Time_Value_of_Money

Time value of money Which would you prefer -- $1000 today or $1000 in 4 years' Certainly, $10,00 today. Your response motivated by the ‘Time value of money’ You already recognized that there is TIME VALUE TO MONEY!! Significance of TIME Why is TIME such an important element in your decision' TIME allows you the opportunity to postpone consumption and earn INTEREST Interest Interest is a return on a deposit. Interest paid on the principal borrowed is simple interest. Interest paid on any previous interest, as well as on the principal borrowed is compound interest. Simple Interest Assume that you deposit $10,000 in an account earning 5% simple interest for 4 years. The accumulated interest at the end of the 2nd year can be calculated using the formula: SI = P0 (i) (n) SI: Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods SI = $10,000 (0.05) (4) = $2,000 The simple interest on the deposit of $10,000 for 4 years @ 5% is $2000. Compound Interest How is compound interest different from simple interest' Shierly Winters deposits $10,000 @ compounded interest rate of 5% per annum for 4 years. She wants to know how much it will becomes after 4 years. She earned $500 interest on your $10000 deposit over the first year. This is the same interest you would earn under simple interest. In the second year the interest would be FV1 = P0 (1+i)1 = $10,000 (1.05) = $10,500 FV2 = FV1 (1+i)1 = P0 (1+i)(1+i) = $10,000(1.05)(1.05) = P0 (1+i)2 = $10,000(1.05)2 = $11,025 In the second year she earned an interest of Rs.525 as interest. The EXTRA $25 is the compound interest over simple interest. Accordingly, the future value at the end of the 4th year would be = P0(1+i)(1+i)(1+i)(1+i) = $10,000(1.05)(1.05)(1.05)(1.05) This is equivalent to P0 (1+i)4 = $12,155 = $10,000 (1.05)4 = $12,155 The formula is: FV = PV(1+i)n Her deposit of $10000 grows to $12155 at the end of the fourth year. In the figure notice the difference in the simple and the compound interest depicted by the first two bars for the 5% rate of interest. The “Rule-of-72” How long does it take to double $10,000 at a compound rate of 12% per year (approx.)' Approx. Years to Double = 72 / i% 72 / 12% = 6 Years [Actual Time is 6.12 Years] Time value I give you 1000 dollars. You deposit in a bank. Bank will give you 5% interest per annum. After 2 years, it becomes $1102.50. The Present Value is $ 1000 and Future Value will be $1102.50. The difference between the present value and the future value is ‘time value of money’. Money loses value with the passage of time due to factors such as inflation. Inflation is a phenomenon, characterised by rising prices in the economy. Inflation is principally caused by the mismatch between generation of incomes and production of goods and services in the economy. The time value of money has the following possibilities: • Present value of a single amount • Future values of a single amount • Future Value of an Uneven Cash Flow • Present value of an annuity • Future value of an annuity Present value of single amount Shierly Winters wants to know how large of a deposit to make so that the money will grow to $25,000 in 4 years at a interest rate of 7%. The general formula for calculating Present Value is : where PV is the present value FVn is the future value of the investment in the year ‘n’ i is the rate of interest and n is the number of years = 25,500 / (1+0.07) 4 = $19,072 Shierly Winters need to deposit $19072 now to realize $25000 after 4 years. The $19072 is the present value of the future sum of $25000. Future value of a single amount: Now Shierly Winters deposits $10000 for a period of 6 years @ 8% per annum. How much she get on maturity of the deposit' Where, FVn is the future value in the year ‘n’ PV is the present value i is the rate of interest and n is the number of years FV6 = $ 10000 (1 + 0.08)6 = $ 15869 Shierly Winter’s deposit will be worth $15869 at the end of the 6th year. Present Value of Cash Flow Shierly Winters calculates that she will receive a set of cash flows for the next 6 years. She wants to know what it is worth now' She can proved she knows the rate of interest. Assuming a rate of 8% the worth can be calculated. Year Cash Flow 1 2500 2 1500 3 2500 4 1800 5 2000 6 2200 The future earnings are converted to their respective present values by discounting them by the discount factor. The discounting is done as follows: = $9,656. The present worth of her earnings is $9,656. Year Cash Flow Discount Factor Present Value (FV) (df) (FV x df) 1/(1+I)n 1 2500 0.93 2314.81 2 1500 0.86 1286.01 3 2500 0.79 1984.58 4 1800 0.74 1323.05 5 2000 0.68 1361.17 6 2200 0.63 1386.37 Present Value 9656.00 Future Value of Cash Flow Shierly Winters plans to deposits her yearly savings for the next 5 years in a bank. She wants to know what her investment will be worth at the end of 5 years' Year Savings $ 1 1000 2 1500 3 2000 4 2200 5 2500 To know the future value, each year’s savings has to be converted to their respective future value. For example $1000 of 1st year has to be compounded to the 5th year. i.e. = 1000 * (1 + 0.08)^5 = 1000 * (1.47) = 1470 Year Cash Flow Compounding Factor Future Value (PV) (PV x df) (1+I)n 1 1000 1.469 1469.3 2 1500 1.360 2040.7 3 2000 1.260 