- 1. Numeracy Skills Part 1. Fundamentals of Mathematics Anthony J. Evans Associate Professor of Economics, ESCP Europe www.anthonyjevans.com London, February 2015 (cc) Anthony J. Evans 2015 | http://creativecommons.org/licenses/by-nc-sa/3.0/
- 2. Description • Fundamentals of Mathematics is usually a pre-term course that provides a basis for the numerical literacy required by other courses on an MBA programme • This course is intended to be a short refresher for students wishing to gain general confidence with numbers, and will provide an opportunity to practice the types of numeracy tests used in graduate recruitment • I will assume that you have little or no mathematical training so basic terminology and methods will be explained 2
- 3. Agenda 1. Looking at proportions 2. Basic algebra 3. Compound Annual Growth Rates (CAGR) 4. Back of the Envelope Calculations (BotEC) 3
- 4. Proportions 25 100 25% 0.25= = Fractions A quotient of numbers Percentages “Percent” means “per 100” Decimal Relating to powers of 10 • There are three equivalent ways to express a proportion 4
- 5. Percentages Use Dynamic % Formula Finding the proportion of a given fixed sizeStatic % Finding the proportional change between two values measured over different time periods 5 € Part = % ×Whole € %Δ = Absolute Δ Original value
- 6. Percentages • 20 is half of 40 We can write this in different ways: 20 is 50% of 40 20 is 1/2 of 40 20 is 0.5 of 40 6 € Part = % ×Whole
- 7. 1. What’s 20% of 48,200? Percentages 7
- 8. 1. What’s 20% of 48,200? Percentages 8 € Part = % ×Whole € = 0.2 × 48,200 € = 9,640
- 9. Percentages 2. How much would you save? Was £20 Now 10% off 9
- 10. Percentages 2. How much would you save? Was £20 Now 10% off 10 € Part = % ×Whole € = 0.10 × 20 € = £2
- 11. Percentages 3. Which product is cheaper? Was £12 Now 40% off Was £18 Now 50% off 11
- 12. Percentages 3. Which product is cheaper? Was £12 Now 40% off Was £18 Now 50% off 12 = 12 - (0.40) * 12 = 12 - 4.80 = £7.20 = 18 - (0.50) * 18 = 18 - 9 = £9.00 As before… = (0.60) * 12 = £7.20 = (0.50) * 18 = £9.00 What’s the new %?
- 13. Percentages 4. Suppose the profits of a certain company go from £365 000 in January to £425 000 in February. What is the % increase in their profits? 13
- 14. Percentages 4. Suppose the profits of a certain company go from £365 000 in January to £425 000 in February. What is the % increase in their profits? 14 € %Δ = Absolute Δ Original value € = 425,000 − 365,000 365,000 € = 0.164 € =16.4%
- 15. Percentages 5. The number of first year students at a certain university studying Law was 127 in 1996 and 114 in 1997. What was the % decrease? 15
- 16. Percentages 5. The number of first year students at a certain university studying Law was 127 in 1996 and 114 in 1997. What was the % decrease? 16 € %Δ = Absolute Δ Original value € = 127 −114 127 € =10.2%
- 17. Percentages 6. An antique jug is now worth 25% more than when it was first bought. The original price was £40. How much is it worth now? 17
- 18. Percentages 6. An antique jug is now worth 25% more than when it was first bought. The original price was £40. How much is it worth now? 18 € %Δ = Absolute Δ Original value € %Δ × Original value = Absolute Δ € 0.25 × 40 = Absolute Δ € =10 Rearrange the formula… € ∴New value = 40 +10 = 50
- 19. Percentages 6. An antique jug is now worth 25% more than when it was first bought. The original price was £40. How much is it worth now? 19 € =1.25 × 40 What’s the new %? Calculate 125% of £40 € = 50
- 20. Percentages 7. The price of a certain model of car goes up by 8%. It used to cost £7,800. What does it cost now? 20
- 21. Percentages 7. The price of a certain model of car goes up by 8%. It used to cost £7,800. What does it cost now? 21 Rearrange the formula… What’s the new %? € =1.08 × 7,800 Calculate 108% of £7,800 = £8,424 € 0.08 × 7,800 = Absolute Δ € = 624 € ∴New value = 7800 + 624 = 8,424
- 22. Percentages 7. The price of a certain model of car goes up by 8%. It used to cost £7,800. What does it cost now? Note: here’s the algebra… New value = Original value + Absolute change New value = 7,800 + (0.08 x 7,800) New value = 7,800 x (1 + 0.08) New value = 7,800 x 1.08 = £8424 22
