January 2019 - Infinity is Really Big

January 30, 2019January 31, 2019

New Video: Immunization, Part 1

It is very important for any business, including any insurance company, to be able to meet their financial liability obligations.

A major class of liabilities that insurance companies must plan for are the claims from their customers. For example, if a person buys car insurance, then the insurance company would be obligated to pay for, among other things, some or … Read the rest

January 28, 2019July 26, 2019

Does the Square Root of Two Exist?

The proof that the square root of two is an irrational number is considered to be one of the most elegant in all of mathematics.

As a formal “If…, then…” statement, it reads: If $x^{2}=2,$ then $x$ is not a rational number. A more informal statement of this is “the square root of two is irrational”.

Rational Numbers

But what is a … Read the rest

January 24, 2019January 28, 2019

New Video: Rebalancing a Bond Portfolio for a New Macaulay Duration

Rebalancing a portfolio is an important kind of financial planning action. It can be done to increase your rate of return or to change the susceptibility of the value of your investments to changes in yield rates.

For a coupon bond, the “duration” or “length” of the bond is a way to measure its susceptibility to … Read the rest

January 22, 2019July 18, 2019

Deconstructing the Mean Value Theorem, Part 3

The Mean Value Theorem implies that there is at least one tangent line with the same slope as the secant line.

Pete Brady strolled out of the U.S. Postal Service office just off Weaver Lake Road in Maple Grove, Minnesota at exactly 10 am.

Coffee and important package in hand, he set off on his 161 mile (259 kilometer) journey … Read the rest

January 18, 2019January 28, 2019

New Video: Derivatives of Bond Prices w.r.t. Parallel Shift in Yield Curve

The series of payments that arises from a coupon bond is an example of a cashflow that can be assigned a “duration“.

A coupon bond can be purchased as an investment that pays “coupons”, which are fixed amounts of money, at certain equally-spaced times during the term of the investment, plus a redemption … Read the rest

January 16, 2019January 28, 2019

Financial Mathematics Terms: Effective Versus Nominal Rates of Interest

One of the most confusing topics for many people when starting to learn about financial mathematics is the various ways of measuring compound interest.

Terms like “effective interest rate”, “nominal interest rate”, “continuous interest rate”, and “real interest rate” show up in various articles and textbooks. Exploring the distinctions and relationships between effective and nominal interest rates is the … Read the rest

January 15, 2019January 28, 2019

New Video: Macaulay Duration is a Decreasing Function of the Coupon Rate r

Early today I posted my 166th video on problem solving for actuarial exam 2/FM (financial mathematics) on YouTube.

Wow! One-hundred and sixty-six videos on financial mathematics for actuaries?!?! Yes! This has been an almost two-year YouTube project so far. Here’s the video that I posted earlier today, about an advanced topic called Macaulay duration… Read the rest

January 15, 2019July 13, 2019

Deconstructing the Mean Value Theorem, Part 2

*The Mean Value Theorem says there’s at least one point where the slope of the tangent line on a smooth curve equals the slope of the secant line between the endpoints.*

At its heart, the Mean Value Theorem is fairly intuitive. It essentially states that a smooth curve has at least one tangent line whose slope is equal to the

… Read the rest

January 11, 2019January 28, 2019

Financial Mathematics Fundamentals: Time Value of Money, Interest Rates, and Discount Factors

The word “discount” has the rare ability to cause peoples’ hearts to race for joy.

I certainly get excited when items are on sale, though I am wary of situations where the original price might be high to begin with.

The mathematics of discounts can be confusing for many people. I once had a student who was very fast at … Read the rest

January 10, 2019July 4, 2019

Financial Mathematics Tools: Substitutions, Quadratic Formula, Logarithms, and “Functional-Thinking”

Useful pre-university mathematics is not restricted to arithmetic, percentages, and basic algebra.

Functional-thinking is one key to doing many other useful things. Tools such as substitutions, the quadratic formula, and logarithms also come in handy.

These tools are used in engineering and science. For example, in the evaluation of useful integrals (such as those that come up in … Read the rest