Financial Math

1. Financial Math Currency, Interest and Depreciation Mr. Morrow 2/21/2013 – 2/26/2013

2. - Warm Up - Suppose one night you go out to dinner downtown. The sales tax on food is 5.02%. The final bill (before tip) came to 52.43. How much was your dinner before tax? Two baseball players, A and B, are in their first half of the season. Player A’s batting average was higher than play B’s batting average. During the second half of the season, player A’s batting average was higher than player B’s (again). For the entire season, player B’s batting average was higher than that of player A. Is that possible? Assume it takes 6 hours to fully cook your 8lb Thanksgiving turkey. You only have 253 minutes that you can set aside to cook though. How much turkey should you get if that is all of the time you have?

3. - Warm Up - Suppose one night you go out to dinner downtown. The sales tax on food is 5.02%. The final bill (before tip) came to $52.43. How much was your dinner before tax? Two baseball players, A and B, are in their first half of the season. Player A’s batting average was higher than play B’s batting average. During the second half of the season, player A’s batting average was higher than player B’s (again). For the entire season, player B’s batting average was higher than that of player A. Is that possible? Assume it takes 6 hours to fully cook your 8lb Thanksgiving turkey. You only have 253 minutes that you can set aside to cook though. How much turkey should you get if that is all of the time you have?

4. - Currency Conversion - Well we all saw last week that: 1 = 1 So can anyone explain to me how this can actually be true…?? 1 ≠ 1 … $1 ≠ € 1 Yea it couldn’t have anything to do with the big heading at the top…Currency Several aspects affect the conversion rate between countries: Buying/ Selling Investments in various countries Strength/ Success of country’s economy But this is economics, who wants to learn about that in a Math class……

5. - Currency Conversion - For this section we will be looking at the idea of, Exchange rates- The price of one country's currency expressed in another country's currency. In other words, the rate at which one currency can be exchanged for another.  Foreign Currency Exchange Rate Currency We Buy We Sell Currency We Buy We Sell USD 1.632 1.494 EUR 1.207 1.106 AUD 1.839 1.660 CAD 1.724 1.549 CZK 31.208 27.645 DKK 9.026 8.072 JPY 150.853 134.585 NZD 2.363 2.113 PLN 4.876 4.299 NOK 9.826 8.787 ZAR 12.593 11.208 SEK 11.947 10.683 CHF 1.777 1.589 TRY 2.495 2.227 AED 6.021 5.410 How many UK pounds (£) will you get for your $150?

6. - Currency Conversion - Currency Conversion Formula: We need to use the “We Buy” figure of 1.632 as the bureau de change is “buying” us British Pounds in exchange for US Dollars.

7. - Practice - • Assume the currency conversion rates between the US and four countries are shown below: US ($)Aus($A) Sterling (£) Yen (¥) Euro (€) 1 1.633 0.610 134.490 1.352 Convert: • $50 US to Yen • £15 to $US • $1500 Australian to Euro • ¥125 to Sterling

8. - Practice - • Assume the currency conversion rates between the US and four countries are shown below: US ($)Aus($A) Sterling (£) Yen (¥) Euro (€) 1 1.633 0.610 134.490 1.352 Convert: • $50 US to Yen

9. - Practice - • Assume the currency conversion rates between the US and four countries are shown below: US ($)Aus($A) Sterling (£) Yen (¥) Euro (€) 1 1.633 0.610 134.490 1.352 Convert: • 2. £15 to $US

10. - Practice - • Assume the currency conversion rates between the US and four countries are shown below: US ($)Aus($A) Sterling (£) Yen (¥) Euro (€) 1 1.633 0.610 134.490 1.352 Convert: • 3. $1500 Australian to euros

11. - Practice - • Assume the currency conversion rates between the US and four countries are shown below: US ($)Aus($A) Sterling (£) Yen (¥) Euro (€) 1 1.633 0.610 134.490 1.352 Convert: • ¥125 to Sterling

12. - A Case of Robbery - Let’s look back at when we converted $150 to . If we had to travel back to America with the same , how much $ UD do we have? $150 right….?? We go to the bureau de change to convert the money again: Ugh… I left with $150 and come home with $137.31 but didn’t buy anything. Who stole my missing $12.69??

