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## 3.4 FSMQ Mathematical Principles for Personal Finance (9996)

### Mathematical Principles for Personal Finance (9996)

The content of this unit covers three areas: the value of money over time, indices used to measure key financial information and tables and diagrams of financial information.

### The value of money over time

 The value of money varies over time. Imagine you were asked if you would like to be given a £1000 now or in ten years time. What would be your response? Even if you didn’t spend the money for ten years it would be better if you had the money now: you could invest it and it would be worth more at the end of the ten years. If, for example, you were able to invest it at 4% interest per year, after 10 years it would be worth £1480. Of course, in that period due to inflation, depending on what you spend the £1000 on it is likely to cost you more. However, some goods come down in price over time: this is often true, for example, for computer equipment. A question you need to consider then is, what is the cost of what you might want to buy likely to be at the end of the ten year period relative to what it costs now? Understanding how money varies over time is, therefore, a very important idea to consider when making all manner of financial decisions

#### What you need to learn

 Financial aspect Mathematical understanding This includes The key idea of present and future values present value, future value, Interest rates: AER calculating the annual effective interest, r, rate given a nominal interest rate, i $r = \Big( 1 + \frac{i}{n} \Big ) ^n - 1$ where $\space n \space$ is the number of compounding periods per year Calculating the future value of a present sum (using ideas of compound interest) $FV = PV(1+r)^n$ where $\space r \space$ is the interest rate expressed as a decimal and $\space n \space$ is the number of time periods understanding as a geometric series $a, \space ar^{2}, \space ar^{3}, \space ...ar^{\mbox{n}-1}, \space ar^{\mbox{n}}$ Use of recurrence relations* eg $P_{n + 1} = P_n(1 + r)$ Calculating the present value of a future sum $PV = \frac{FV}{(1+r)^n}$
*You should understand and be able to use recurrence relations in a range of financial situations, such as iteratively calculating the balance on a credit card, the balance remaining on an outstanding mortgage loan, the accumulating amount in a savings account when you make regular savings and so on.
 Continuous compounding understanding that the idea of continuous compounding leads to exponential functions ie considering the case where $P = P_0 \Big( 1 + \frac{r}{n} \Big ) ^{nt}$ is the amount after $\space t \space$ years for an initial investment of $\space P_0 \space$ when the interest is compounded $n$ times per year, and $\displaystyle \space n \rightarrow \infty \space$ giving $\space P = P_0e^{rt}$ APR (annual percentage rate) Assume no arrangement or exit fees. Use of the simplified version formula for APR in straightforward cases. ie $\mbox{C} = \displaystyle \sum_{k=1}^{m} \left(\frac{A_k}{(1+i)^{(t_k)}}\right)$ where $\space i \space$ is the APR expressed as a decimal, $\space k \space$ is the number identifying a particular instalment, $\space A_k \space$ is the amount of the instalment $\space k \mathrm{,}\space t_k \space$ is the interval in years between the payment of the instalment and the start of the loan. For simple cases only: for example, (i) for a single loan repaid in full after a fixed period in which case $\space \mbox{C} = \frac{A}{(1 + i)^n}$ where $\space n \space$ is the number of years between the advance of the loan and its repayment. (ii) for a loan repaid in a small number of instalments (eg 2,3 or 4). ie working with an equation of the form $\mbox{C} = \frac{A_1}{1 + i} + \frac{A_2}{(1+i)^2} + \frac{A_3}{(1+i)^3}+\frac{A_4}{(1+i)^4}$ In this case you will be expected to either substitute values into the resulting equation for confirmation, or solve for $\space i \space$ using the interval bisection method Applications to financial areas such as: loans credit cards motgages savings Personal Taxation Complex calculations involving multiple rates To include income tax, national insurance and value added tax. Capital gains tax, including the effect of indexation on the taxable gain.

### Indices used to measure key financial information

 When you make a financial decision you need to have measures available that allow you to make sense of data. For example, as you found in section 1 when considering how the value of money varies over time, it is useful, if you are considering borrowing money and investigating which loan you should take that you make sure you have details of the APR (annual percentage rate) for each possibility so that you can compare like with like. In this section you will learn how indices such as the retail price index and the FTSE 100 share index are developed so that you can quickly understand financial information such as how the cost of living is varying or how share prices are increasing or decreasing

