economic data calculus and economics basics of economic models advanced calculus and economics ...

Econ 299Quantitative Methods in Economics

Economic DataCalculus and EconomicsBasics of Economic ModelsAdvanced Calculus and Economics Statistics and EconomicsEconometric Introduction

Lorne Priemaza, M.A.

[email protected]

1. Data Description, Presentation, and Manipulation

1.1 Data Types and Presentations1.2 Real and Nominal Variables1.3 Price Indexes1.4 Growth Rates and Inflation1.5 Interest Rates1.6 Aggregating Data: Stocks and Flows1.7 Seasonal AdjustmentAppendix 1.1 Exponentials and Logarithms

Why do economists need data?

1) Describe EconomyCurrent and past data Increases and decreasesThis information can influence decisions

ie: GDP, interest rate, unemployment, price, debt, etc.

2) Test TheoryDoes variable A affect variable B?

ie: Smokers and the cost to healthcare

ie: Married couples and health

1 Data Types

Data is essential for economists. Data can be categorized by:

1) How it is collected:time series data cross-sectional datapanel data

2) How it is measured:nominal datareal data

Time Series Data

-Collects data on one economic agent (city/person/firm/etc.) over time-Frequency can vary (yearly/monthly/ quarterly/weekly/daily/etc.)

-ie: Canadian GDP, GMC stock value, your height, U of A tuition, world population

Alberta’s Tuition – Time Series

University Tuit 99/00 Tuit 00/01 Tuit 01/02 Tuit02/03

Alberta 3551.00 3770.00 3890.00 4032.00

British Columbia 2295.00 2295.00 2181.00 2661.00

Calgary 3650.00 3834.00 3975.00 4120.00

Concordia 1668.00 1668.00 1668.00 1668.00

Lethbridge 3360.00 3470.00 3470.00 3470.00

Manitoba 3005.00 2796.00 2807.00 2818.00

McGill 1668.00 1668.00 1668.00 1668.00

Ottawa 3760.00 3892.00 4009.00 4085.00

Year GDP (In current US$, in trillions)

2007 3.52

2008 4.56

2009 5.06

2010 6.04

2011 7.49

2012 8.46

2013 9.49

2014 10.4

China GDP – Time Series

Source: World Development Indicators, The World Bank, www.worldbank.org

Final Fantasy Quality - Time Series Data# Year Rating

1 1987 7.5

2 1988 6.5

3 1990 7.3

4 1991 8.3

5 1992 7.1

6 1994 8.7

7 1997 9.4

8 1999 9.2

Source: www.thefinalfantasy.com

# Year Rating

1 1987 9

2 1988 5

3 1990 7

4 1991 10/12

5 1992 4

6 1994 11

7 1997 8

8 1999 7

Source: the truth

Time Series: One AgentMany Time

Periods

Cross Sectional Data

-Collects data on multiple economic agents (locations/persons/firms/etc) at one time-Taken at one specific point in time (September report, January report, etc.)

-ie: current stock portfolio, hockey player stats, provincial GDP comparison, last year’s grades

99/00 Tuition – Cross Sectional


Alberta 3551.00 3770.00 3890.00 4032.00


Calgary 3650.00 3834.00 3975.00 4120.00

Concordia 1668.00 1668.00 1668.00 1668.00

Lethbridge 3360.00 3470.00 3470.00 3470.00

Manitoba 3005.00 2796.00 2807.00 2818.00

McGill 1668.00 1668.00 1668.00 1668.00

Ottawa 3760.00 3892.00 4009.00 4085.00

Canadian Provincial Corporate Tax 2015

- Cross Sectional Data

BC Alberta Saskatchewan Manitoba Ontario Nova Scotia

11% 12% 12% 12% 11.5% 16%

Source: Canada Revenue Agency (http://www.cra-arc.gc.ca/tx/bsnss/tpcs/crprtns/prv/menu-eng.html),

NDP platform (Alberta)*Refers to the higher rate; not applicable to small business

Timothy A. Student’s Weekly Time Spent Studying for Midterms

- Cross Sectional Data

Course English 101 Philosophy 262

Llama Studies 371

Economics 282

Economics 299

Hours 6 12 2 11 25

Cross Sectional: Many Agents

One Time Period

Panel Data

-Combination of Time Series and Cross-sectional Data

-Many economic agents-Many time periods

-More difficult to use-Often required due to data restrictions-also referred to as pooled data

Pooled Tuition


Alberta 3551.00 3770.00 3890.00 4032.00


Calgary 3650.00 3834.00 3975.00 4120.00

Concordia 1668.00 1668.00 1668.00 1668.00

Lethbridge 3360.00 3470.00 3470.00 3470.00

Manitoba 3005.00 2796.00 2807.00 2818.00

McGill 1668.00 1668.00 1668.00 1668.00

Ottawa 3760.00 3892.00 4009.00 4085.00

1.1 Data Types

Exercise: What kind of data is:1) Election Predictions 10 days before

an election?2) MacLean’s University Rankings?3) Yearly bank account summary?4) University Transcript after your 4th

year?