2519.4 4 2200 1.166 2566.1 5 2500 1.080 2700.0 Future Value 11295.6 Therefore, after 5 years, she will have $11295.6. That is the future value of her cash flow. Effective Interest Rate Very often people are carried away with the stated interest rate when they avail a loan. In fact a loan carrying 12% interest for a housing loan may be cheaper than a loan carrying an interest of 11.25%. It depends on the frequency of compounding. Therefore, to assess the actual cost of a loan, one should calculate the effective interest rate. The effective interest rate is the actual interest rate that the borrower pays for his loan or receives for his deposit. The effective interest rate is different from the nominal interest rate. The nominal interest rate is specified for the loan. Shierly Winters has $5,000 to invest for 4 Years at an annual interest rate of 8%. The table shows the effective interest for different frequencies of compounding for a nominal interest of 8%. Year Interest Amount Effective Rate of Interest (%) Annual 400.00 8.00 Biannual 408.00 8.16 Quarterly 412.16 8.24 Monthly 415.00 8.30 Daily 416.39 8.33 The effective interest rate is calculated using the formula given below: Where, i is the rate of interest m is the frequency of compounding during a year Shierly winters will receive $5,400 if the interest is compounded annually and $5,412.16 if compounded quarterly. The effective interest for quarterly compounding of 8.24% is arrived at using the formula: Annuity Annuity is a series of equal payments or receipts occurring over a specified number of equidistant periods. Car Loan Payments, Insurance Premiums, Mortgage Payments and Retirement Savings are some of the examples of annuity. Present value of an annuity Shierly after reviewing her budget decides that she can pay $ 1200 a month for a period of 3 years towards purchase of a new car. Car loans are available from M/s. Auto Mart which offers car loans at 8% interest. Shierly would like to know how much she can borrow so that she can decides which car to buy. Problems such as these, involve deciding the present value of the future annuity payments, which in Shirley’s case is $1200 per month. The formula for calculating the present value of an annuity is: = $37,110.20 Shierly can avail a loan of $37,110.10. Now, she can decide which car to buy. Shierly by paying $1200 per month is amortizing a loan of $37,110.10 over the period of the three years. Future value of an annuity Shierly is saving at the rate of $500 per month for a period of 15 years for her retirement. The interest on the deposit is 10%. How much will she receive on maturity of her deposit' Since Shierly is depositing money every month, the first instalment earns interest for 180 months, second instalment earns interest for 179 months and so on. To know the future value of her deposit the formula is: Her annuity is the monthly contribution. The rate of interest ‘r’ which is usually expressed per annum, in this case 10 per cent has to be converted to each period in to which the year is divided. In this case it is 12. Therefore, the monthly interest would be 0.10/12 = 0.00833 per cent. This rate is used in the formula above. = 500 x ((1 + 0.00833) 180 – 1) / 0.00833 = 500 x 414.32 = $ 207159 The maturity value of Shirley’s deposit will be $207159. Suppose the interest is calculated on the deposit once in six months, rather than every month, then appropriate changes have to be made in the interest rate, the annuity amount and the number of periods of the deposit. Since the year is divided into two periods, the interest rate has to be divided by two and the period has to be multiplied by two, and the deposits has to be aggregated for six months. Thus, ‘r’ will be 5 per cent, ‘n’ will be 30 and annuity will be $3000. = 3000 x ((1 + 0.05) 30 – 1) / 0.05 = 3000 x 66.44 = $ 199316.54 Capital Budgeting Shirley Winters decides to start a Boutique in her neighborhood as there isn’t one close by and therefore she thinks that it has a good chance of succeeding. With the help of her friend who has experience in running a boutique she estimated the cost of setting it up and finds it requires an investment of $ 18000 which she could ill afford. While discussing her plans with a friend, she was told if she could establish the financial soundness of the investment, she has a good chance of getting a loan from a financial institution. Now she has to prove the worth of her investment. This can be done through capital budgeting. Capital Budgeting Analysis is a process of evaluating investments in capital assets to determine whether future benefits of this project be large enough to justify the investment given the risk involved. Shirley Winters must know the cost of obtaining funds to make the long-term investments in new product lines, new equipment and other assets. Cost of Capital represents the rate a business must pay for each source of funds. Why use the Cost of Capital' Because we know the Shierly wouldn't do the project, which earns profits below the cost of capital. She would lose money. Hopefully Shirley’s Boutique would earn much more than the cost of capital. The cost of capital is the minimum acceptable rate of return for long-term investments. The discount rate is usually the cost of capital i.e. interest rate. The Three Stages of Capital Budgeting Analysis We must focus much of our attention on present values so that we can understand how expenditures today influence values in the future. An approach to looking at present values of projects is the discounted cash flow (DCF) technique. Capital Budgeting involves three stages:  Decision Analysis (for Knowledge Building)  Option Pricing (to Establish Position)  Discounted Cash Flow (DCF) (for making the Investment Decision) These processes helps to reduce the uncertainty in the investment decision. 