February 2019 - Infinity is Really Big

February 26, 2019February 27, 2019

New Video: Dynamic Pascal’s Triangle

Made with Mathematica

The top 6 rows of Pascal’s triangle. Each row starts and ends with a “1”. Any number in-between the ends is the sum of the two numbers most close to it on the preceding line.

Pascal’s triangle is an infinite triangle of numbers with many internal patterns.

Start by creating a list of “1’s” in an upside-down … Read the rest

February 22, 2019February 22, 2019

New Video: Immunization, Part 6

Checking Conditions for (Lack of) Redington Immunization in a Spreadsheet

Spreadsheets are extremely useful tools for many business, academic, and personal applications.

Examples of situations where spreadsheets are useful include: budgeting (totals, averages, piecharts), descriptive statistics (means, standard deviations, correlations, graphs), inferential statistics (t-statistics, p-values, ANOVA regression tables), finance (monthly payments, internal rates of return), simulation (random number generation, discrete … Read the rest

February 20, 2019February 21, 2019

New Video: Immunization, Part 5

Redington immunization is a concept that can be defined in terms of derivatives of present value functions.

Given asset cashflows $A_{0},A_{1},A_{2},\ldots,A_{n}$ and liability cashflows $L_{0},L_{1},L_{2},\ldots,L_{n},$ each set of flows occurring at times $t=0,1,2,\ldots,n,$ the expressions $P_{A}(i)=\displaystyle\sum_{t=0}^{n}A_{t}(1+i)^{-t}$ and $P_{L}(i)=\displaystyle\sum_{t=0}^{n}L_{t}(1+i)^{-t}$ represent the present values of these cashflows, as functions of an arbitrary periodic interest rate $i.$

If we let $h(i)=P_{A}(i)-P_{L}(i)=\displaystyle\sum_{t=0}^{n}C_{t}(1+i)^{-t},$ where $C_{t}=A_{t}-L_{t},$ then we say the liabilities are Redington immunized … Read the rest

February 12, 2019February 12, 2019

New Video: Immunization, Part 4

In “New Video: Immunization, Part 3“, I showed that if a liability cashflow is not immunized by an asset cashflow, we might be able to make modifications to change this state of affairs.

In particular, the time of payment of one of the asset amounts could be changed to achieve, at least, Redington immunization. To be even … Read the rest

February 8, 2019December 1, 2019

Why Does Slicing a Cone Give an Ellipse?

The intersection of the pink plane with the blue cone is an ellipse (shaded orange when y > 0 and brown when y < 0). The length of either red line added to the length of either yellow line is constant.

Currently my favorite YouTube channel is “3Blue1Brown“, created by Grant Sanderson.

With excellent visuals and high production quality, Grant both shows and explains mathematics in a way that … Read the rest

February 4, 2019February 12, 2019

New Video: Immunization, Part 3

Immunization of a liability cashflow by an asset cashflow can help to cushion a company when interest rates change, but is it always possible?

My most recent video is “Actuarial Exam 2/FM: Liabilities Not Immunized by Assets in Spite of PV and Duration Matching” (Financial Math for Actuarial Exam 2 (FM), Video #171. October 2018 SOA Sample Exam, Problem #127). … Read the rest

February 1, 2019February 1, 2019

New Video: Immunization, Part 2

In the previous post, “New Video: Immunization, Part 1“, I introduced the idea of Redington immunization of liability cashflows by asset cashflows.

It was a post that delved into the details of my previous YouTube video “Actuarial Exam 2/FM Prep: Use Redington Immunization to Find a Ratio of Assets in a Portfolio” embedded … Read the rest