1.2 Real and Nominal Variables

1. Nominal variables

Measured using current pricesProvides a measure of current value

Ie: a movie today costs $12.

1.2 Real and Nominal Variables

2. Real variables

Measured using base year pricesProvides a measure of quantity (removing the effects of price change over time)

Ie: a movie today costs $4.00 in 1970 dollars

A Movie in 1970

In 1970, a movie cost $0.50BUT

$0.50 then was a lot more than $0.50 now.

Nominal Comparison:Movie prices have increased by a factor of 24 ($0.50 -> $12)

Real Comparison:Movie prices have increased by a factor of 8 ($0.50 -> $4)

GDP example

Gross Domestic Product -Monetary value of all goods and services produced in an economy

How do nominal and real GDP differ?

Nominal GDP

-Current monetary value of all goods produced:

∑ quantities X prices

-changes when prices change-changes when quantities change

The Problem with Nominal GDP

Assume: prices quadruple (x4)production is cut in half (x

1/2)

Nominal GDP (year 1) = 1 X 1 = 1Nominal GDP (year 2) = 0.5 X 4 = 2

-although production has been devastated, GDP reflects extreme growth

Real GDP

-Base year value of all goods currently produced:

∑ quantities X prices base year

-doesn’t change when prices change

-changes when quantities change

The Solution of Real GDP

Assume: prices quadruple (x4)production is cut in half (x

1/2)

Real GDP (year 1) = 1 X 1 = 1Real GDP (year 2) = 0.5 X 1 = 0.5

-real GDP accurately reflects the economy

Price Indexes (Indices)

-Used to convert between real and nominal terms

-different indexes for different variables or groups of variables

Ie: GDP Deflator2002 = 100 (base

year)2010 = 125 (World

Bank)The “price” of GDP has risen

25% between 2002 and 2010

GDP – Converting Between Real and Nominal

GDP Real x 100

Index PriceGDP Nominal

100 x Index Price

GDP NominalGDP Real

General Conversion Equations

Real x 100

Index Price Nominal

100 x Index Price

Nominal Real

Example: Tuition


Alberta (Nominal) 3551 3770 3890 4032

Tuition Price Index* 100 103 106.1 109.273

Real Tuition (1999 dollars)=(Nominal Tuition/Price Index)100 3551 3660 3667 3689.85

British Columbia (Nominal) 2295 2295 2181 2661

Tuition Price Index* 100 103 106.1 109.273

Real Tuition (1999 dollars)=(Nominal Tuition/Price Index)100 2295 2228 2056 2435.19

*Based on 3% yearly inflation typical to years listed

1.3 How to Calculate Price Indexes-simple price index

-weighted sum of individual prices of a good or group of goods

Simple Price Index = ∑price X weight

Example #1

John is constructing a price index to reflect his entertainment spending

John values two activities equally: seeing movies and eating hot dogs

The prices of movies and hot dogs have moved as follows:

Year MoviesHotDogs

Price Price

2000 $12.00 $1.00

2001 $20.00 $1.00

2002 $10.00 $2.00

Example #1

Simple Price Index = ∑price X weight

Year Movies Hot DogsSimple Price

Index

Price Weight Price Weight

2000 $12.00 0.5 $1.00 0.5 $6.50

2001 $20.00 0.5 $1.00 0.5 $10.50

2002 $10.00 0.5 $2.00 0.5 $6.00

Exercise: If John valued hot dogs three times as much as movies, what would the price indexes become?

1.3.1 Normalizing Price Indexes

-price indexes themselves are meaningless

“The price of GDP was 78.9 this year”

-price indexes help us:1) Compare between years2) Convert between real and

nominal-to compare more easily, we

normalize to make the index equal 100 in the base year

Normalizing the price index:

= 100 in base year

100 x Index Price Raw

Index Price RawIndex Price Normalized

year baset

t

For example, if GDP was 310 in 1982, dividing every year’s GDP by 310 and then multiplying by 100 normalizes GDP to be 100 in 1982.

Example #1a - Normalized

Take 2000 as the base year:

Year Movies Hot DogsSimplePrice Index

Normalized Price Index


2000 $12.00 0.5 $1.00 0.5 $6.50 100

2001 $20.00 0.5 $1.00 0.5 $10.50 162

2002 $10.00 0.5 $2.00 0.5 $6.00 92.3

Does the base year chosen affect the outcome?

