Learning Material Sample
Investment principles and risk5. The principles of the time value of moneyLearning outcome 4 Apply the principles of the time value of money
We start this chapter with an audiovisual presentation to help explain various calculations used in financial planni...
Shortened demo course. See details at foot of page. ...a future targetThe present value of capital needed now to achieve a target in the future How to discount
A sum of money is likely to be worth more at a point in the future than it is now because, if invested, it can accumulate interest over the given timeframe.
If interest is added to the sum of money (i.e. reinvested), when the next interest payment is made, it will be based on a percentage of the original sum invested plus the interest already accumulated. This is known as 'compounding interest'. It is important to understand the basic definitions used throughout the time value of money calculations before looking at the calculations themselves. ... Shortened demo course. See details at foot of page. ...ormulae (manually)3. Use a financial calculator (Hewlett Packard 10b recommended) Although we will work through this section showing how calculations can be carried out using methods 2 and 3, you should be aware that compound interest tables exist. If these tables have been input into a Microsoft Excel spreadsheet, for example, you can use the formulae functions and will have the ability to vary data input relevant to each scenario. What are the four basic elements of the time value of money calculations? Answer : Purchase course for answer The basic formula for the compound interest accumulation of a capital sum is:
FV = PV (1 + r) n Where: FV is the future value PV is the present value r is rate of interest n is time period This and the other variations of the formula which we will use throughout this chapter can be calculated manually using a basic calculator, however the order in which the operations (each step of the calculation) are carried out is important and if these are not completed in the correct order you will end up with an incorrect result. A simple way of remembering the order of operations is by using a mnemonic, BOMDAS . B – Brackets first O – Operations also known as indices which are powers (e.g. 10 2 ) and roots (e.g. √10) MD – Multiply and divide before you add and subtract working from left to right AS – Finally add and subtract working from left to right Some versions of this mnemonic use “I” instead of “O” (BIMDAS); in this version 'operations' has simply been substituted for the word 'indices'. Using a worked example, the basic compoun... Shortened demo course. See details at foot of page. ...rs (n = 3) . To write this out long-hand the formula would be:FV = £15,000(1 + 0.035) x (1 + 0.035) x (1 + 0.035) FV = £16,631 (rounded down) There is a short cut that can help speed up this process using a basic calculator: Stages Instructions Worked Example Stage 1 Type in the value of 1 + r Enter 1.035 Stage 2 Press multiply twice (x) Press x twice Stage 3 Press equal. This then gives the value of 1 + r 2 , the equivalent to 2 time periods Press equal. Result should be 1.071225 Stage 4 Press equal for each additional time period required Press equal once more for year 3. Result should be 1.108717875 Stage 5 Press multiply and enter the amount of the capital invested (PV) Multiply by 15,000 Stage 6 Press equal to obtain your result Press equal. The result should be 16,630.76812 The calculation described above would be input as follows: 1.035 x = x 15,000 = 16,630.76812 Calculate the accumulated value of a lump sum of £12,000, invested for 5 years at an interest rate of 5% Answer : Purchase course for answer The Hewlett Packard 10b calculator has the following functions that are used for compound interest calculations. Not all of the following keys will be needed with each calculation and we will guide you through which function keys to press for each type of mathematical sum we use:
Hewlett Packard 10b: Function Keys for Compound Interest Calculations Function Key Description N Number of annual compounding periods 1/YR Annual nominal rate of interest PV Present value, or initial investment at the beginning of the first period PMT Amount of payments, which can be at the beginning or end of each period FV Future value at the end of the last period Shift + P/YR Stores the number of compounding periods per year ... Shortened demo course. See details at foot of page. ... the algebraic formula.In order to raise a number to a power (e.g. 10 3 is 10 to the power of 3), most calculators will have an x y or ^ button and you may need to use the shift key to make these functions work. For example, inputting “1.05x y 5”, or “1.05^5” should give an answer of 1.276281563. Your calculator may give the answer to a greater or lesser number of decimal places, but your results should agree with those for at least 5 or 6 digits from the left. It is worth taking some time to get used to how the functions on your calculator work and to check your method against worked examples, in order to ensure that you are using the functions on your calculator correctly, and that you end up with the correct answer.
There are four components to the compound interest formula, and as with most algebraic equations, any three of the components can be used to calculate the fourth. Basically you enter all the known information into the formula and then balance each side of the equation until you can isolate the fourth element and therefore find your solution.
How to find the present value (PV) This is the amount that needs to be invested now to achieve a target amo... Shortened demo course. See details at foot of page. ...bsp; = 20,000/ 1.07 10= 20,000/ 1.967 = 10,166.98 The lump sum that John would need to invest now to achieve a target of £20,000 in 10 years' time, assuming a rate of return of 7% per annum, is therefore £10,167. Calculate the lump sum required now to produce an amount of £15,000 in 8 years' time, assuming an interest rate of 5% per annum Answer : Purchase course for answer This calculation can be used to ascertain the rate of return required to achieve a target lump sum in the future, where you know the lump sum available now and the length of time in which to achieve it. It could also be used to calculate the rate of return a client has achieved on an investment over a given period of time.
