Finance in Math

Thursday, 12 may 2016. 5 : 45 pm . i wanna make new post math about finance

Finance - Grade 10 8.1 Introduction Should you ever find yourself stuck with a mathematics question on a television quiz show, you will probably wish you had remembered the how many even prime numbers there are between 1 and 100 for the sake of R1 000 000. And who does not want to be a millionaire, right? Welcome to the Grade 10 Finance Chapter, where we apply maths skills to everyday financial situations that you are likely to face both now and along your journey to purchasing your first private jet. If you master the techniques in this chapter, you will grasp the concept of compound interest, and how it can ruin your fortunes if you have credit card debt, or make you millions if you successfully invest your hard-earned money. You will also understand the effects of fluctuating exchange rates, and its impact on your spending power during your overseas holidays! 8.2 Foreign Exchange Rates Is $500 (”500 US dollars”) per person per night a good deal on a hotel in New York City? The first question you will ask is “How much is that worth in Rands?”. A quick call to the local bank or a search on the Internet (for example on http://www.x-rates.com/) for the Dollar/Rand exchange rate will give you a basis for assessing the price. A foreign exchange rate is nothing more than the price of one currency in terms of another. For example, the exchange rate of 6,18 Rands/US Dollars means that $1 costs R6,18. In other words, if you have $1 you could sell it for R6,18 - or if you wanted $1 you would have to pay R6,18 for it. But what drives exchange rates, and what causes exchange rates to change? And how does this affect you anyway? This section looks at answering these questions. 8.2.1 How much is R1 really worth? We can quote the price of a currency in terms of any other currency, but the US Dollar, British Pounds Sterling or even the Euro are often used as a market standard. You will notice that the financial news will report the South African Rand exchange rate in terms of these three major currencies. So the South African Rand could be quoted on a certain date as 6,7040 ZAR per USD (i.e. $1,00 costs R6,7040), or 12,2374 ZAR per GBP. So if I wanted to spend $1 000 on a holiday in the United States of America, this would cost me R6 704,00; and if I wanted £1 000 for a weekend in London it would cost me R12 237,40. This seems obvious, but let us see how we calculated that: The rate is given as ZAR per USD, or ZAR/USD such that $1,00 buys R6,7040. Therefore, we need to multiply by 1 000 to get the 53 8.2 CHAPTER 8. FINANCE - GRADE 10 Table 8.1: Abbreviations and symbols for some common currencies. Currency Abbreviation Symbol South African Rand ZAR R United States Dollar USD $ British Pounds Sterling GBP £ Euro EUR e number of Rands per $1 000. Mathematically, $1,00 = R6,0740 ∴ 1 000 × $1,00 = 1 000 × R6,0740 = R6 074,00 as expected. What if you have saved R10 000 for spending money for the same trip and you wanted to use this to buy USD? How much USD could you get for this? Our rate is in ZAR/USD but we want to know how many USD we can get for our ZAR. This is easy. We know how much $1,00 costs in terms of Rands. $1,00 = R6,0740 ∴ $1,00 6,0740 = R6,0740 6,0740 $ 1,00 6,0740 = R1,00 R1,00 = $ 1,00 6,0740 = $0,164636 As we can see, the final answer is simply the reciprocal of the ZAR/USD rate. Therefore, R10 000 will get: R1,00 = $ 1,00 6,0740 ∴ 10 000 × R1,00 = 10 000 × $ 1,00 6,0740 = $1 646,36 We can check the answer as follows: $1,00 = R6,0740 ∴ 1 646,36 × $1,00 = 1 646,36 × R6,0740 = R10 000,00 Six of one and half a dozen of the other So we have two different ways of expressing the same exchange rate: Rands per Dollar (ZAR/USD) and Dollar per Rands (USD/ZAR). Both exchange rates mean the same thing and express the value of one currency in terms of another. You can easily work out one from the other - they are just the reciprocals of the other. If the South African Rand is our Domestic (or home) Currency, we call the ZAR/USD rate a “direct” rate, and we call a USD/ZAR rate an “indirect” rate. In general, a direct rate is an exchange rate that is expressed as units