Example #1b - Normalized

Take 2002 as the base year:

Year Movies Hot DogsSimplePrice Index

Normalized Price Index


2000 $12.00 0.5 $1.00 0.5 $6.50 108

2001 $20.00 0.5 $1.00 0.5 $10.50 175

2002 $10.00 0.5 $2.00 0.5 $6.00 100

Note: Raw and normalized PI’s WORK the same, normalized PI’s are just easier to visually interpret

Example #2 – Tuition

If instead of using inflation for our tuition deflator, we use the education deflator, we can first normalize it to 1999/2000:

YearRaw Education

Index* CalculationNormalized Price

Index

1999/2000 149.3 149.3/149.3 X 100 100

2000/2001 155.6 155.6/149.3 X 100 104

2001/2002 160.6 160.6/149.3 X 100 108

2002/2003 165.8 165.8/149.3 X 100 111

*Cansim series V735564, January Data

Example: Tuition – Converting from real to nominal

UniversityTuition 1999/00

Tuition00/01

Tuition 01/02

Tuition02/03

Alberta 3551 3770 3890 4032

Tuition Deflator 100 104 108 111

Real Tuition (1999 dollars) 3551 3625 3602 3632.43

Calgary 3650 3834 3975 4120

Tuition Deflator 100 104 108 111

Real Tuition (1999 dollars) 3650 3687 3681 3711.71

1.3.1.1 Changing Base Years

-base years can be changed using the same formula learned earlier

-in the formula, always use the price indexes from the SAME SERIES (same base year)

1.3.2 Common Price Indexes

-Up until this point, price index weights have been arbitrary

-Arbitrary weights leads to bias, difficulty in recreating data, and difficulty in interpreting and comparing data

-One common price index (which the Consumer Price Index uses) is the Laspeyres Price Index

(The Paasche Price Index is the other common index used)

1.3.2 Laspeyres Price Index

-uses base year quantities as weights-still = 100 in base year

(automatically normalized

LPIt = ∑ pricest X quantitiesbase year

---------------------------------- X 100 ∑ pricesbase year X quantitiesbase

year

-tracks cost of buying a fixed (base year) basket of goods (ie: CPI)

Example: Movies and Karaoke

Year Movies Karaoke

Price Quantity Price Quantity

1 10 20 20 10

2 11 15 25 15

3 12 25 15 20

4 15 5 15 20

5 11 10 20 15

Example: Laspeyres (Base year 1)

Laspeyres Price Index

Cost in year t Cost in Base Year

Year of base year basket of base year basket Laspeyres Price Index

1 (10*20)+(20*10) 400 (10*20) + (20*10) 400 400/400 X 100 100

2 (11*20)+(25*10) 470 (10*20) + (20*10) 400 470/400 X100 118

3 (12*20)+(15*10) 390 (10*20) + (20*10) 400 390/400 X 100 97.5

4 (15*20)+(15*10) 450 (10*20) + (20*10) 400 450/400 X 100 113

5 (11*20)+(20*10) 420 (10*20) + (20*10) 400 420/400 X 100 105

2 Price Index Calculation Methods1) Using individual prices and

quantities-Same as before

2) Using basket costsPaQb

Price of basket b in year aP2012Q1997

Price in 2012 of what was bought in 1997

Method 1 – Individual Prices and Quantities




1 (10*20)+(20*10) (10*20) + (20*10) 400/400 X 100 100

2 (11*20)+(25*10) (10*20) + (20*10) 470/400 X100 118

3 (12*20)+(15*10) (10*20) + (20*10) 390/400 X 100 97.5

4 (15*20)+(15*10) (10*20) + (20*10) 450/400 X 100 113

5 (11*20)+(20*10) (10*20) + (20*10) 420/400 X 100 105

Method 2 – Basket Costs




1 400 400 400/400 X 100 100

2 470 400 470/400 X100 118

3 390 400 390/400 X 100 97.5

4 450 400 450/400 X 100 113

5 420 400 420/400 X 100 105

Method 2 Example

Every year, Lillian Pigeau likes to travel.

The first year, she went to Maraket,

the second year to Ohm, and the third year to

Moose Jaw. The costs of those trips

are as follows:

Year Maraket OhmMooseJaw

1 $800 $1,000 $650

2 $900 $1,100 $550

3 $600 $1,200 $700

Method 2 – Laspeyres (Year 1 Base Year)

100

)100(800$

800$

)100(

)100(

1

1

11

111

LPI

LPI

QP

QPLPI

QP

QPLPI

bb

btt


1 $800 $1,000 $650

2 $900 $1,100 $550

3 $600 $1,200 $700


5.112

)100(800$

900$

)100(

)100(

2

2

11

122

LPI

LPI

QP

QPLPI

QP

QPLPI

bb

btt


1 $800 $1,000 $650

2 $900 $1,100 $550

3 $600 $1,200 $700


75

)100(800$

600$

)100(

)100(

3

3

11

133

LPI

LPI

QP

QPLPI

QP

QPLPI

bb

btt


1 $800 $1,000 $650

2 $900 $1,100 $550

3 $600 $1,200 $700

1.3.2.1 Chained Price Indexes

-A chained price index gives a measure of an aggregate good’s price from one year/term to the next-chained price indexes are less affected by a base year-chained price indexes can better capture substitution away from goods