Again we take the four elements of the compound interest formula and rebalance the equation to solve the four th element when the other three elements are known. Example David invested £20,000 into a unit trust 5 years ago and the fund is now worth £35,000. What compound rate of return has he made on his investment? F... Shortened demo course. See details at foot of page. ...dic;y symbol on your calculator. If your calculator does not have this function it can be written as (1.75) 1/5 , using the x y function on the calculator.1.1184 = 1 + r r = 1.1184 – 1 r = 0.1184 As the answer required is an interest rate or rate of return, the answer should be expressed as a percentage to two decimal places. In order to do this, multiply the answer by 100. The compound interest rate achieved by David’s investment is 11.84%. An investor needs £30,000 in 6 years' time and has £20,000 to invest. What rate of return will they need to attain their target? Answer : Purchase course for answer All the calculations we have used so far have assumed that interest is paid annually in arrears; in other words, that it is paid once at the end of the time period. Where interest is paid at the start of a period or upfront, the compound interest formula is amended as follows:
FV = PV(1 + r) n+1 We add 1 onto the number of time periods as the investment will effecively receive an extra year’s interest. Example If we take the example used previously where we had £15,000 invested for 3 years at an interest rate of 3.5%, but this time the interest is paid in advance, the calculation would be: PV = £15,000, r = 0.035 and n = 3 FV = 15,000(1 + 0.035) 3+1 = 15,000 x 1.035 4 = 15,000 x 1.1475 = 17,212.845 The lump sum value at the end of the 3-year term would be £17,212.85 when the interest in paid in advance, compared to £16,630.77 wh... Shortened demo course. See details at foot of page. ...nterest will be addedThe interest accumulation factor (r) reduces proportionately to the increased frequency of compounding during the year. In the example above (where the nominal annual rate is 10%), when this is paid quarterly (r) becomes 0.025 (2.5%). This can be expressed as 1 + r/n The multiplier is multiplied by itself once for each conversion period. This can be expressed as (1 + r/n ) n Where interest is paid more frequently than annually, instead of using the nominal rate of interest, you should calculate the effective rate of interest (EAR). The formula for this is as follows: EAR = (1 + r/n) n – 1 The effective rate of interest is also known as the 'annual percentage rate' (APR), which is generally used for loans, or the 'annual equivalent rate' (AER), used for deposits. Calculate the APR for a loan where the nominal rate of interest is 28.95% charged on a monthly basis. Answer : Purchase course for answer You may need to change the compounding rate during the accumulation period. One example of this is where a bonus rate is paid for an initial period and a lower rate is payable for the remainder of the term. The formula would be expressed as follows:
FV = PV (1 + r1) n1 x (1 + r2) n2 etc Where: FV is the future value PV is the present value r1 is rate of interest paid in the first period n1 is the number of time periods the initial rate is paid... Shortened demo course. See details at foot of page. ... is as follows:PV = £25,000, r1 = 0.05 and n1 = 2, r2 = 0.065 and n2 = 3 FV = 25,000(1 + 0.05) 2 x (1 + 0.065) 3 = 25,000 x 1.05 2 x 1.065 3 = 25,000 x 1.1025 x 1.207949625 = 33,294.11154 The value at the end of the term would be £33,294.11. Calculate the future value of an investment of £20,000 which will pay interest at 4.5% for 3 years, followed by 7% for 3 years Answer : Purchase course for answer Here we will consider how to calculate:
The amount accrued from a series of regular payments The present value of a series of payments Note – these calculations are not tested in R02 but are included for information. Setting aside a regular amount each year to accumulate a lump sum over a period of time is the equivalent to the sum of a series of single payments that are each invested for a different set period. Example Brenda wants to invest £200 per annum for 10 years, the first payment being at the end of year 1. Assuming an interest rate of 6.25% per annum, how much will she have accumulated? This is considered here in tabular form: End of year Amount Value at end of year 10 1 200 200 (1.0625) 9 345.14 2 200 200 (1.0625) 8 324.83 3 200 200 (1.0625) 7 &nb... Shortened demo course. See details at foot of page. ...bsp;Description 12 Shift + P/YR = 12.00 Divides year into monthly payment periods 50,000 FV = 50,000 Stores future value 8 Shift + xP/R = 96.00 Stores number of payments (8 years is 96 months 0 PV = 0.00 Enter zero as there is no initial lump sum payment 8 Shift + EFF% = 8.00 Enters effective interest rate Shift + NOM% = 7.72 Converts to nominal interest rate Shift + BEG/END = BEGIN First monthly contribution is assumed to be immediate. If at the end of the first month, ensure that begin is not shown PMT = - 375.64 Answer shows a negative sign because it represents a monthly outflow of cash The formula used above can be amended to calculate the sum of money needed to make regular payments plus interest over a fixed term and a fixed rate of interest. This is called an annuity ( A ), and the general formula is:
A = P(1-(1+r) -n /r) The main difference from the previous formula is the order of the symbols and that 1+r is now being raised to the power of –n . The same logic can be applied for other frequencies of payment such as half yearly, quarterly and daily. Calculate the amount that can be withdrawn per month over a period of 10 years, where the starting balance is £50,000 and the interest rate is 3.5% per annum? Answer : Purchase course for answer As we have seen in previous chapters, inflation can have a huge impact on the future purchasing power of money. Real returns are adjusted to account for inflation, and can be important not only when looking at the impact of inflation on money in the future, but also when comparing returns over different time periods.
The formula that links real returns to nominal returns is: ... Shortened demo course. See details at foot of page. ...portfolio would be:3.5% + 5% = 8.5% An approximate nominal return of 8.5% is required to achieve the target return. You have calculated that Maria needs to achieve a real return of 5.5% per annum in order to achieve an investment target. Assuming that inflation is running at 3%, calculate the nominal rate of interest required. Answer : Purchase course for answer Your results and the estimated s... Shortened demo course. See details at foot of page. ...ate.Estimated study time 2.1 hours |
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