Time Value of Money by Binam Ghimire

Learning Objectives Concept of TVM Represent the cash flows occurred in different time periods using cash flow time line Calculate the present value and future value of given streams of cash flows with and without using table Identify the impact of time period and required rate of return on present value and future value Prepare amortised schedule for amortised term loan Compare interest rates quoted over different time intervals Understand the real and nominal cash flows

Source: Total Access http://www.totalaccess.co.uk/Case_Studies/big_ben

Warm Up: The School Maths Adding a percent to an amount A computer costs £ 420 + VAT 15%. What is the cost after VAT is added? Reverse percentage The price of a computer is £230 after VAT of 15%. What is the price before VAT is added?

Time value of Money: Concept and Significance Theory behind TVM: “One pound today is worth more than one pound tomorrow” Deals with values of cash flows occurred at different points in time Every sum of money received now has a value for reinvestment

Time value of Money: Concept and Significance Why is APR a legal requirement? 36.5% p.a.* buy today pay after 12 months * Condition apply see page 115 for detail On page 115: payable 0.1 % interest per day Image source: google

Time value of Money: Concept and Significance Having Cash NOW is valuable Inflation People’s preference to current consumption TVM Usage: Firms, Households, Banks, Insurance Cos. TVM in Corporate Finance: Economic Welfare of Shareholders Investment Decisions Financing Decisions

Cash Flow Time Line Cash Flow Time Line is an important tool used to understand the timing of cash flows. It is a graphical presentation of cash flows occurring at different points of time, and is helpful for analysing the time value of cash flows

Cash Flow Time Line The corresponding cash flows are placed below the scale as shown in the following time line of cash flows: The time line of cash flow is also used to denote the interest rate that each cash flow earns

Future Value Future Value (FV) is the sum of investment amount and interest earned on the investment Interest may be earned simple and compound Accordingly FV can be calculated using simple and compound interest rate methods

Future Value: Simple Interest Interest earned only on the original investment is Simple Interest. For example, a savings account with a bank in which bank posts the interest every year. The interest for principle £P deposited at r % interest p.a. for t number of years will equal to I (where I = p x n x r ÷ 100). So the FV of an investment will be P + I We aim to maximise the benefit from any monetary transaction. So in real world, interest is paid (savings) or charged (loan) more than once during the tenure of the savings or loan. As such the Simple Interest does not exist

Future Value : Compound Interest Compound Interest is interest earned on principle plus interest from the previous period FV of £ 100,000 investment for one year, rate of interest: 12% and when interest is 1) simple interest 2) compounded monthly and daily are shown in figure next

Future Value : Compound Interest Effect

Future Value : Compound Interest different intervals Hence, with Compound interest, FV grows when the frequency of interest payment is higher. (i.e. total amount of interest is growing every next period). Note that when interest is per annum and to be calculated for mth times within a year the formula changes to: And if it is for more than one year, the formula is Where, n=no. of years Effective interest rate formula £100 at 12% p.a. at the end = 112 simple £100 at 12% p.a. half yearly = £100 x 0.12/2 = 6 + 100 = 106 £106 x 0.12/2 = 112.36 Devise a formula £100 x (1.06)^2 £112.36 So the effective interest rate is 12.36% Now let us make a formula for this 100 x (1+ APR) = 100 x (1+0.12/2)^2 APR = (1 + r/m)^m -1

Compound Interest different intervals has offered 5 % interest charged half yearly has offered 4.5% interest charged every 3 months The name of the banks are used only to illustrate the APR and the numbers are imaginary. Logo source: website of respective banks. APR = (1 + r/m)^m -1 You want to borrow money Which bank will you prefer and why?

Future Value : Compound Interest Table Below FV of £ 1 at the end of various numbers of years have been calculated for different rates of interest: The table is known as FV Compound Interest table [(1+r)t]

Future Value : Graphical View FV of a sum of money has positive relation with the time period and interest rate

Future Value: FV, Time Period and Interest rate FV increases with rate of interest and time The higher the interest rate higher will be the FV More the number of years the more will be the FV Higher the no. of frequency of payment (within the year) higher will be the FV

Future Value : Compound Interest in Excel In Microsoft Excel, FV function wizard can be used to calculate FV Check the relationship between FV and r and t by making line chart in Microsoft Excel for r=5, 10 & 15 percent for t= 0,2,4,6,8,10,12 & 14 years

Present Value Present value (PV) is today’s value of a future cash flow FV is only available in the future Can we get the FV now? i.e. can we convert the FV into PV? You may if you sacrifice 1) the interest that you could have earned had you invested the money and 2) adjust for the time period that you don’t want to wait. Hence, The sacrifice is reflected in the formula above. When t and r get bigger PV gets smaller

Present Value: Discount Rate You bought a television. The payment term is single payment of £ 2,000 to be made after 2 years. If you can earn 8% on your money how much money should you set aside today in order to make final payment when due in 2 years? Note that to calculate PV, we discounted FV at the interest rate r. The calculation is therefore termed a discounted cash flow. The interest rate is known as Discount rate

Present Value: Discount Factor The PV formula may be converted as follows The expression 1/(1+r)t basically measures the PV of £1 received in year t. The expression is known as Discount Factor. Using the PV of £ 1, a table can be constructed for FV to be received after t years at different discount rates