-to form a chained price index, one must first form each year’s “link”, then multiply links together

1.3.2.1 Laspeyres Chain Link (LCL)

-uses last term quantities as weights-still = 100 in base year

LCLt-1,t = ∑ pricest X quantitiest-1

---------------------------------- ∑ pricest-1 X quantitiest-1

-tracks cost of buying a last term’s quantities

1.3.2.1 Laspeyres Chained Price Index (LCPI)

LCPI1=100LCPI2=LCPI1 x LCL1, 2

LCPI3=LCPI2 x LCL2, 3

LCPI3=LCPI1 x LCL1, 2 x LCL2, 3

and so on…

Example: Laspeyres Chained Index


Cost in year t Cost in t-1

Year of t-1 basket of t-1 basketLink Index

1 N/A N/A N/A N/A N/A 100

2 (11*20)+(25*10) 470 (10*20) + (20*10) 400 1.175 117.5

3 (12*15)+(15*15) 405 (11*15) + (25*15) 540 0.75 88

4 (15*25)+(15*20) 675 (12*25) + (15*20) 600 1.125 99

5 (11*5)+(20*20) 455 (15*5) + (15*20) 375 1.21 120

1.3.3 Splicing Price Indexes

-As time goes on, base years change-Prices and quantities of horses and

cars in the 1960’s are a little different than today

-This creates price indexes with different base years, spanning different periods

-Sometimes these differing price indexes need be spliced together

1.3.3 Splicing Price Indexes

1) Find a year with price indexes from BOTH series & calculate a conversion factor

Conversion factor = Price Index (new base)--------------------------------------------------

Price Index (old base)New = index you want to fill inOld = index you want to convert2) Multiply old index by conversion

factor to fill in new index

Ie: Price Index (Computers)

Year Price Index Price Index Calculations Price Index

(1989=100) (1992=100) (1992=100)

1988 120 120 X (110/92) 143

1989 100 100 X (110/92) 120

1990 95 95 X (110/92) 114

1991 92 110

1992 100

1993 95

1994 95

Exercise: How would the full price index look with 1989 as the base year?

1.4 Growth Rates and Inflation

Growth Rates are important concepts in economics.

Inflation = growth rate of CPI (all items)

Growth = { (Xt – Xt-1)/ Xt-1 } X 100= { ln(Xt) – ln(Xt-1) } X 100

Note: g = (Xt – Xt-1)/ Xt-1

1.4 Growth Rates Example

UBC tuition in 2001/2002: $2181. In 2002/2003 it was $2661

Growth = { (2661-2181)/2181 } X 100 = 22.01%

Growth = { ln(2661) – ln(2181) } X 100 = 19.89%

U of A tuition in 2001/2002: $3890. In 2002/2003 it was $4032

Growth = { (4032-3890)/3890 } X 100 = 3.65%

Growth = { ln(4032) – ln(3890) } X 100 = 3.59%

1.4 Log Growth Restrictions

Growth = {ln(Xt) – ln(Xt-1)} X 100

The log growth formula is only appropriate when growth is small.

If the log growth formula reveals large growth, use the normal growth formula instead

Why two growth formulas? (proof)

If g is SMALL g ≈ ln (1+g)ln(1+g) = ln [1+(Xt-Xt-1)/Xt-1]

= ln [(Xt-1+Xt-Xt-1)/Xt-1]= ln [Xt/Xt-1]= ln [Xt] – ln[Xt-1]

Therefore g ≈ ln [Xt] – ln[Xt-1]or (Xt-Xt-1)/Xt-1 ≈ {ln [Xt] – ln[Xt-1]}

Log Review

1) Division Ruleln(A/B) = ln(A) – ln(B)

2) Multiplication Ruleln(AB) = ln(A) + ln (B)

3) Power Ruleln(Ab) = b X ln (A)

Note:ln (A+B) ≠ ln (A) + ln (B)

Example: Relative Growth Rate

gA/B = [ln(At/Bt) – ln(At-1/Bt-1)] X 100= [ln(At)-ln(Bt)-

{ln(At-1)-ln(Bt-1)}] X 100= [ln(At)-ln(At-1)-

{ln(Bt)-ln(Bt-1)}] X 100= [ln(At)-ln(At-1)] X 100 – {ln(Bt)-ln(Bt-1)} X 100

gA/B = growth of A – growth of B


Recall that:Real = nominal /(price index/100)

Ie: Real price=nominal price/(PI/100)

Therefore:Real growth = nominal growth – PI growth

For example:Real price change = nominal price change

-inflation


If tuition was $5000 last year and $5100 this year, how much did real tuition change if inflation is 3%?