of Home Currency per 54 CHAPTER 8. FINANCE - GRADE 10 8.2 units of Foreign Currency, i.e., Domestic Currency / Foreign Currency. The Rand exchange rates that we see on the news are usually expressed as Direct Rates, for example you might see: Table 8.2: Examples of exchange rates Currency Abbreviation Exchange Rates 1 USD R6,9556 1 GBP R13,6628 1 EUR R9,1954 The exchange rate is just the price of each of the Foreign Currencies (USD, GBP and EUR) in terms of our Domestic Currency, Rands. An indirect rate is an exchange rate expressed as units of Foreign Currency per units of Home Currency, i.e. Foreign Currency / Domestic Currency Defining exchange rates as direct or indirect depends on which currency is defined as the Domestic Currency. The Domestic Currency for an American investor would be USD which is the South African investor’s Foreign Currency. So direct rates from the perspective of the American investor (USD/ZAR) would be the same as the indirect rate from the perspective of the South Africa investor. Terminology Since exchange rates are simple prices of currencies, movements in exchange rates means that the price or value of the currency changed. The price of petrol changes all the time, so does the price of gold, and currency prices also move up and down all the time. If the Rand exchange rate moved from say R6,71 per USD to R6,50 per USD, what does this mean? Well, it means that $1 would now cost only R6,50 instead of R6,71. The Dollar is now cheaper to buy, and we say that the Dollar has depreciated (or weakened) against the Rand. Alternatively we could say that the Rand has appreciated (or strengthened) against the Dollar. What if we were looking at indirect exchange rates, and the exchange rate moved from $0,149 per ZAR (= 1 6,71 ) to $0,1538 per ZAR (= 1 6,50 ). Well now we can see that the R1,00 cost $0,149 at the start, and then cost $0,1538 at the end. The Rand has become more expensive (in terms of Dollars), and again we can say that the Rand has appreciated. Regardless of which exchange rate is used, we still come to the same conclusions. In general, • for direct exchange rates, the home currency will appreciate (depreciate) if the exchange rate falls (rises) • For indirect exchange rates, the home currency will appreciate (depreciate) if the exchange rate rises (falls) As with just about everything in this chapter, do not get caught up in memorising these formulae - that is only going to get confusing. Think about what you have and what you want - and it should be quite clear how to get the correct answer. Activity :: Discussion : Foreign Exchange Rates In groups of 5, discuss: 1. Why might we need to know exchange rates? 2. What happens if one countries currency falls drastically vs another countries currency? 55 8.2 CHAPTER 8. FINANCE - GRADE 10 3. When might you use exchange rates? 8.2.2 Cross Currency Exchange Rates We know that the exchange rates are the value of one currency expressed in terms of another currency, and we can quote exchange rates against any other currency. The Rand exchange rates we see on the news are usually expressed against the major currency, USD, GBP and EUR. So if for example, the Rand exchange rates were given as 6,71 ZAR/USD and 12,71 ZAR/GBP, does this tell us anything about the exchange rate between USD and GBP? Well I know that if $1 will buy me R6,71, and if £1.00 will buy me R12,71, then surely the GBP is stronger than the USD because you will get more Rands for one unit of the currency, and we can work out the USD/GBP exchange rate as follows: Before we plug in any numbers, how can we get a USD/GBP exchange rate from the ZAR/USD and ZAR/GBP exchange rates? Well, USD/GBP = USD/ZAR × ZAR/GBP. Note that the ZAR in the numerator will cancel out with the ZAR in the denominator, and we are left with the USD/GBP exchange rate. Although we do not have the USD/ZAR exchange rate, we know that this is just the reciprocal of the ZAR/USD exchange rate. USD/ZAR = 1 ZAR/USD Now plugging in the numbers, we get: USD/GBP = USD/ZAR × ZAR/GBP = 1 ZAR/USD × ZAR/GBP = 1 6,71 × 12,71 = 1,894 Important: Sometimes you will see exchange rates in the real world that do not appear to work exactly like this. This is