12-11-2017 MBA – 401:5 Accounting for Decision Makers (MBA 401) Lecture Title No. 5 Time Value of Money Document map: No. MBA401/5/PPT Version: #1 Status: Draft Owner: Extralearn

Understanding Time Value? 12-11-2017 Understanding Time Value? We say that money has a time value because it can be invested with the expectation of earning a positive rate of return In other words, “a £ received today is worth more than a £ to be received tomorrow” That is because today’s £ can be invested so that we have more than one £ tomorrow

Terminologies of Time Value 12-11-2017 Terminologies of Time Value Present Value - An amount of money today, or the current value of a future amount Future Value - An amount of money on a future time Period - A length of time or duration (year, month, week, day, or even hour) Interest Rate - The compensation paid to a lender (or investor) for the use of funds expressed as a percentage for a period (normally stated as an annual rate)

Abbreviations PV: Present value FV: Future value 12-11-2017 Abbreviations PV: Present value FV: Future value Pmt: Per period payment amount N: Either the total number of cash or The number of a specific period i: The interest rate per period

12-11-2017 Timelines A timeline is a graphical device used to clarify the timing of the cash flows for an investment Each tick represents one time period PV FV 1 2 3 4 5 Today

Calculating the Future Value 12-11-2017 Calculating the Future Value Suppose that you have an extra £100 today that you wish to invest for one year. If you can earn 10% per year on your investment, how much will you have in one year? -100 ? 1 2 3 4 5

Calculating the Future Value 12-11-2017 Calculating the Future Value Suppose that at the end of year 1 you decide to extend the investment for a second year. How much will you have accumulated at the end of year 2? -110 ? 1 2 3 4 5

Generalising Future Value 12-11-2017 Generalising Future Value Recognizing the pattern that is developing, we can generalize the future value calculations as follows: If you extended the investment for a third year, you would have:

12-11-2017 Compound Interest Note from the example that the future value is increasing at an increasing rate In other words, the amount of interest earned each year is increasing Year 1: £ 10 Year 2: £ 11 Year 3: £ 12.10 The reason for the increase is that each year you are earning interest on the interest that was earned in previous years in addition to the interest on the original principle amount

Compound Interest - Graph 12-11-2017 Compound Interest - Graph

12-11-2017 Magic Compounding On Nov. 25, 1626 Peter Minuit, a Dutchman, reportedly purchased Manhattan from the Indians for £24 worth of beads and other trinkets. Was this a good deal for the Indians? This happened about 371 years ago, so if they could earn 5% per year, then in 1997 they would have: If they could have earned 10% per year, in 1997, they would have: That’s about £ 54,563 Trillion

Present Value Calculations 12-11-2017 Present Value Calculations So far, we have seen how to calculate the future value of an investment But we can turn this around to find the amount that needs to be invested to achieve some desired future value:

Present Value - Example 12-11-2017 Present Value - Example Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about £100,000 on her 18th birthday to pay for four years of college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal?

12-11-2017 Annuities An annuity is a series of nominally equal payments equally spaced in time Annuities are very common: Rent Mortgage payments Car payment Pension income The timeline shows an example of a 5-year, £100 annuity 100 100 100 100 100 1 2 3 4 5

Value Additivity How do we find the value (PV or FV) of an annuity? 12-11-2017 Value Additivity How do we find the value (PV or FV) of an annuity? First, you must understand the principle of value additivity: The value of any stream of cash flows is equal to the sum of the values of the components In other words, if we can move the cash flows to the same time period we can simply add them all together to get the total value

12-11-2017 PV of Annuity We can use the principle of value additivity to find the present value of an annuity, by simply summing the present values of each of the components:

12-11-2017 PV of Annuity Using the example, and assuming a discount rate of 10% per year, we find that the present value is: 1 2 3 4 5 100 62.09 68.30 75.13 82.64 90.91 379.08

12-11-2017 PV of Annuity Actually, there is no need to take the present value of each cash flow separately We can use a closed-form of the PVA equation instead:

12-11-2017 PV of Annuity We can use this equation to find the present value of our example annuity as follows: This equation works for all regular annuities, regardless of the number of payments

12-11-2017 FV of Annuity We can also use the principle of value additivity to find the future value of an annuity, by simply summing the future values of each of the components:

12-11-2017 FV of Annuity Using the example, and assuming a discount rate of 10% per year, we find that the future value is: } 146.41 133.10 121.00 = 610.51 at year 5 110.00 100 100 100 100 100 1 2 3 4 5