Real growth = nominal growth – inflation={(5100-5000)/5000}X100 -3= (100/5000)X100 - 3= 2-3= -1%

Example: Multiplicative Growth Rate

gAB= [ln(AtBt) – ln(At-1Bt-1)] X 100= [ln(At)+ln(Bt)-

{ln(At-1)+ln(Bt-1)}] X 100= [ln(At)-ln(At-1)+

{ln(Bt)-ln(Bt-1)}] X 100= [ln(At)-ln(At-1)] X 100 +

{ln(Bt)-ln(Bt-1)} X 100 gAB = growth of A + growth of B

Example: Multiplicative Growth Rate

Recall that:Per Capita GDP = GDP/Population

THEREFOREGDP = Per Capita GDP X Population

THEREFOREGDP growth = per capita GDP growth +

population growth

If each person produces 1% more, and population grows by 2%, overall GDP growth is 3%

1.5 Interest Rates

Interest rates are important in economics, as they show the opportunity cost of a project.

Different interest rates apply to different situations.

Different interest rates are available to different people.

1.5 Interest Rate Examples (Aug 2015)

Saving: 1 Year GIC: 0.85%1 Year Cashable GIC: 0.4%3 Year GIC: 1.05%3 Year Cashable GIC: 0.5%Bank Account: 0.0%BorrowingBank of Canada Rate: 0.5%1 year closed Mortgage: 2.89%1 year open Mortgage: 6.3%

1.5 Different Interest Rates

Bank of Canada rate for banksIs less than

Chartered Banks’ rates for best customers

Is less thanTypical Bank Rate

Is less thanRisky Investor Bank Rate

More risk = higher rate

1.5.2 Real Vrs. Nominal Rates

Super Savings Bank Account: 2% interest

Cash on hand: $1002 DVD players: Basic: $100

DVD PlaybackDeluxe: $102DVD/VCD/SVCD/AVI/DVD±R/CD/CD±R3D Blu-Ray, Wi-Fi, Memory Card Slot,

Picture Viewer, Stop Memory, Shiny Red Colour

1.5.2 Real Vrs. Nominal Rates

You want the deluxe, so you invest for a year, cash on hand in a year: $102

But, due to 3% inflation, the DVD players now cost: $103 (basic) $105.06 (deluxe)

Now you can’t afford eitherYou’ve LOST buying power

1.5.2.1 Calculating real interest

rreal = (1+rnom) --------- -1

(1+inf)

rreal= real interest raternom= nominal interest rateinf = inflation

1.5.2.1 Easy Interest Formula

rreal = (1+rnom-1-inf) ---------------- (cross multiply to

get…)(1+inf)

rreal+ rreal*inf = rnom-inf (rreal*inf is small)

rreal = rnom – inf

Last example: rreal = 2%-3%=-1%

1.5.2.1 Depressing Interest Facts

Very few safe investments offer a return greater than inflation.

You are losing buying power

Is buying today a better move?

WHY SAVE?

Example: Calculating currency interestYou can invest in Canada, the US, or

Mexico. Investment opportunities are 4%, 5%, and 15% respectively.

However, country currency inflation is 2%, 3% and 14%

Real interest rate then becomes:Canada: 4%-2%=2%US: 5%-3%=2%Mexico: 15%-14% = 1%

1.5.3.1 Annual Compounding

Investment: $100 Interest rate: 2%

Year Calc. Amount

1 100 100.00

2 100*1.02 102.00

3 100*1.022 104.04

4 100*1.023 106.12

5 100*1.024 108.24

Derived Formula:

S = P (1+r)t

S = value after t years

P = principle amount

r = interest ratet = years

1.5.3.2 More Frequent Compounding

If interest is compounded m times a year, 1/m of the interest is paid each time

Modified Formula:

S = P (1+[r/m])mt

S = value after t years P = principle amount

r = interest rate t = yearsm = times compounded (monthly = 12,

etc)

Infinite Compounding: S = Pert

Compounding Comparison

Year Yearly Biyearly Monthly Weekly Daily

0 $100.00 $100.00 $100.00 $100.00 $100.00

1 $110.00 $110.25 $110.47 $110.51 $110.52

2 $121.00 $121.55 $122.04 $122.12 $122.14

3 $133.10 $134.01 $134.82 $134.95 $134.98

4 $146.41 $147.75 $148.94 $149.13 $149.17

5 $161.05 $162.89 $164.53 $164.79 $164.86

6 $177.16 $179.59 $181.76 $182.11 $182.20

7 $194.87 $197.99 $200.79 $201.24 $201.36

8 $214.36 $218.29 $221.82 $222.38 $222.53

9 $235.79 $240.66 $245.04 $245.75 $245.93

10 $259.37 $265.33 $270.70 $271.57 $271.79

More frequent compounding gives greater returns.