- 23. Percentages 8. The price of a certain model of car goes down by 8%. It used to cost £7,800. What does it cost now? 23
- 24. Percentages 8. The price of a certain model of car goes down by 8%. It used to cost £7,800. What does it cost now? 24 What’s the new %? € = 0.92 × 7,800 Calculate 92% of £7,800 € = £7,176
- 25. Percentages vs. percentage points • “We produced only 28% more faults than the industry average” 25A percentage is a portion of the whole (where the whole isn’t necessarily 100) A percentage point is a unit of measurement that is calculated as a portion of 100
- 26. Percentages vs. percentage points • “We produced only 28% more faults than the industry average” • Actually, it was 28 percentage points higher • To calculate the percentage difference, you have to divide 28 by 35 • 80% more faults than the national average 26A percentage is a portion of the whole (where the whole isn’t necessarily 100) A percentage point is a unit of measurement that is calculated as a portion of 100
- 27. Examples of the conflation of percentages and percentage points Nationwide has upped the cost of its fixed-rate deals by up to 0.86%, and state-owned Northern Rock has raised its five-year fixed rates by 0.2%, both with effect from tomorrow* 27*“Buyers face hike in mortgage rates as inflation fears mount” The Guardian, 11th June 2009, **“Lenders rush to raise fixed-rate mortgages “ The Times 12th June 2009 On Wednesday, Times Online revealed that Nationwide Building Society, Britain's third biggest lender, was putting up rates by up to 0.86 percentage points today, the biggest hike in mortgage rates for months. A five-year fix has jumped from 4.78 per cent to 5.64 per cent**
- 28. Agenda 1. Looking at proportions 2. Basic algebra 3. Compound Annual Growth Rates (CAGR) 4. Back of the Envelope Calculations (BotEC) 28
- 29. Algebra • Algebra can be useful since it denotes numbers as symbols (e.g. a, b, c etc) – This helps us to find general rules and arithmetic laws – It allows us to recognise unknown numbers – It allows functional relationships 29
- 30. Algebra 9. The price of a widget is £1 plus half of the total price. How much would you have to pay to buy one? ? 30
- 31. Algebra 9. The price of a widget is £1 plus half of the total price. How much would you have to pay to buy one? ? 31 € PW =1+ (0.5 × PW ) € PW − 0.5PW =1 € 0.5PW =1 € PW = £2
- 32. Algebra 10.If you buy a computer for £680, how much VAT have you paid? 32 Note: £680 includes the VAT, therefore we ask two questions: 10a. What is the price before VAT gets added? 10b. What was the VAT?
- 33. Algebra 10.If you buy a computer for £680, how much VAT have you paid? 33 € 680 = RRP + 0.175RRP € 680 = RRP(1+ 0.175) € 680 1.175 = RRP € £578.82 = RRP Note: £680 includes the VAT, therefore we ask two questions: 10a. What is the price before VAT gets added? 10b. What was the VAT? € = 0.175 × 578.72 € = £101.28
- 34. Algebra 11.If you buy a computer for £5,000, how much VAT have you paid? 34
- 35. Algebra 11.If you buy a computer for £5,000, how much VAT have you paid? Need to calculate the original price: • 5,000/(1+0.175) = P • P = £4255.32 Solution: 0.175 * 4255.32 • VAT = £744.68 35
- 36. Algebra 12. A jacket costs £185.00 inc vat. What is the cost excluding vat? 36http://www.thehogman.co.uk/www.thehogman.co.uk/info.php?p=26&pno=0
- 37. Algebra 12. A jacket costs £185.00 inc vat. What is the cost excluding vat? 37http://www.thehogman.co.uk/www.thehogman.co.uk/info.php?p=26&pno=0
- 38. Agenda 1. Looking at proportions 2. Basic algebra 3. Compound Annual Growth Rates (CAGR) 4. Back of the Envelope Calculations (BotEC) 38
- 39. CAGR: Average Growth • A company boasts that they’ve achieved average growth of 25% in the last two years. Are you impressed? 39 Year Value Percentage Return 2004 £1,000 2005 £2,000 + 100% 2006 £1,000 - 50% Average growth: %25 2 %50%100 = - But growth = 0
- 40. CAGR: Average Growth • A company boasts that they’ve achieved average growth of 25% in the last two years. Are you impressed? 40 Year Value Percentage Return 2004 £1,000 2005 £2,000 + 100% 2006 £1,000 - 50% Average growth: %25 2 %50%100 = - But growth = 0