usually because some financial institutions add other costs to the exchange rates, which alter the results. However, if you could remove the effect of those extra costs, the numbers would balance again. Worked Example 8: Cross Exchange Rates Question: If $1 = R 6,40, and £1 = R11,58 what is the $/£ exchange rate (i.e. the number of US$ per £)? Answer Step 1 : Determine what is given and what is required The following are given: • ZAR/USD rate = R6,40 • ZAR/GBP rate = R11,58 56 CHAPTER 8. FINANCE - GRADE 10 8.2 The following is required: • USD/GBP rate Step 2 : Determine how to approach the problem We know that: USD/GBP = USD/ZAR × ZAR/GBP. Step 3 : Solve the problem USD/GBP = USD/ZAR × ZAR/GBP = 1 ZAR/USD × ZAR/GBP = 1 6,40 × 11,58 = 1,8094 Step 4 : Write the final answer $1,8094 can be bought for £1. Activity :: Investigation : Cross Exchange Rates - Alternate Method If $1 = R 6,40, and £1 = R11,58 what is the $/£ exchange rate (i.e. the number of US$ per £)? Overview of problem You need the $/£ exchange rate, in other words how many dollars must you pay for a pound. So you need £1. From the given information we know that it would cost you R11,58 to buy £1 and that $ 1 = R6,40. Use this information to: 1. calculate how much R1 is worth in $. 2. calculate how much R11,58 is worth in $. Do you get the same answer as in the worked example? 8.2.3 Enrichment: Fluctuating exchange rates If everyone wants to buy houses in a certain suburb, then house prices are going to go up - because the buyers will be competing to buy those houses. If there is a suburb where all residents want to move out, then there are lots of sellers and this will cause house prices in the area to fall - because the buyers would not have to struggle as much to find an eager seller. This is all about supply and demand, which is a very important section in the study of Economics. You can think about this is many different contexts, like stamp-collecting for example. If there is a stamp that lots of people want (high demand) and few people own (low supply) then that stamp is going to be expensive. And if you are starting to wonder why this is relevant - think about currencies. If you are going to visit London, then you have Rands but you need to “buy” Pounds. The exchange rate is the price you have to pay to buy those Pounds. Think about a time where lots of South Africans are visiting the United Kingdom, and other South Africans are importing goods from the United Kingdom. That means there are lots of Rands (high supply) trying to buy Pounds. Pounds will start to become more expensive (compare this to the house price example at the start of this section if you are not convinced), and the 57 8.3 CHAPTER 8. FINANCE - GRADE 10 exchange rate will change. In other words, for R1 000 you will get fewer Pounds than you would have before the exchange rate moved. Another context which might be useful for you to understand this: consider what would happen if people in other countries felt that South Africa was becoming a great place to live, and that more people were wanting to invest in South Africa - whether in properties, businesses - or just buying more goods from South Africa. There would be a greater demand for Rands - and the “price of the Rand” would go up. In other words, people would need to use more Dollars, or Pounds, or Euros ... to buy the same amount of Rands. This is seen as a movement in exchange rates. Although it really does come down to supply and demand, it is interesting to think about what factors might affect the supply (people wanting to “sell” a particular currency) and the demand (people trying to “buy” another currency). This is covered in detail in the study of Economics, but let us look at some of the basic issues here. There are various factors affect exchange rates, some of which have more economic rationale than others: • economic factors (such as inflation figures, interest rates, trade deficit information, monetary policy and fiscal policy) • political factors (such as uncertain political environment, or political unrest) • market sentiments and market behaviour (for example if foreign exchange markets perceived a currency to be overvalued and starting selling the currency, this would cause the currency to fall in value - a self fulfilling expectation). Exercise: Foreign Exchange 1. I want to buy an IPOD that costs £100, with the exchange rate currently at £1 = R14. I believe the exchange rate will reach R12 in a month. (a) How much will the MP3 player cost in Rands, if I buy it now? (b) How much will I save if the exchange rate drops to R12? (c) How much will I lose if the exchange rate moves to R15? 