1.5.3.3 Effective Rate of Interest

Which is the better investment: 25% compounded annually or 24% compounded monthly?

rE = effective rate of interest if

compounded annually

P (1+rE)t = P (1+[r/m])mt

Solving for rE, we get:

rE = (1+[r/m])m-1

1.5.3.3 Effective Rate of Interest

Which is the better investment: 25% compounded annually or 24% compounded monthly?

rE = (1+[r/m])m-1

= (1+[0.24/12])12-1= (1+0.02)12 -1= (1.02)12 -1= 1.268-1= 26.8%

1.5.3.3 Annualizing Monthly Inflation

infann = (1+infmon)12-1

In one month of 2005, gas prices rose from 98 to 112 cents a liter.

Infmon = [(112-98)]/98 X 100 = 14.3%

If this continued throughout the year, inflation would reach:

infann = (1+0.143)12-1 = 397%

Some sketchy investments (some mutual funds sold by a “friend”) use this misleading calculation often.

1.5.3.3 Short Term Loans

infann = (1+infday)365-1

Cheezy loan inc. offers 0.1% daily interest on payday loans. They advertise that a one-day payday loan of $1000 only costs $1!

However, yearly this becomes:

infann = (1+0.001)365-1

= (1.001)365-1= (1.44-1)= 44% interest!

Effective Interest Rate Formulas

If interest/return is expressed yearly, but paid out multiple times per year, effective interest/return is:

1)1( mE m

rr

If interest/return is expressed more frequently (monthly, etc), effective interest/return is:

1)1( mE rr

1.5.3.4 Present Value

How much do I have to invest now to have a given sum of money in the future?

PV = S/[(1+r)t]

PV = present value (money invested now)

S = sum needed in futurer = interest ratet = years**Note: time can be in months (or any time period) if interest rate

is also in months (or any time period)

1.5.3.4 Tuition Example

You and your spouse just got pregnant, and will need to pay for university in 20 years. If university will cost $30,000 in real terms in 20 years, how much should you invest now? (long term GIC’s pay 5%)

PV = S/[(1+r)t]= $30,000/[(1.05)20]= $11,307

How does this change if it’s more than a one-time investment/payment?

(ie: $100 per year for 5 years, 7% interest)

PV= 100+100/1.07 + 100/1.072 + 100/1.073

+ 100/1.074

= 100 + 93.5 + 87.3 + 81.6 + 76.3 = $438.7

OrPV = A[1-(1/{1+r})t] / [1- (1/{1+r})]PV = A[1-xt] / [1-x] x=1/{1+r}PV = 100[1-(1/1.07)5]/[1-1/1.07] =

$438.72

1.5.3.4 Continued Deposits

PV = A[1-(1/{1+r})t] / [1- (1/{1+r})]PV = A[1-xt] / [1-x] x=1/{1+r}A = value of annual paymentr = annual interest ratet = number of annual payments

Note: if specified that the first payment is delayed until the end of the first year, the formula becomes

PV = A[1-xt] / r x=1/{1+r}

1.5.3.4 Annuity Formula

1.5.3.4 Example

You won the lottery. Which is greater? $800,000 now or $100,000 for the next 10 years at 5% real interest?

PV = A[1-(1/{1+r})t] / [1- (1/{1+r})]PV = $100,000 {(1-[1/1.05]10)/(1-

[1/1.05])} = $100,000 {0.386/0.0476} = $100,000 {8.11} = $811,000

Take the money over 10 years(Surprising how many take the lump

sum)

1.5.4 Calculating average returns

Arithmetic Mean-Averaging items that are added together(University grades, income, rent)

Ie: 3 numbers: 7, 15, and 20Average= (7+15+20)/3 = 14


Geometric Mean-Averaging items that are multiplied together(Interest rates, inflation)

Ie: 3 numbers: 7, 15, and 20Geo Mean= (7x15x20)1/3 = 12.81

(Generally more useful than arithmetic)


Year Account GIC Investment

1 0.03 0.015 -0.500

2 0.03 0.020 -0.100

3 0.03 0.025 0.100

4 0.03 0.040 0.150

5 0.03 0.050 0.500

Arithmetic Mean 0.03 0.030 0.030

Consider three investment opportunities: a stable bank account with 3% interest, an escalading GIC, or a risky investment, all with the “same” return:



1 0.03 0.015 -0.500

2 0.03 0.020 -0.100

3 0.03 0.025 0.100

4 0.03 0.040 0.150

5 0.03 0.050 0.500

Arithmetic Mean 0.03 0.030 0.030

Geometric Mean 0.03 0.027 -0.031

Although each investment has the same arithmetic mean, the geometric means clearly rank the investments.

1.5.4 Investment Results


1 103 101.50 50

2 106.09 103.53 45

3 109.273 106.12 49.5

4 112.551 110.36 56.925

5 115.927 115.88 85.3875

Assume an initial investment of $100:

1.5.4 Investments and means

When investing with compound interest:

ALWAYS CONSIDER GEOMETRIC MEANS

As Arithmetic means are meaningless.

(Even though they’re sometimes reported.)

