MMW 04-05

Interest

sum of money paid for the use of another person's money

Principal

amount of money borrowed or invested.

Rate

percent taken from the principal.

Term

period or time of the loan

Amount

sum of the principal and the interest

principal (P); The interest rate (r); the time (t); total amount or the final amount (F)

Simple interest is calculated on the

principal (P)

original amount of the loan

interest rate (r)

refers to the percent taken from the principal

time (t)

refers to the term of the loan

total amount or the final amount (F)

amount paid at the end of the term

SIMPLE DISCOUNT INTEREST

Interest deducted in advance

Simple Discount Situation

A scenario where a future amount of money due within a year can be paid in advance if agreed upon by both debtor and creditor.

Future Value

amount of money due on a certain future date, usually within a year

Present Value (Proceeds)

The amount to be paid today

less than

Present Value (Proceeds) is _____ the future value due to the deduction of a discount.

Discount

The amount subtracted from the future value, calculated based on time and the discount rate.

Discount Loan

interest is deducted at the time the loan is obtained.

interest period

will involve two dates

date the loan is given & end date or the maturity date

2 dates of interest period

Exact or Actual Time

actual number of days between two dates

Exact or Actual Time

exact or actual number of days in any given month

Approximate Time

considered that there were 30 days in each month or 360 days in one year

Ordinary Interest Using Exact Time (Banker's Rule); Ordinary Interest Using Approximate Time; Exact Interest Using Exact Time; Exact Interest Using Approximate Time

4 methods of computing interest between dates

ordinary and exact interest

2 kinds of simple interest

ordinary and exact interest

interest use the same formula for solving the simple interest but they differ on using the time.

ordinary simple interest

uses 360 days as the equivalent number of days in a year.

Exact simple interest

uses an exact number of days in a year which is 365 (or 366 for leap year)

days

exact & ordinary simple interest are only applicable if the unit of time used is in ___

Compound Interest

interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods

Compound Amount

total accumulated amount at the end of the period

Simple Interest

fixed amount paid only on the principal at the end of the period.

Term Deposit

usually earns simple interest over a fixed period.

Compound Interest

Interest that accrues and is added to the accumulated interest from previous periods

Present value

amount you would have to invest now at a certain interest rate so that you would end up with some predetermined future sum of money

effective interest rate (ER)

equivalent to interest rate compounded annually

effective interest rate (ER)

can be used to compare two rates with different compounding periods

Apportionment

fair division method used to divide a whole (identical, indivisible objects) into various parts or among units that may be entitled to unequal shares

goal of apportionment

determine a method to allocate the total number of items to be apportioned in a fair manner

Determining the number of representatives a state (city or province) can have in the legislative department

most known apportionment problem

Hamilton Method

proposed by Alexander Hamilton to assign voting seats in the House of Representatives to each represented state

Hamilton Method

This method is based on standard divisor and standard quota of the population

largest-remainder method

Hamilton method of apportionment is actually a ____

Hamilton Method

sets the divisor as the proportion of the total population per house seat.

Hamilton Method

After each state's population is divided by the divisor, the whole number of the quotient is kept and the fraction dropped. This will result in surplus house seats. The first surplus seat is assigned to the state with the largest fraction after the original division. the next is assigned to the state with the second-largest fraction and so on.

Hamilton Method

1. Calculate each group's standard quota.

2. Round each standard quota down to the nearest integer (the lower quota). Initially, each group receives its lower quota.

3. Distribute any leftover items to the groups with the largest fractional parts until all items are distributed.

Modified Divisor (MD)

chosen to adjust seat assignments when the initial assigned numbers do not match the total seats.

higher divisor, lower divisor

If the total seats assigned are too high, choose a ____. If the total seats are too low, choose a _____.

Jefferson Method, Adams Method, and Webster Method

use the modified divisor to fairly distribute seats

Jefferson method

avoids the problem of an apportionment resulting in a surplus or a deficit of House seats by using a divisor that will result in the correct number of seats being apportioned.