2. Study the following exchange rate table: Country Currency Exchange Rate United Kingdom (UK) Pounds(£) R14,13 United States (USA) Dollars ($) R7,04 (a) In South Africa the cost of a new Honda Civic is R173 400. In England the same vehicle costs £12 200 and in the USA $ 21 900. In which country is the car the cheapest if you compare it to the South African Rand ? (b) Sollie and Arinda are waiters in a South African Restaurant attracting many tourists from abroad. Sollie gets a £6 tip from a tourist and Arinda gets $ 12. How many South African Rand did each one get ? 8.3 Being Interested in Interest If you had R1 000, you could either keep it in your wallet, or deposit it in a bank account. If it stayed in your wallet, you could spend it any time you wanted. If the bank looked after it for you, then they could spend it, with the plan of making profit off it. The bank usually “pays” you to deposit it into an account, as a way of encouraging you to bank it with them, This payment is like a reward, which provides you with a reason to leave it with the bank for a while, rather than keeping the money in your wallet. 58 CHAPTER 8. FINANCE - GRADE 10 8.4 We call this reward ”interest”. If you deposit money into a bank account, you are effectively lending money to the bank - and you can expect to receive interest in return. Similarly, if you borrow money from a bank (or from a department store, or a car dealership, for example) then you can expect to have to pay interest on the loan. That is the price of borrowing money. The concept is simple, yet it is core to the world of finance. Accountants, actuaries and bankers, for example, could spend their entire working career dealing with the effects of interest on financial matters. In this chapter you will be introduced to the concept of financial mathematics - and given the tools to cope with even advanced concepts and problems. Important: Interest The concepts in this chapter are simple - we are just looking at the same idea, but from many different angles. The best way to learn from this chapter is to do the examples yourself, as you work your way through. Do not just take our word for it! 8.4 Simple Interest Definition: Simple Interest Simple interest is where you earn interest on the initial amount that you invested, but not interest on interest. As an easy example of simple interest, consider how much you will get by investing R1 000 for 1 year with a bank that pays you 5% simple interest. At the end of the year, you will get an interest of: Interest = R1 000 × 5% = R1 000 × 5 100 = R1 000 × 0,05 = R50 So, with an “opening balance” of R1 000 at the start of the year, your “closing balance” at the end of the year will therefore be: Closing Balance = Opening Balance + Interest = R1 000 + R50 = R1 050 We sometimes call the opening balance in financial calculations Principal, which is abbreviated as P (R1 000 in the example). The interest rate is usually labelled i (5% in the example), and the interest amount (in Rand terms) is labelled I (R50 in the example). So we can see that: I = P × i (8.1) and Closing Balance = Opening Balance + Interest = P + I = P + (P × i) = P(1 + i) 59 8.4 CHAPTER 8. FINANCE - GRADE 10 This is how you calculate simple interest. It is not a complicated formula, which is just as well because you are going to see a lot of it! Not Just One You might be wondering to yourself: 1. how much interest will you be paid if you only leave the money in the account for 3 months, or 2. what if you leave it there for 3 years? It is actually quite simple - which is why they call it Simple Interest. 