12-11-2017 FV of Annuity Just as we did for the PVA equation, we could instead use a closed-form of the FVA equation: This equation works for all regular annuities, regardless of the number of payments

12-11-2017 FV of Annuity We can use this equation to find the future value of the example annuity:

12-11-2017 Annuities Due Thus far, the annuities that we have looked at begin their payments at the end of period 1; these are referred to as regular annuities A annuity due is the same as a regular annuity, except that its cash flows occur at the beginning of the period rather than at the end 5-period Annuity Due 100 100 100 100 100 5-period Regular Annuity 100 100 100 100 100 1 2 3 4 5

12-11-2017 PV of Annuity Due We can find the present value of an annuity due in the same way as we did for a regular annuity, with one exception Note from the timeline that, if we ignore the first cash flow, the annuity due looks just like a four-period regular annuity Therefore, we can value an annuity due with:

12-11-2017 PV of Annuity Due Therefore, the present value of our example annuity due is: Note that this is higher than the PV of the, otherwise equivalent, regular annuity

12-11-2017 FV of Annuity Due To calculate the FV of an annuity due, we can treat it as regular annuity, and then take it one more period forward: Pmt Pmt Pmt Pmt Pmt 1 2 3 4 5

FV of Annuity Due The future value of our example annuity is: 12-11-2017 FV of Annuity Due The future value of our example annuity is: Note that this is higher than the future value of the, otherwise equivalent, regular annuity

12-11-2017 Deferred Annuities A deferred annuity is the same as any other annuity, except that its payments do not begin until some later period The timeline shows a five-period deferred annuity 100 100 100 100 100 1 2 3 4 5 6 7

12-11-2017 PV of Deferred Annuity We can find the present value of a deferred annuity in the same way as any other annuity, with an extra step required Before we can do this however, there is an important rule to understand: When using the PVA equation, the resulting PV is always one period before the first payment occurs

12-11-2017 PV of Deferred Annuity To find the PV of a deferred annuity, we first find use the PVA equation, and then discount that result back to period 0 Here we are using a 10% discount rate PV2 = 379.08 PV0 = 313.29 100 100 100 100 100 1 2 3 4 5 6 7

12-11-2017 PV of Deferred Annuity Step 1: Step 2:

12-11-2017 FV of Deferred Annuity The future value of a deferred annuity is calculated in exactly the same way as any other annuity There are no extra steps at all

12-11-2017 Un-even Cash flows Very often an investment offers a stream of cash flows which are not either a lump sum or an annuity We can find the present or future value of such a stream by using the principle of value additivity

An Example Un-even Cash flows 12-11-2017 Un-even Cash flows An Example Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment? 100 200 300 1 2 3 4 5

Another Example Un-even Cash flows 12-11-2017 Un-even Cash flows Another Example Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year? 300 500 700 1 2 3 4 5

Non-annual Compounding 12-11-2017 Non-annual Compounding So far we have assumed that the time period is equal to a year However, there is no reason that a time period can’t be any other length of time We could assume that interest is earned semi-annually, quarterly, monthly, daily, or any other length of time The only change that must be made is to make sure that the rate of interest is adjusted to the period length

Non-annual Compounding 12-11-2017 Non-annual Compounding Suppose that you have £1,000 available for investment. After investigating the local banks, you have compiled the following table for comparison. In which bank should you deposit your funds?

Non-annual Compounding 12-11-2017 Non-annual Compounding To solve this problem, you need to determine which bank will pay you the most interest In other words, at which bank will you have the highest future value? To find out, let’s change our basic FV equation slightly: In this version of the equation ‘m’ is the number of compounding periods per year

Non-annual Compounding 12-11-2017 Non-annual Compounding We can find the FV for each bank as follows: First National Bank: Second National Bank: Third National Bank: Obviously, one should choose the Third National Bank

Continuous Compounding 12-11-2017 Continuous Compounding There is no reason why we need to stop increasing the compounding frequency at daily We could compound every hour, minute, or second We can also compound every instant (i.e., continuously): Here, F is the future value, P is the present value, r is the annual rate of interest, t is the total number of years, and e is a constant equal to about 2.718

Continuous Compounding 12-11-2017 Continuous Compounding Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. What is the future value of your £1,000 investment? This is even better than daily compounding The basic rule of compounding is: The more frequently interest is compounded, the higher the future value

Continuous Compounding 12-11-2017 Continuous Compounding Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. If you plan to leave the money in the account for 5 years, what is the future value of your £1,000 investment?