13. - Commission - It is generally the case that agents who change money between currencies will charge for this service. There charges are often called commission. There are two main ways in which agents charge such a commission. An agent offers to exchange $US to other currencies at the published daily rate. Their commission is $5 per transaction or 1%, whichever is greater with the commission being paid is in $US. Two customers wish to convert the following amounts of $US to Italian lire on a day when the exchange rate was 1760 lire to the dollar: a) $20 b) $20,000 $20 - $5 = $15 $20,000 - $5 = $19,995 $15(1760) = 26,400 lire $19,800(1760) = 34,848,000 lire

14. - Practice - A bank offers the following exchange rates for $1 Australian in relation to French franc: ‘We buy: 3.5959, we sell: 3.5138’. A customer wishes to exchange $1,200 Australian for francs. How many francs will the customer receive? If the customer then immediately exchanges these francs for Australian dollars, how much will see receive? What is the effective exchange rate? Assume that all currency amounts are rounded to the nearest whole number to complete each transaction. The bank is buying Australian from the customer so the rate for the first transaction is the ‘buying rate’ of 3.5138. If the customer immediately reconverts the amount back to Australian dollars, the bank will use the ‘selling rate’ of 3.5138. This amounts to a commission of 1,200 – 1,174 = $26 on two transactions.

15. - Walk Out…and take home to finish - Suppose you are planning your honeymoon and need to determine how much money you are going to spend for the entire trip. Fill in the blanks to the following ‘Trip Scenario’ then determine the cost of your Honeymoon. Sunday night we are leaving from Dulles Airport to _________ . The flight is going to cost $2,500 USD. We are going to stay there for 2 nights at their Trump Hotel which has breakfast, lunch and dinner included in the cost for __ 500. The next morning we are planning to rent a car for __ 750 (Assume no gas purchases). We drive from _________ to _________ where we will stay at their Hilton for one night at __ 150. Wednesday morning we board a flight from _________ to _________ for __ 3,250. Being the last stop on our trip we want this to be really memorable. We take a guided tour of the area for __ 100, buy __ 350 worth of souvenirs, partake in __ 1,200 worth of activities, and our romantic meals for those 2 days add up to __ 800. Our flight home Saturday morning costs __ 3,250.

16. - Exchange Rates (2/8/13) - We Buy We Sell US $ 1.63 1.49 €-Euro1.21 1.11 British Pound-£9.61 8.73 Canadian $1.72 1.55 Australian $1.84 1.66 Indian Rupee44.5 41.4 Philippine Peso57.4 53.7 Swiss Franc1.78 1.59 South African Rand 12.6 11.2 New Zealand $2.36 2.11 Jamaica $137.1 132.4 Japanese Yen 150.9 134.6 Russian Ruble 44.1 39.8 Brazilian Real 2.88 2.63 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙

17. - Warm Up - A bank offers the following exchange rates for 1 German mark in relation to the European Euro: ‘We Buy: 1033, we sell:1019’. If a customer wishes to exchange the following amounts, find: The number of Euros the customer receives, correct to the nearest number. The number of German marks that will result if the amount in i. is immediately returned to German marks. The effective commission on the two transactions. a) (1000) b) (1608) c) (1217)

18. - Interest - • When money is borrowed or lent, the borrower usually pays the lender for the service. The amount charged is generally called interest. There are commonly 2 methods for calculating interest: • Simple Interest • Compound Interest • Simple Interest – Calculated as a percentage of the amount borrowed (Principal) • I = interest paid • C = principal (amount originally invested) • r = the interest rate (%) • n = the number of time periods that the loan or investment lasts. • r and n must refer to the same period of time. • This formula can be altered to show total amount of Future money earned (or borrowed).

19. - Interest - If $600 is invested in an account and earns $136.50 simple interest over 3.5 years, find the annual interest rate and future value. F = C + I = 600 + 136.5 = $736.50

20. - Practice - If 1,450 crowns are invested at a monthly interest rate of 0.08% and earn 8.12 crowns simple interest, find the number of months of the investment. F = C + I = 1,450 + 8.12 = 1,458.12

21. - Simple (Graphically) - Simple interest is the most basic type of return. Suppose we deposit $100 into an account with 50% simple (annual) interest, we could represent it like this: You start with a principal of $100 and earn $50 each year. However this ‘new’ green money is stagnant – it can’t grow! With simple interest, the $50 just sits there. Only the original $100 can do ‘work’ to generate money For simple interest you can kind of imagine It as ‘speed’. Here you’ll earn 50% of your Principal in the course of a year. (your ‘speed of money growth’).

