128MTSM study material
Quarantine of the corona virus will be stretched, so we have to do the lessons remotely. I had to adapt
the content of the lesson to this, thus, it took me some time. However, I would ask all of you to
comment on the content of the lesson. So what specifically would you like to learn more, try or
practice. What problems to try to solve.
Please address the following tasks one by one, and so communicate with me about them. I don't want
you to spend a long time over all tasks before asking questions and handling uncertainties.
Task 1 - Supplementary pension savings:
Attached is an Excel file with a model in which iterative calculation is used.
The calculations in this file are described below.
Usually, we would first discuss the model, what to model, what you know from your background and
country. Due to the “remote lessons” necessity, however, I had to make the selection myself.
The calculations relate to Pension Savings supported by the State, under the auspices of the State. The
State contributes partially to the participants, according to the amount of their own savings
contribution. I would like you to choose a similar (maybe savings) program, e.g. in your country,
with state or other support. It does not have to be a pension scheme, the states also support other
similar products, e.g. building savings (for purchase of an apartment or house, its equipment…).
And you can process it - in a similar way as the example below. If you have questions and/or
encounter problems, I assume that we will communicate about them and resolve them…
Instructions:
The file "Supplementary pension savings.doc" contains a description of current Czech pension
savings, how it works for participants.
The file "Supplementary pension savings.xls" contains the following:
On the "Suppl. Pension Savings 2013” tab there are input data in cells with a yellow background. You
can change it to try the calculation with different parameters.
The “Rules for State Contribution” table calculates the state contribution based on the amount of the
participant's contribution. The state contributes nothing if the participant's contribution is x<300CZK.
If the participant pays x = CZK 300 a month, the state will contribute CZK 90. If the participant pays
more than x>300CZK but at most x<=1000CZK, the state contribution is 90CZK plus 20% of what x
exceeds 300CZK (i.e. plus 20% of (x-300)). If the participant pays x>=1000CZK, the contribution
from the state is 230CZK, it does not increase. The last two columns "Auxiliary Subsidy Value" and
"Monthly Subsidy by Deposit" calculate the state contribution in cell M6 (and E6) based on the actual
contribution of the participant "Regular Monthly Deposit" in cell E5. Yearly Fund Valorization YFV
in cell E8 is another input value. The 6-month Fund Valorization M6FV evaluation in cell E9 is
calculated based on equality (1 + M6FV) * (1 + M6FV) = (1 + YFV). This value is used to
approximate a yearly even series of payments - it is replaced by one total payment in the middle of the
year, with only half a year interest.
The lower extensive table contains a year-by-year calculation. Review the formulas/calculations/row
for yr=2 or higher, then compare the formulas to yr=1. The Monthly Contribution/Deposit column,
when multiplied by 12 it produces the total annual deposit, then the total contribution for all years
since the beginning of the savings. This is followed by the amount inclusive the valorization for
previous years except the latest year, the appreciation for the latest year and the total amount including
valorization for the latest year. The calculation of amounts on the basis of State contributions is
analogous. The column "Total: S" contains the total amount in the pension savings account at the end
of the relevant year.
To the right of the next “year” column is the calculation used to determine “profit” as a percentage
(profitability). First we write/choose an (estimated, expected or required) interest rate (corresponding
to profitability) in column “i” - the meaning of this value is: what interest rate we would like to get or
would have to get in the bank (or in alternative investment) to get some amount; in the end we will
aim at the same amount as in pension savings. The interest coefficient u=1+i follows. The SI column
calculates the amount that we would have received in an alternative investment (e.g. a bank account)
with the selected interest rate. It is a formula you can find for financial calculations on the Internet, or
even in financial calculations in Excel. The principle is that we replace 12 monthly deposits with one
aggregate deposit in the middle of the year. The annual deposit P thus bears interest for 1 to n years
plus half the last year (member u^0.5). The column “S-SI” contains the difference between the amount
obtained (in a given year) from pension savings and from alternative investment. The (S-SI)/S/year
column is a bit of a trick, among other things, it provides the necessary numerical stability for the later
iterative calculation of the interest rate "i". While S-SI is the absolute difference in return, the
expression (S-SI)/S is the relative difference in return on both investments. Divided by the number of
years, i.e. the current length of savings, “(S-SI)/S/years” is the annual rate of return (interest rate
difference) of both investments. The last column “~i” is the sum of the original estimated interest rate
(profitability) and the difference in interest rates, so we are essentially trying to approach the interest
rate of pension savings. Try to enter the value "i" on some line (i.e. for some year, e.g. year=3) and
decrease or increase this value "i" according to the calculated "~i".
Attention: In the following, iterative calculations are needed. If an error message appears, reading
“There are one or more circular references…” we need to allow for it. Go to File tab, select “Options”,
then “Formulas, and check the “Enable iterative calculation” checkbox.