Jefferson method

uses a modified standard divisor (MSD) which is chosen by trial and error until the sum of the lower quotas is equal to the required number of allocations.

Jefferson method

The lower quota is the final apportionment obtained at the final value of the modified divisor.

Jefferson method

In this method, the MSD < SD.

Adams method

rounds up the quotients obtained after dividing the population with the standard divisor

Adams method

Pick a modified divisor, d, that is slightly more than the standard divisor.

Webster method

modified version of the Hamilton method

Webster method

After the state populations are divided by the divisor, those with quotients that have fractions of 0.5 or above are awarded an extra seat. States with a quotient with a fraction below 0.5 have the fraction dropped.

Webster method

Round all of the shares to the nearest whole number

Webster method

modify the standard divisor (by making it larger or smaller) until the rounded shares add up to the correct total

Huntington-Hill method

modified version of the Webster method, but it uses a slightly different rounding method.

Huntington-Hill method

rounds at the geometric mean.

geometric mean of two numbers

square root of their product

round down

If SQ < G.M.

round up

If SQ > G.M.,

Standard quota

population of the group/number of group representatives

Average Constituency

refers to the quotient of population of the group and the number of group representatives

Absolute Unfairness of Apportionment

refers to the absolute value of the difference between the larger average constituency and the smaller one

Relative Unfairness of Apportionment

refers to the quotient between the absolute unfairness of the apportionment and the average constituency of the group receiving the apportionment.

Principle of Apportionment

When adding a new representative to a group, the representative is assigned to the group in such a way as to give the smallest relative unfairness of apportionment.

Voting

tool used by groups of people to make a collective decision.

Plurality Method of Voting

simply means the choice with the most votes wins.

Majority Method of Voting

at least 1 vote more than half of the total number of votes

Majority System

most common voting system applied to an election with only two candidate

more than half (>50%)

winner in the majority system requires _____ of the people voting for an issue or a candidate.

Unanimous Method of Voting

everyone votes for the same choice

Plurality Voting

most commonly used and easiest method to use when there are more than two candidates.

Plurality Voting

Each voter votes for one candidate. The candidate receiving the most votes is declared the winner.

Plurality with Elimination Voting

Each voter votes for one candidate. If a candidate receives a majority of votes, that candidate is declared the winner. If no candidate receives a majority, eliminate the candidate with the fewest votes and hold another election. (If there is a tie for the fewest votes, eliminate all candidates tied for the fewest votes.) Repeat this process until a candidate receives a majority.

Plurality with Elimination Voting

The candidate with the fewest number of first-place votes is first eliminated. In case there are two alternatives with the same lowest votes, then both are to be eliminated. The remaining candidates are re-ranked with the assumption that voters' preference do not change from round to round

Borda Count Method of Voting

Voters rank candidates from the most favorable to the least favorable. Each last-place vote is awarded one point, each next-to-last-place vote is awarded two points, each third-from-last-place vote is awarded three points, and so forth. The candidate receiving the most points is the winner of the election

Pairwise Comparison Method of Voting

Voters rank the candidates. A series of comparisons in which each candidate is compared with each of the other candidates follows. If candidate A is preferred to candidate B, A receives one point. If candidate B is preferred to candidate A, B received 1 point. If the candidates tie, each receives ½ point. After making all comparisons among the candidates, the candidate receiving the most points is declared the winner.

Majority criterion

fairness criteria if candidate who receives a majority of the first-place votes is the winner.

Monotonicity criterion

fairness criteria If candidate A wins an election, then candidate A will also win the election if the only change in the voters' preferences is that supporters of a different candidate change their votes to support candidate A.

Condorcet criterion

fairness criteria A candidate who wins all possible head-to-head matchups should win an election when all candidates appear on the ballot.

Independence of irrelevant alternatives

fairness criteria If a candidate wins an election, the winner should remain the winner in any recount in which losing candidates withdraw from the race

Arrow's Impossibility Theorem

There is no voting method involving three or more choices that satisfies all four fairness criteria.

Nominal rate

Rate at which a given amount is invested to produce another amount after a given period of time