22. - Compound Interest - As we just saw, simple interest implies that every time interest is paid, it is withdrawn by the investor. This leaves the principal untouched throughout the period of the loan. However, it is much more usual for investors to add (compound) the interest to the principal so that the principal grows during the term of the investment. Compound Interest – Calculated as a percentage of the amount borrowed (Principal) = Future amount in the account after n periods C = principal (amount originally invested) r = the interest rate (%) n = the number of compounding periods This formula can be altered to show total amount of Interest earned.

23. - Practice - $1,200 is placed in a savings account that pays 8% annual interest compounded annually. Find the amount in the account and the interest paid after 10 years. How much interest was actually earned? I = $2,590.71 – $1,200 = $1,390.71

24. - Compound (Graphically) - Simple interest should make you squirm. Why can’t our interest earn money? We should use the bond payouts ($50/year) to buy more bonds. Compound growth means your interest earns interest. Einstein called it “one of the most powerful forces in nature”, and it’s true. When you have a growing thing, which creates more growing things, which creates more growing things… your return adds up fast. We earn $50 from year 0 – 1, just like with simple interest. But in year 1-2, now that our total is $150, we can earn $75 this year (50% * 150) giving us $225. In year 2-3 we have $225, so we earn 50% of that, or $112.50. This is an interesting viewpoint. The $100 just mindlessly cranks out $50 “factories”, which start earning money independently (notice the 3 blue arrows from the blue principal to the green $50s). These $50 factories create $25 factories, and so on. The pattern seems complex, but it’s simpler in a way as well. The $100 has no idea what those zany $50s are up to: as far as the $100 knows, we’re only making $50/year. With simple interest, we kept the same pace forever ($50/year — pretty boring). With annually compounded interest, we get a new trajectory each year.

25. - Compound-(ing) Interest - How can we earn more money if our rate, duration of investment and principal stay the same? Instead of just compounding once a year we alter how many times we compound per year: = Future amount in the account after n periods C = principal (amount originally invested) r = the interest rate (%) n = the number of compounding periods k = the number of times that interest is paid for each period n This formula can be altered to show total amount of Interest earned.

26. - Practice - $1,200 is placed in a savings account that pays 8% interest compounded quarterly. Find the amount in the account and the interest paid after 10 years. How much interest was actually earned? I = $2,649.65 – $1,200 = $1,449.65 $58.94 more than compounding it annually (Gotta love free money!)

27. - Effective Interest Rates - In order to be able to compare different loan or deposit offers we need to determine the effective rate– the annual rate that will produce the same amount of interest per year as the nominal rate r compounded k times per year. r is not a percentage but a decimal now. Example: If the nominal rate is 8% compounded semiannually, determine the effective rate.

28. - What is time, time, time, baby just solve for me once more - If this is our given equation: How would we solve for n? We would solve for time just as we did with our exponential problems (I mean who would’ve figured since this is exponential growth) with our calculators.

29. - Practice - How long will it take $2,000 to grow to $4,300 if invested at 8% with interest compounded quarterly? Homework:pg. 622 (odd) pg. 624 (odd problems, for letters just do a,c,e) pg. 629/630 (same as 624) pg. 635/636 (odd)

30. - Warm Up - $2,500 is placed in a savings account that pays 6.5% interest compounded monthly. Find the amount in the account and the interest paid after 20 years. How much interest was actually earned? If $500 is invested in an account and earns $75.50 simple interest over 2.75 years, find the annual interest rate and future value. How long will it take $2,000 to grow to $4,300 if invested at 8% with interest compounded quarterly?

