} = 1, and {\sigma _{12}} = {\rho _{12}}{\sigma _1}{\sigma _2}; i.e., the covariance between assets 1 and 2 is equal to the correlation between 1 and 2 multiplied by the product of the standard deviations for 1 and 2.  When assets 1 and 2 are uncorrelated, this implies that {\rho _{12}} = 0, and in that case,

{\sigma _p} = \sqrt {w_1^2\sigma _1^2 + w_2^2\sigma _2^2} .

Problem Set 2 helpful hints

Problem Set 2 is available from the course website at http://fin4366.garven.com/spring2022/ps2.pdf; its due date is Tuesday, February 1.

Problem Set 2 consists of two problems. The first problem requires calculating expected value, standard deviation, and correlation, and using this information as inputs into determining expected returns and standard deviations for 2-asset portfolios. The second problem involves using the standard normal probability distribution to calculate the probabilities of earning various levels of return by investing in risky securities and portfolios; see pp. 17-23 of the http://fin4366.garven.com/spring2022/lecture4.pdf lecture note for coverage of that topic.

Z Table Extra Credit Assignment (due at the start of class on Tuesday, February 1)

Here’s an extra credit opportunity for Finance 4366. Working on your own (i.e., this is not a group project; credit will only be given for spreadsheets that are uniquely your own), build your own “z” table in Excel (patterned after the table located at http://fin4366.garven.com/stdnormal.pdf); the top row should have values ranging from 0.00 to 0.09, and the first column should have z values ranging from -3.0 to +3.0, in increments of 0.1).

Quite conveniently, Excel has the standard normal distribution function built right in; e.g., if you type “=normsdist(z)”, Excel returns the probability associated with whatever z value that you provide. Not surprisingly, if you type “=normsdist(0)”, .5 is returned since half of the area under the curve lies to the left of the expected value E(z) = 0. Similarly, if you type “=normsdist(1)”, then .8413 is returned because 84.13% of the area under the curve lies to the left of z = 1. Perhaps you recall from your QBA course that 68.26% of the area under the curve lies between z = -1; this “confidence interval” of +/- 1 one standard deviation away from the mean (E(z)=0) is calculated in Excel with the following code: “=normsdist(1)-normsdist(-1)”, and so forth.

The grade you earn on this extra credit assignment will replace your lowest quiz grade; that is if your lowest quiz grade is lower than your extra credit grade. The deadline is the start of class on Tuesday, February 1.

You can turn your spreadsheet for this extra credit assignment in at the link labeled “Z Table Extra Credit Assignment” under the Assignment tab on Canvas.

Calculus, Probability and Statistics, and a preview of future topics in Finance 4366

Probability and statistics, along with the basic calculus principles covered last Thursday, are foundational for the theory of pricing and managing risk with financial derivatives, which is what this course is all about. During tomorrow’s class meeting, we will discrete and continuous probability distributions, calculate parameters such as expected value, variance, standard deviation, covariance, and correlation, and apply these concepts to measure expected returns and risks for portfolios comprising risky assets. During Thursday’s class meeting, we will take a deeper dive into discrete and continuous probability distributions, in which the binomial and normal distributions will be showcased.

On Tuesday, February 1, we will introduce and describe the nature of financial derivatives, and motivate their study with examples of forwards, futures, and options. Derivatives are so named because they derive their values from one or more underlying assets. Underlying assets typically involve traded financial assets such as stocks, bonds, currencies, or other derivatives, but derivatives can derive value from pretty much anything. For example, the Chicago Mercantile Exchange (CME) offers exchange-traded weather futures and options contracts (see “Market Futures: Introduction To Weather Derivatives“). There are also so-called “prediction” markets in which derivatives based upon the outcome of political events are actively traded (see “Prediction Market“).

Besides introducing financial derivatives and discussing various institutional aspects of markets in which they are traded, we’ll consider various properties of forward and option contracts, since virtually all financial derivatives feature payoffs that are isomorphic to either or both schemes. For example, a futures contract is simply an exchange-traded version of a forward contract. Similarly, since swaps involve exchanges between counter-parties of payment streams over time, these instruments essentially represent a series of forward contracts. In the option space, besides traded stock options, many corporate securities feature “embedded” options; e.g., a convertible bond represents a combination of a non-convertible bond plus a call option on company stock. Similarly, when a company makes an investment, so-called “real” options to expand or abandon the investment at some future is often present.