Present Value : Discount Factor Table Below PV of £ 1 at the end of various numbers of years have been calculated for different rates of interest: The table is known as PV Discount Factor table [1/(1+r)t]

Present Value: Graphical View The PV of a sum of money has inverse relation with the time period and interest rate

Present Value: PV, Time Period and Interest rate PV increases when rate of interest falls. The lower the interest rate higher will be the PV Similarly, PV increases when the time period decreases In the example of TV purchase above, note that £1,714.68 invested for 2 years at 8% will prove just enough to buy your computer

Present Value: PV, Time Period and Interest rate The longer the time period available for payment the less you need to invest today (i.e. lower PV value). For example, if the payment for TV was only required in the third year, the PV would be: The relationship between PV and Interest Rate is also the same i.e. higher interest rate lower PV

Multiple Cash Flow: FV and PV for uneven Cash Flows Most real word investments will involve many cash flows over time. This is also known as Stream of cash flows. Calculation of FV and PV of a stream of cash flows is more important and common in finance To calculate the FV of a stream of uneven cash flows, we may calculate what each cash flow is worth at that future date, and then add up the values To find the PV of a stream of uneven cash flows, we may calculate what each FV is worth today and then add up the values

Multiple Cash Flow: Future Value of Uneven Cash Flow A security provides the following Cash Flows If the interest rate is 10% the FV will be The third column above is the factor from FV of £ 1 table

Multiple Cash Flow: Present Value of Uneven Cash Flow The PV for the same Cash Flow above assuming same rate as discount rate will then be The third column above is the factor from PV of £ 1 table

Multiple Cash Flow: FV and PV for even Cash Flows In finance, it is more common to find a stream of EQUAL cash flows at same time intervals. For example, a home mortgage or a car loan might require to make equal monthly payments for the life of the loan. Any such sequence of equally spaced, level cash flows is called an Annuity

Multiple Cash Flow: Annuity and Annuity Due Annuity may be Ordinary Annuity and Annuity Due In case of an ordinary annuity, each equal payment is made at the end of each interval of time throughout the period In case of Annuity Due, equal payments are made at the beginning of each interval throughout the period

Multiple Cash Flow: Annuity and Annuity Due For example, if an individual promises to pay £ 1,000 at the end of each of three years for amortisation of a loan, then it is called an ordinary annuity. If it were the annuity due, each payment would be made at the beginning of each of the three years

Multiple Cash Flow: Future Value of an Annuity Future Value for the Ordinary Annuity (FVA) will be

Multiple Cash Flow: Future Value of an Annuity Formula, where C is the Cash payment from annuity If payment value is £ 1, it would be: The table showing future value of £ 1 for various years at different interest rates is called FV Annuity table

Multiple Cash Flow: Future Value of an Annuity Due Future value for the Annuity Due (FVAD) will be Hence,

Multiple Cash Flow: Perpetuity, Delayed Annuity &Annuity

Multiple Cash Flow: Perpetuity If the payment stream lasts forever, it is Perpetuity Example: Consol For a perpetual stream of £C payment every year the PV = C/r So a perpetual stream of £1 payment every year the PV = 1/r

Multiple Cash Flow: Delayed Perpetuity Delayed perpetuity is one in which the payment (C) only starts after some time t PV of a delayed perpetuity for £ 1 payment starting after time t will be

Multiple Cash Flow: Present Value of an Annuity Annuity is basically a difference between an Annuity (immediate) and a delayed perpetuity PV of an Annuity (t year) will therefore be PV of immediate Perpetuity – PV of Delayed Perpetuity starting after t year This is also called t-year annuity factor. The PV table constructed for various time t and interest rate r for payment £1 each year is called PV Annuity factor table

Multiple Cash Flow: PV of an Annuity and Annuity Due For Payment £ C every year, PV of t year annuity is therefore Payment x annuity factor or Or:

Multiple Cash Flow: PV of an Annuity and Annuity Due For Payment £ C every year, PV of t year annuity is therefore Payment x annuity factor or Present value of Annuity Due simply requires the use of the formula: PV of ordinary annuity x (1+r)

Multiple Cash Flow: PV of an Annuity and Annuity Due

Multiple Cash Flow: Amortised Loans Loan that is to be repaid in equal periodic installments including both principal and interest is known as Amortised Loan Let us suppose a loan of £ 10,000 is to be repaid in four equal installments including principal and 10 percent interest per annum The lender needs to set the payments so that a present value of £ 10,000 is received Present Value = Mortgage Payment x t year annuity Factor

Multiple Cash Flow: Amortised Loans Mortgage Payment = The loan amortisation schedule is shown next

Multiple Cash Flow: Amortised Loan Schedule

Cash Flow and Inflation Note that cash flows are affected by the level of inflation in the country Generally, the rate of interest quoted in the market are nominal interest rate which does not take into consideration the effect of price changes. The real interest rate can be calculated using the formula:

Cash Flow and Inflation: Example What is the real interest rate when you deposit £ 1,000 in a bank at interest rate of 5%. Suppose that the Inflation is 5% as well What will it be if interest rate is 6% and inflation is only 2 %? Discounting real payment by real interest rate and Nominal Payment by Nominal interest rate will always give the same answer. We must not mix up real and nominal

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