31. - Depreciation - A reduction in the value of an asset with the passage of time, due in particular to wear and tear. There are 3 terms generally associated with depreciation: 1. Book Value – The estimated value of an item at any point in time. 2. Written Off Value – When the book value is equal to zero. 3. Scrap Value – At the end of an item’s effective (or useful) life, its book value is called its scrap value. Depreciation can be calculated by any of the following methods (in this course): 1. Flat Rate (straight-line) 2. Reducing Balance 3. Unit Cost

32. - Flat Rate (Straight-line) - What do you think this section of deprecation relates to in what we’ve been learning about? ∙ Arithmetic Growth (Straight-line function) If an asset's value declines at a fixed rate over time then it is said to experience straight line or flat rate depreciation. Book Value ($) BVt BVt = C – Rt (Where R is the fixed rate of depreciation) Original Cost of Asset (Purchase Price) C Book value at time T = t Time until the asset is written off BVt Time t (periods) T

33. - Practice - James bought the MacBook Pro 4 years ago. The current book value of it is $850. If the computer. If the computer has been depreciating at $87.5 p.a. how much did James pay for the laptop?

34. - Practice - Henrico High School purchased our classroom desks for the school 15 years ago at $5,000. Today, all of the desks are worth $3,200. Assuming a flat rate of depreciation, determine the annual depreciation of the desks.

35. - Reducing Balance - Sometimes equipment wears out rather fast but then remains useful for an extended period of time due to investments in maintaining it… ∙ Businesses may choose to replace equipment rather than maintain it while other businesses will choose the alternative. ∙ Reducing balance depreciation results in an exponential equation where 0 ≤ r ≤ 1. ∙ The equation often takes the form: Where… is the book value at some future time t C is the original cost or purchase price r is the fixed % depreciation rate t is the time since the asset was purchased

36. - Reducing Balance - BVt C Book value at time T BVt Scrap value t T

37. - Practice - Glen Allen towing just bought two new tow trucks for $35,000 each. They decide that one of the trucks will be depreciating at the flat rate of 15% of the purchase price, while the other will be depreciating at 20% using a reducing balance method. It is estimated that the trucks will have a useful life of 6 years. 1. Construct a depreciation schedule for each truck 2. Sketch the graphs showing the relationship between the truck’s book value and the time since they were purchased. 3. At the end of 6 years, Glen Allen towing analyzes the depreciation behavior for both methods. Is there a preference as to which method they should use in the future?

38. - Practice - 1. Construct a depreciation schedule for each truck 1st Truck: 2nd Truck: 2. Sketch the graphs showing the relationship between the truck’s book value and the time since they were purchased. Flat rate Book value is equal Reducing balance 1 2 6 4 3 5

39. - Unit Cost - • Unlike the flat rate or the reducing balance methods of calculating depreciation, unit cost depreciation is not a time based calculation • Unit cost recognizes that the life of an asset can be estimated by its actual usage • rather then how old it is. • ∙ The challenge is to determine a measure of usage of the asset. • - Number of kilometers driven • - Number of copies made • - Number of hours used • Whatever method is used some estimate of usage needs to be made: • Current depreciation = (amount of actual usage )(rate of depreciation) • Where rate of depreciation = • Total depreciation = Purchase Price – scrap value

40. - Practice - A photocopy machine was purchased for $12,000 and has an estimated life span of 1,000,000 copies at which point its scrap value will be $1,500. During its first year operation 110,000 copies are made. Find the book value at the end of the first year. Book value = purchase price – depreciation (after 110,000 copies) Amount of actual usage = 110,000 copies Estimated total usage = 1,000,000 Total depreciation = $12,000 - $1,500 = $10,500 Depreciation after the first 110,000 copies = Therefore the book value is $12,000 - $1,155 = $10,845

41. - Practice - Our $120,000 Audi has a scrap value of $25,000 once we have driven 250,000 miles. 1. Find the depreciation rate in dollars/ km 2. Find the book value of the car after reaching 50,000 miles Homework: pg. 664 – Grade Revision Exercises (All)

42. - Practice - Our $120,000 Audi has a scrap value of $25,000 once we have driven 250,000 miles. 1. Find the depreciation rate in dollars/ km Total Depreciation = Purchase Price – Scrap Value = $120,000 - $25,000 = $95,000 Therefore our rate = = 0.38 or $0.38/mile 2. Find the book value of the car after reaching 50,000 miles Book Value (@ 50,000) = Purchase Price – Depreciation (@ 50,000) First we need = Thus our depreciation at 50,000 miles is … = $ (95,000) = $ 19,000 Book Value = $120,000 – $19,000 = $101,000