Finally, we want to calculate the interest rate (profitability) of pension savings (varies according to the
length of savings). If we choose/estimate "i" by exactly hitting the interest rate (profitability) of
pension savings, SI will be equal to S, column "S-SI" will be 0, and column "~i" will be equal to
column "i". This equality "i=~i" is achieved by an iterative calculation - we create a cycle by inserting
a value from the column (reference to the column) "~i" into the column "i", e.g. in cell "O20" write a
reference '=T20'. The row is recalculated by an iterative calculation. Then copy (stretch) this row
(links and formulas in the O-T columns) to the other rows. If this copying results in a numerical
instability of the iterative calculation, the message “#NUM!” appears in the cells. Then repeat the copy
from the cells before the line with the message “#NUM!”.
Note: On the “Supplementary Pension Insurance” tab, there is the pension savings model as valid until
2012. It differs in the rules for the amount of the state contribution, see the table “Rules for State
Contribution”.
Q1: Use this Excel model of pension savings to answer the following questions: Is the profitability of
this savings acceptable to me? Would I commit myself (obligate myself, engage myself) to this
savings, and for how long?
Task 2:
Coronavirus is the theme of the day. States and governments take various and different measures. For
the time being, try to describe verbally a (simple enough!!!, not too complicated!!!) model of the
infection spreading, healing and deaths of patients. The spreading should depend on some parameters
- identify and describe those parameters. Example of such a parameter: how many other new people
are infected by an already infected patient per unit of time (probably 1 day), depending on other
parameters such as the level of infection of the patient. Sometimes, parameters may be “averaged”,
attain/have the same value for all people, like the Fund Valorization in SPS model above.
It is not necessary that the model be limited to coronavirus. Similarly, individual opinions and
ideologies spread throughout society. Consider history examples, the principle of human freedom,
liberation from feudal serfdom (“freedom, equality, fraternity”). Or ideologies such as fascism,
Marxism/communism, Nazism. Or various contemporary neo-Marxist ideologies that spread primarily
throughout the ruling circles of Europe and North America. We would need a somewhat different
model for competing ideologies, for ideologies fighting against each other.
Question: Inoculation, vaccination can be used against biological infection. What is the analogy of
vaccination in disseminating opinions or ideologies?
We would like to process the resulting model in Excel. In a similar way to the “Supplementary pension
savings” model, we want to model the development at different times, here rather by days rather than
by years.
Task 3:
In the Czech Republic, protected species of birds, fish-eating – specifically a few of cormorant species
– live in several places, in areas with ponds and reservoirs. One species is threatened with extinction
due to changes in water management conditions. The threat of extinction is a consequence of the lack
of food and consequently the inability to feed and bring out the young. Suggest some measures to save
this species in nature (other than relocation to the zoo), and design a model (simple enough!) that
calculates the functioning (over time) of the proposed measures.
Task 4:
Suggest what problem you would like to model.
Task 5:
Q1: How would you define the meaning of the word “society”?
Q2: How would you define what is “democratic”? How to distinguish democratic from nondemocratic? (It would be more difficult to define the meaning of “democracy” than the attribute
“democratic”.)
Q3: How would you define what is “progressive”? How to distinguish progressive from nonprogressive?
Note: you can search the Internet, but then think about it and add your ideas. If you email me answers,
don't copy what you found on the Internet or in books, but give your opinions, ideas, and eventually
comments on what you found as an official or common opinion.
Task 6:
In the lessons I often assign “exercises” for “creative thinking/reasoning”. Well, here are some such
exercises. If you feel like it, have a nice time searching for solutions. Can you make a “model of your
reasoning”, before and after resolving the respective exercises? How does your model, as created
before the exercise is resolved, changes after you have found the solution?
Exercise 1: We have two pots/containers, the first with 1 liter of milk, the second with 1 liter of tea.
We also have a spoon/ladle available.
Step 1: Pick up a full spoon of milk from the first pot, and pour it into the second pot of tea.
Step 2: Mix the contents in the second pot thoroughly to make the milk and tea mixture homogeneous.
Step 3 Take the same full spoon of mixture from the second pot and pour it into the first pot with milk.
Question 1: Is there more tea in the milk in the first pot or is there more milk in the tea in the second
pot? Explain the result, justify it to a layman, without (lengthy) mathematical calculations.
Excercise 2: We have 9 points in a regular 3x3 square grid - see the drawing.
Draw these 9 points on a sheet of paper. The task is to connect all 9 points with a continuous polyline
consisting of several straight lines, drawn in one stroke without raising the pen/pencil, for example as
follows:
or:
or:
In the examples, the polyline consists of 5 straight lines.
or:
Q1: Is it possible to connect all 9 points with a continuous polyline consisting of only 4 lines? If so,
how?
Q2: Is it possible to connect all 9 points with a continuous polyline consisting of only 3 lines? If so,
how?
Q3: Is it possible to connect all 9 points with a continuous polyline consisting of just one line? If so,
how?