1. Three months is 1/4 of a year, so you would only get 1/4 of a full year’s interest, which is: 1/4 × (P × i). The closing balance would therefore be: Closing Balance = P + 1/4 × (P × i) = P(1 + (1/4)i) 2. For 3 years, you would get three years’ worth of interest, being: 3 × (P × i). The closing balance at the end of the three year period would be: Closing Balance = P + 3 × (P × i) = P × (1 + (3)i) If you look carefully at the similarities between the two answers above, we can generalise the result. In other words, if you invest your money (P) in an account which pays a rate of interest (i) for a period of time (n years), then, using the symbol (A) for the Closing Balance: Closing Balance,(A) = P(1 + i · n) (8.2) As we have seen, this works when n is a fraction of a year and also when n covers several years. Important: Interest Calculation Annual Rates means Yearly rates. and p.a.(per annum) = per year Worked Example 9: Simple Interest Question: If I deposit R1 000 into a special bank account which pays a Simple Interest of 7% for 3 years, how much will I get back at the end? Answer Step 1 : Determine what is given and what is required • opening balance, P = R1 000 • interest rate, i = 7% • period of time, n = 3 years We are required to find the closing balance (A). Step 2 : Determine how to approach the problem We know from (8.2) that: Closing Balance,(A) = P(1 + i · n) 60 CHAPTER 8. FINANCE - GRADE 10 8.4 Step 3 : Solve the problem A = P(1 + i · n) = R1 000(1 + 3 × 7%) = R1 210 Step 4 : Write the final answer The closing balance after 3 years of saving R1 000 at an interest rate of 7% is R1 210. Worked Example 10: Calculating n Question: If I deposit R30 000 into a special bank account which pays a Simple Interest of 7.5% ,for how many years must I invest this amount to generate R45 000 Answer Step 1 : Determine what is given and what is required • opening balance, P = R30 000 • interest rate, i = 7,5% • closing balance, A = R45 000 We are required to find the number of years. Step 2 : Determine how to approach the problem We know from (8.2) that: Closing Balance (A) = P(1 + i · n) Step 3 : Solve the problem Closing Balance (A) = P(1 + i · n) R45 000 = R30 000(1 + n × 7,5%) (1 + 0,075 × n) = 45000 30000 0,075 × n = 1,5 − 1 n = 0,5 0,075 n = 6,6666667 Step 4 : Write the final answer n has to be a whole number, therefore n = 7. The period is 7 years for R30 000 to generate R45 000 at a simple interest rate of 7,5%. 8.4.1 Other Applications of the Simple Interest Formula 61 8.4 CHAPTER 8. FINANCE - GRADE 10 Worked Example 11: Hire-Purchase Question: Troy is keen to buy an addisional hard drive for his laptop advertised for R 2 500 on the internet. There is an option of paying a 10% deposit then making 24 monthly payments using a hire-purchase agreement where interest is calculated at 7,5% p.a. simple interest. Calculate what Troy’s monthly payments will be. Answer Step 1 : Determine what is given and what is required A new opening balance is required, as the 10% deposit is paid in cash. • 10% of R 2 500 = R250 • new opening balance, P = R2 500 − R250 = R2 250 • interest rate, i = 7,5% = 0,075pa • period of time, n = 2 years We are required to find the closing balance (A) and then the montly payments. Step 2 : Determine how to approach the problem We know from (8.2) that: Closing Balance,(A) = P(1 + i · n) Step 3 : Solve the problem A = P(1 + i · n) = R2 250(1 + 2 × 7,5%) = R2 587,50 Monthly payment = 2587,50 ÷ 24 = R107,81 Step 4 : Write the final answer Troy’s monthly payments = R 107,81 Worked Example 12: Depreciation Question: Seven years ago, Tjad’s drum kit cost him R12 500. It has now been valued at R2 300. What rate of simple depreciation does this represent ? Answer Step 1 : Determine what is given and what is required • opening balance, P = R12 500 • period of time, n = 7 years • closing balance, A = R2 300 We are required to find the rate(i). Step 2 : Determine how to approach the problem We know from (8.2) that: Closing Balance,(A) = P(1 + i · n) Therefore, for depreciation the formula will change to: Closing Balance,(A) = P(1 − i · n) 62 CHAPTER 8. FINANCE - GRADE 10 8.5 Step 3 : Solve the problem A = P(1 − i · n) R2 300 = R12 500(1 − 7 × i) i = 0,11657... Step 4 : Write the final answer Therefore the rate of depreciation is 11,66%

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