1.5.4 An Easy Method For Solving for the Geo. Mean:

By definition:(1+rgeo)T = (1+r1)(1+r2)(1+r3)…(1+rT)

(1+rgeo) = [(1+r1)(1+r2)(1+r3)…(1+rT)]1/T

rgeo= [(1+r1)(1+r2)(1+r3)…(1+rT)]1/T -1

It is EXTREMELY important to add 1 to each interest rate.

1.5.4 Geometric Note

If there is NO compounding… the arithmetic mean will be an

appropriate measure of average returns

Ie) A person invests $1000 each year, takes it all out, and then invests $1000 next year.

Ie) A person invests in a poor GIC that does not compound

1.6 Aggregating Data – Stocks and Flows

Sometimes data needs to be AGGREGATED – changed from one form (time period) to another.

ie) monthly tuition payments => yearly tuition payments

How to aggregate depends on whether the variable is a STOCK or a FLOW

ie) I pay $500 a month in tuition. Therefore yearly tuition is $500 (the average). -FALSE

1.6.1Stocks and Flows

Stock : a set, tangible value at a period of time

Flow: a change to a stock variable

ie) Tuition:Total tuition paid – stock variableMonthly tuition payment – flow

variableTotal tuition paid = ∑ Monthly tuition

payment

1.6.1 -Stocks and Flows

Stock : a set, tangible value at a timeFlow: a change to a stock variable

ie) Capital:Kt = Kt-1 + It – Dt

K = Capital – stockI = investment – flowD = depreciation - flow



ie) Final Mark:Final Markt = Final Markt-1 + Bribe

effectt +Scalingt

Final Mark = stockBribe Effect = flowScaling = flow



ie) Your markM=a1+a2+a3+a4+midterm+lab+fin

alM =end mark (stock)A =mark gained by assignmentMidterm =mark gained by midtermLab =mark gained through lab

componentfinal =mark gained through final (all

flows)


Jedit = Jedit-1 – Darkt + Traint + Redeemt – Aget – Battlet -66t

Jedi = number of Jedi (stock)Dark = Jedi turning to dark side (flow)Train = New Jedi’s trained (flow)Redeem = Dark Jedi’s returning (flow)Age = Jedi’s dying of old age (flow)Battle = Jedi’s dying in battle (flow)66 = Jedi’s killed by Emperor's order

(flow)

1.6.1 -Stocks and Flows Exercises


What are the stocks and flows in:1) Your Bank Account2) Yearly Debt3) Flirting with a girl/guy

1.6.1 -Stocks and Flows Summary

Type of Variable

Stock Flow

Major Characteristic

Measured at a point in time

Measured over a period (between points in time)

Examples Debts, wealth, housing, stocks, capital, tuition

Deficits, income, building starts, investment, payments

Aggregation Method

Average orUse values from the same time each year

Sum(Average if annualized)

Stock or Flow?

Monthly Savings

Flow – ADD

Temperature

Stock – Average

Population

Stock – Average

Births

Flow - ADD

Stock or Flow?

Vacancy Rate

We’ve had 10% vacancy a month.

a)That’s 120% vacancy a year (flow)

Or

b) That’s an average of 10% vacancy for the year. (stock)

Stock or Flow?

Building Starts

500 new buildings have started each month

a)That’s 6000 new buildings a year (flow)

Or

b) That’s an average of 500 new buildings this year (stock)

Stock or Flow?

Money Supply

Canada’s money supply each month has been $200 billion

a)That’s $2.4 trillion a year (flow)

Or

b) The money supply was $200 billion that year (stock)

Stock or Flow?

Investment

“Each month I invest $500 in elevators inc. It’s bound to go up sometime!”

a)That’s an investment of $6,000 a year (flow)

Or

b) That’s an average yearly investment of $500 (stock)

Stock or Flow?

Consumption

“My grocery bill is $300 a month

a)That’s an bill of $3,600 a year (flow)

Or

b) That’s average yearly groceries of $300 (stock)

Stock or Flow?

Job creation

“Our new evaporated water factory will create 2,000 new jobs every month. Now that’s the magic of government!”

a)24,000 jobs will be created this year (flow)

Or

b) Government “magic” creates 2,000 jobs this year! (stock)

1.6.2 – The User Cost of Capital

Two methods of determining costs of durable goods (goods not consumed in 1 time period):

1) Purchase price-actual sticker price paid for good-one time price, ignores durability

2) User cost of Capital-value of services received

over time-implicit rental rate

1.6.2 – Simple Choice Example

You buy a used printer (that only lasts one year) for $20, to print 2,000 pages. Ink and paper cost you $50, and photocopying (“renting”) would cost $0.02 a sheet.

Buying = $20 + $50 = $70Photocopying = 2,000 * $0.02 = $40

You would “rent” instead of buy…but most printers last MORE than one year


Economist’s user cost of capital:

“How much would you be willing to pay per term (ie: year) to rent capital that you could buy for $X?”