### What you need to learn

 Financial aspect Mathematical understanding This includes Understanding of an index as a ratio that describes the relative change in a variable (eg price) compared to a certain base period (eg one specific year). As applied in particular to measures of inflation such as the Retail Price Index (RPI), Consumer Price Index (CPI) and Average Earning Index. The index at any time tells you what percentage the variable is of its respective value at the base time. The value of the index at the base time is 100. Calculations using measures of inflation, including annual changes to pensions and tax allowances. Calculating contributions made by individual items to indices, eg calculating contributions made by the prices of commodities in different shops and regions to a consumer price index. Weighted averages for example, carrying out calculations such as finding the effective costs of a commodity which varies in price between shops. Eg the commodity costs £5 in shop A and £6 in shop B. 0.4 of customers buy the commodity from shop A whereas 0.6 buy it from shop B. The effective cost of the commodity to be used in calculating an index is given by 0.4 × £5 + 0.6 × £6 = £5.60 Understanding the idea of calculating a composite index by combining indices using weighting eg in calculating a price index the index of each commodity multiplied by its weighting is totalled and this sum is divided by the sum of all the commodities weights. Calculating and using different index formulae For the indices below the following apply: is the price of commodity at time is the quantity of commodity at time represents the base period so for example represents the quantity of commodity at the base period Laspeyres index formula (weighted by quantities in the base period) Paasche index formula (weighted by quantities in the calculation period) Fischer index formula The geometric mean of the Laspeyres index formula and the Paasche index formula using indices to understand change understanding ideas of fixed base indices and chain indices fixed base index chain index understanding that for a fixed-base index quantities at time t are compared with the base period ( t = 0 ) understanding that in a chain index comparisons are always made between subsequent points and therefore take account of changes between the start and end points. percentage change calculating average changes using the arithmetic mean eg quarterly change figures from monthly figures understanding and working with basic principles as well as applying to areas of finance such as: FTSE 100 share index
inflation including: consumer price index, retail price index

### Making sense of data over time

 Data you may want to use to make financial decisions is often presented as time-series data, that is a particular measure is given every month, quarter or annually. Sometimes, particularly when the data fluctuates a lot, this may have been processed so that you can identify trends over time. For example, share prices can fluctuate from day to day, as can the FTSE 100 share index. To understand the underlying trend over time it useful to average the data before considering this. Other issues you may need to consider include seasonal variation and cyclical patterns.

#### What you need to learn

 Financial aspect Mathematical understanding This includes Considering data, either primary or secondary (such as financial indices), over time. Interpreting trends. The types of data considered should relate to data or indicators likely to be met when making personal financial decisions: for example, prices of stocks and shares (including 100 share index), interest rates, exchange rates and so on. Time series data in unprocessed form and understanding variability and how this may be random, seasonal or cyclical in nature. Representation graphically and identifying linear trends Inspection of data tables and graphs. Data over different time intervals, for example daily, weekly, quarterly etc. Finding linear equations to model data using gradient and intercept and algebraic substitution. To smooth short-term fluctuations, time-series data can be averaged so that longer-term trends can be identified. For example, indices such as the 100 share index can fluctuate from day-to-day, but over a month or two there may be a distinct trend. Seasonal and cyclical variations may also be more easily detected by such smoothing. Moving averages: for data points $\space p_1, \space p_2, \space ... \space$ the simple moving average, $\space x_m \space$ at interval m takes account of n data points $\displaystyle x_m = \frac{p_m \space + \space p_{m-1} \space + \space p_{m-2}\space + .... \space p_{m(n-1)}}{n}$ Calculating successive values of the simple moving average using $\displaystyle \bar{x}_{m+1} = \bar{x}_m - \frac{P_{m-(n - 1)}}{n} + \frac{P_{m + 1}}{n}$ Interpretation of situations which may include seasonal and cyclical variations. Understanding that the simple weighted average based on a relatively large number of data points can be considered to lag behind the trend of the data The problem of lag in moving averages can be addressed using weighting. The linear weighted moving average (over n intervals) weights the current data with weight n, the previous day with weight (n – 1) and so on. $\displaystyle x_m = \frac{np_m + (n-1) \space p_{m-1} + (n-2) \space p_{m-2}\space + .... \space p_{m - (n-1)}}{ n + (n-1) + (n-2) + .... + 2 + 1 }$ recognising the denominator as a triangular number with sum $\displaystyle \frac{n(n+1)}{2}$

### Tables and diagrams of financial information

 Much basic financial information is presented in as simple a form as possible, for example using indices such as the Retail Price Index and FTSE 100 index which you learned about in section 2. Other information is often quoted in tabular or diagrammatic form giving simplified data and measures so that you can quickly compare like with like. You will have met such ideas in other walks of life; for example, you are probably aware of school performance tables and how attempts have been made to look for measures of "value added" in pupil performance rather than taking raw scores that don't allow for the ability of pupils at entry to different schools. In this section you will learn how to make sense of a range of information presented in tables and diagram relating to personal finance. For example, you will learn to interpret information about how an investment might perform or how to compare financial products.It is not the intention that you should learn specific financial measures other than those highlighted in previous sections but that you should be able to work with and interpret financial information presented in tables and diagrams when basic terms are defined.

#### You need to learn:

• to be able to extract and understand data from tables and diagrams
• to work with the data carrying out calculations using basic mathematics, such as calculating with percentages
• to interpret the original data and results of your calculations in terms of the financial situation