Perhaps the most important (pre-Midterm 1) idea that we’ll introduce is the concept of a so-called “arbitrage-free” price for a financial derivative. While details will follow, the basic idea is that one can replicate the payoffs on a forward or option by forming a portfolio comprising the underlying asset and a riskless bond. This portfolio is called the “replicating” portfolio, since, by design, it replicates the payoffs on the forward or option. Since the forward or option and its replicating portfolio produce the same payoffs, then they must also have the same value. However, suppose the replicating portfolio (forward or option) is more expensive than the forward or option (replicating portfolio). If this occurs, then one can earn a riskless arbitrage profit by simply selling the replicating portfolio (forward or option) and buying the forward or option (replicating portfolio). However, competition will ensure that opportunities for riskless arbitrage profits vanish quickly. Thus the forward or option will be priced such that one cannot earn arbitrage profit from playing this game.

Additional (fourth) required reading for class on Tuesday, January 25

Just want to give everyone a heads up about an additional (4th) required reading for this Tuesday’s Finance 4366 class meeting. The required readings list now also includes a brief (3-page) teaching note entitled “Calculating (Math) Derivatives”. This reading is required, along with Peter Bernstein’s Harvard Business Review article entitled The New Religion of Risk Management and two other teaching notes I authored, entitled Normal and standard normal distribution and Mean and Variance of a Two-Asset Portfolio.

As I point out in the introduction of “Calculating (Math) Derivatives”,

“In Finance 4366, competency with basic math and stat principles is essential. The Finance 4366 math tutorial is designed to ensure competency with math principles that are needed for understanding how to price financial derivatives (such as options and futures contracts) and designing risk management strategies using derivatives. The 2-part stat tutorial (to be taken up during the second week of class) is designed to ensure competency with stat principles which are used throughout the course.

In this teaching note, I briefly delve further into the logical foundations for calculating math derivatives.”

On the ancient origin of the word “algorithm”

The January 25th assigned reading entitled “The New Religion of Risk Management” (by Peter Bernstein, March-April 1996 issue of Harvard Business Review) provides a succinct synopsis of the same author’s 1996 book entitled “Against the Gods: The Remarkable Story of Risk“. Here’s a fascinating quote from page 33 which explains the ancient origin of the word “algorithm”:

“The earliest known work in Arabic arithmetic was written by al­Khowarizmi, a mathematician who lived around 825, some four hun­dred years before Fibonacci. Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “al­Khowarizmi” fast. That’s where we get the word “algo­rithm,” which means rules for computing.”

Note: The book cover shown above is a copy of a 1633 oil-on-canvas painting by the Dutch Golden Age painter Rembrandt van Rijn.

Origin of the “Product Rule”, and Visualizing Taylor polynomial approximations

This blog entry provides a helpful follow-up for a couple of calculus-related topics that we covered during today’s Mathematics Tutorial.

  1. See page 12 of the above-referenced lecture note.  There, the equation for a parabola (y = {x^2}) appears, and the claim that \frac{{dy}}{{dx}} = 2x is corroborated by solving the following expression:
    In the 11-minute Khan Academy video at https://youtu.be/HEH_oKNLgUU, Sal Kahn takes on the solution of this problem in a very succinct and easy-to-comprehend fashion.
  2. In his video lesson entitled “Visualizing Taylor polynomial approximations”, Sal Kahn replicates the tail end of today’s Finance 4366 class meeting in which we approximated y = ex with a Taylor polynomial centered at x=0 (as also shown in pp. 18-23 of the Mathematics Tutorial lecture note).  Sal approximates y = ex with a Taylor polynomial centered at x=3 instead of x=0, but the same insight obtains in both cases, which is that the accuracy of Taylor polynomial approximations increases as the order of the polynomial increases.

Week 2 readings, quiz, and problem set

Here’s a friendly reminder that the following readings are due next Tuesday:

1. The New Religion of Risk Management, by Peter Bernstein
2. Normal and standard normal distribution, by James R. Garven
3. Mean and Variance of a Two-Asset Portfolio, by James R. Garven

Keep in mind that Quiz 2, which is based on these readings, must be completed prior to the start of class next Tuesday.

Going forward, I will typically not post reminders like this concerning Finance 4366 assignment deadlines; however, you’ll be “good to go” in Finance 4366 if you faithfully follow the dates and times listed on Canvas and on the course website.