-implicit rental rate-BUYING the good is equivalent to

renting it for this amount each term

1.6.2 – Factors Affecting User Cost

1)Depreciation – the more that an item depreciates (more it costs to maintain), the less likely one is to buy-higher maintenance=>higher “implicit rent”

2) Opportunity cost of funds – the more that a buyer can earn for his money, the less likely he will be to buy-higher interest rates =>higher “implicit rent”

1.6.2 – Factors Affecting User Cost

3) Capital gains (loses) – a buyer is more likely to purchase a product that keeps its value over time-gains value => lower “implicit rent”-loses value =>higher “implicit rent”


User cost of capital = implicit rental rate

Pkt ( d + r - [Pkt+1 – Pkt]/Pkt )

d = depreciation(more willing to rent a costly item)r = return on alternate investments

(more willing to rent given high returns)[Pkt+1 – Pkt]/Pkt = capital gains/losses

(less willing to rent an item that gains/holds value)

1.6.2 – Simple Choice Example


=Pkt ( d + r - [Pkt+1 – Pkt]/Pkt )

d = 1 (printer explodes)r = 0 (no alternate investments)

[Pkt+1 – Pkt]/Pkt = 0 (no price change)

Implicit rental cost = $70(1+0-0)= $70

Buy : $70 (implicit rental) > $40 (actual rental)

1.6.2 – House Example

You decide to buy a tiny (almost condemned) house for $200,000. The house is so old and decrepit that depreciation is 10%. You can invest in a GIC at 5%, and expect the price of the house to increase to $205,000 over the next year.

1.6.2 – House Example



d = 0.10 r = 0.05

[Pkt+1 – Pkt]/Pkt = [205-200]/200 = 0.025

Implicit rental cost = $200,000(0.10+0.05-0.025)

= $200,000(0.125)= $25,000

1.6.2 – Computer Example

You decide to buy a new supercomputer. The computer originally costs $2,000, and depreciates 25% a year (since you don’t have Norton Internet Security). The purchase price DECREASES 10% each year, and you could alternately invest at 5%

1.6.2 – Computer Example



d = 0.25 r = 0.05

[Pkt+1 – Pkt]/Pkt = -0.10 (decrease)

Implicit rental cost = $2000(0.25+0.05-[-0.10])

= $2000(0.4)= $800

1.6.2 – User Cost of Capital

If you could rent the house for LESS than $25k a year, you should rent

If you could rent the computer for MORE than $800 a year, you should buy.

If Rent > User Cost of Capital, buyIf Rent < User Cost of Capital, rent

1.7 Seasonal Adjustment

“Icon’s ice cream sales fell in November – they should shut down.”“The new federal budget has caused a decrease in student unemployment this May.”“Apple CEO demands raise for increase in sales in December.”“Holes Greenhouse sales fall in March – accountants perplexed.”


Many economics variables often have PREDICTABLE seasonal movements.

Failure to appreciate these movements can lead to wrong assumptions.

Is growth or loss:1) A seasonal effect OR 2) A true change.


-Ice cream sales fall in winter-Students get jobs in May-Christmas boosts sales in December-Flower sales rise for Valentines Day, then fall afterwards-Health Club memberships soar following New Years’ resolutions-Gas sales decrease in winter as certain drivers chose not to drive.


Statistics Canada accounts for seasonal adjustments by publishing two sets of data:

1)Raw (not seasonally adjusted) data2)Seasonally adjusted data

1.7 Dealing with Seasons

In order to make correct conclusions when faced with seasonally adjusted data, one should:

1)Use seasonally adjusted data2)Compare between years (not

between months)

1.7 Dealing with Seasons

Note: Other factors other than seasons can create variable movements:

a) long-term trendsb)Business cyclec) Irregular shocksThese events are not factored out by

seasonal adjustments, but must be identified in a decent study. (ie: plot the trend on a graph and look for patterns)

APPENDIX 1.1 – EXPONENTIALS AND LOGARITHMS

1 0, xIf 3)

1 0, xIf 2)

10 0, xIf 1) ,0

)exp(

...3)2(1

1

)2(1

1

1

11718.2

x

x

xx

x

e

e

ee

xe

e

Two key mathematical concepts used in economics are exponentials and logarithms (which are related concepts)

The features of exponentials are:


0ln(x) 1 xif

0ln(x) 1 xif

0ln(x) 1x0 if

:0for x definedonly is )ln( )2

z,)ln( if 1)

10) basenot e, base torefers always (economics )ln()(log

x

xex

xxz

e

The key features of Logarithms are:


026,22

10 ln(x) 10

x

xe

if

Note that exponentials and logarithms can be interchanged to solve a problem:

From Section 1.4, Log Review:

1) Division Ruleln(A/B) = ln(A) – ln(B)

2) Multiplication Ruleln(AB) = ln(A) + ln (B)

3) Power Ruleln(Ab) = b X ln (A)

Noteln (A+B) ≠ ln (A) + ln (B)

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