- 41. CAGR: Formula • Compound Annual Growth Rate (CAGR) is given by: 1 1 -÷÷ ø ö çç è æ = ÷ ø ö ç è æ n ValueBeginning ValueEnd CAGR 41
- 42. CAGR: Example • In 2002, 845.2m units were shipped globally. Unit shipments are expected to reach 1.404bn in 2006 • What is the CAGR for the global smartcard market? 42 € = 1,404,000 845,200 " # $ % & ' 1 4 " # $ % & ' −1 € =13.5%
- 43. CAGR: Example • In 2002, 845.2m units were shipped globally. Unit shipments are expected to reach 1.404bn in 2006 • What is the CAGR for the global smartcard market? 43 € = 1,404,000 845,200 " # $ % & ' 1 4 " # $ % & ' −1 € =13.5%
- 44. CAGR: Example • TowerGroup estimates that the financial services industry’s global IT spending on outsourcing services will grow from $27.8bn in 2003 to $38.2bn in 2006 • What’s the CAGR? 44 € = 38.2 27.8 " # $ % & ' 1 3 " # $ % & ' −1 € =11.1%
- 45. CAGR: Example • TowerGroup estimates that the financial services industry’s global IT spending on outsourcing services will grow from $27.8bn in 2003 to $38.2bn in 2006 • What’s the CAGR? 45 € = 38.2 27.8 " # $ % & ' 1 3 " # $ % & ' −1 € =11.1%
- 46. Agenda 1. Looking at proportions 2. Basic algebra 3. Compound Annual Growth Rates (CAGR) 4. Back of the Envelope Calculations (BotEC) 46
- 47. BotEC • “Back of the Envelope” means rough calculations • Tests analytic abilities • Requires logical thought process and ease with numbers • Somewhere between a guess and a proof – Demonstrate a structured thought process to arrive at a numerical answer • Many alternative ways to proceed • Have to use assumptions, and therefore justify them 47
- 48. BotEC: TV Sets • You are consulting an advertising agency who wish to launch a major television advert campaign in the US, and they ask you to estimate the potential market size • Proxy: How Many TV Sets in the US? • Two variables – # Households 100m 100m – # TVs per household 2 2.4 • Total # TV Sets 200m 240m 48
- 49. BotEC: TV Sets • You are consulting an advertising agency who wish to launch a major television advert campaign in the US, and they ask you to estimate the potential market size • Proxy: How Many TV Sets in the US? • Two variables – # Households 100m 100m – # TVs per household 2 2.4 • Total # TV Sets 200m 240m 49
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- 51. BotEC: UK Ringtone Market • A French media company is intending on breaking into the UK ringtone market (in 2006), but require an estimate of the size of the market • Split the question up – UK Population 60m – Mobile phone penetration rate 80% – Users who download ringtones 1/3 – Annual av. no. of ring tones 18 – Price per ringtone 1 • Market Size £288m 51
- 52. BotEC: UK Ringtone Market • A French media company is intending on breaking into the UK ringtone market (in 2006), but require an estimate of the size of the market • Split the question up – UK Population 60m – Mobile phone penetration rate 80% – Users who download ringtones 1/3 – Annual av. no. of ring tones 18 – Price per ringtone 1 • Market Size £288m 52
- 53. BotEC: ESCP Europe • A Middle-East consortium wish to enter the market for European business education. They realise that the most important resource in a business school is the quality of the faculty, and they have identified ESCP Europe as being especially world-class (in particular the London campus) • They have hired your team to provide a ball park estimate of the current market value of ESCP Europe 53
- 54. BotEC: Useful Figures USA • Population: 300m (US Census Bureau estimate, 2006) • GNI per capita: US $43,740 (World Bank, 2006) UK • Population: 60.2 million (National Statistics, 2005) • GNI per capita: US $37,600 (World Bank, 2006) India • Population: 1.1 billion (UN, 2005) • GNI per capita: US $720 (World Bank, 2006) China • Population: 1.3 billion (UN, 2005) • GNI per capita: US $1,740 (World Bank, 2006) 54
- 55. • This presentation forms part of a free, online course on analytics • http://econ.anthonyjevans.com/courses/analytics/ 55