1. Decision Analysis Decision-making in the beginning of the project for Shirley is complex because of uncertainty like capital requirements, risks, tax considerations and expected rates of return. She has to understand the existing markets to forecast project revenues, assess competitive impacts of the project, and determine the life cycle of the project. If our capital project involves production, we have to understand operating costs, additional overheads, capacity utilization, and start-up costs. Consequently, we cannot manage capital projects by simply looking at the numbers. We must assess all relevant variables and outcomes within an analytical hierarchy. 2. Option Pricing The second stage is to consider all options for the project. Therefore, before employing discounted cash flow technique we need to build a set of options into our project for managing unexpected changes. Shirley must consider options which she could easily take up lest her original project fails. 3. Discounted Cash Flows Discounted Cash Flow techniques is concerned with the present values of assets. Since capital projects like Shirley’s Boutique provide benefits into the future and to determine the present value of the project, we discount the future cash flows of a project to the present. Discounting refers to taking a future amount and finding its value today. Future values differ from present values because of the time value of money. Financial management recognizes the time value of money because: 1. Inflation reduces values over time; i.e. $ 1,000 today will have less value five years from now due to rising prices (inflation). 2. Uncertainty in the future; i.e. we think we will receive $ 1,000 five years from now, but a lot can happen over the next five years. 3. Opportunity Costs of money; $ 1,000 today is worth more to us than $ 1,000 five years from now because we can invest $ 1,000 today and earn a return. Capital Budgeting Techniques The Capital Budgeting Techniques are a) Project Evaluation and Selection b) Potential Difficulties c) Capital Rationing d) Project Monitoring a) Project Evaluation: The methods of Project Evaluation are 1) Net Present Value (NPV) 2) Internal Rate of Return (IRR) 3) Payback Period (PBP) 4) Profitability Index (PI) Net Present Value (NPV) In order to assess the worth of Shirley’s project, she must determine its present worth. The future cash flows have to be converted into its present worth, called Net Present Worth. By finding out the net present worth of the Investment in the Boutique Shierly will convince the financer whether it is economically viable or not. Shirley submits two proposals one for the Boutique and another for a Restaurant to Grow & Prosper Finance Company The finance company is evaluating two investment proposals. Their respective investment and cash flows are here: Boutique Restaurant Year Investment ($) Cash Flow Year Investment (Rs) Cash Flow 1 250000 40000 1 300000 60000 2 50000 2 80000 3 80000 3 80000 4 100000 4 100000 5 100000 5 100000 The interest for the loan is 14 per cent per annum i.e. the cost of capital. The NPV of the two projects is shown below: Year Investment ($) Cash Flow Net Flow Discount factor Present Value Boutique 1 250000 40000 -210000 0.8772 -184211 2 50000 50000 0.7695 38473 3 80000 80000 0.6750 53998 4 100000 100000 0.5921 59208 5 100000 100000 0.5194 51937 NPV -> 19405 Restaurant 1 300000 60000 -240000 0.8772 -210526 2 80000 80000 0.7695 61557 3 80000 80000 0.6750 53998 4 100000 100000 0.5921 59208 5 100000 100000 0.5194 51937 NPV -> 16174 The cash flow is the returns over the cost. The net flow is the difference between the cash flow and the total investment cost. The discount factors are calculated using the formula : Where, ‘r’ is 0.14 and ‘n’ takes values 1,2,3,4 and 5 and presented in column 5 of the table. The present values are derived by multiplying the net flow by the corresponding discount factor and shown in the last column of the table. Then the present values are added to obtain the Net Present Value. For Grow & Prosper Finance Company to be satisfied about the feasibility of the Projects they should have a positive NPV. When it comes to choosing between the two investment proposals, the project with the higher NPV will be preferred Between the two Projects both projects are economically viable since both have positive NPVs. The investment in the Boutique is preferred over Restaurant because of its higher Net Present Value of the two. Internal rate of return Shirley is happy that her Boutique meets the requirement of Net Present Worth but wants to know what return the project yields in percentage terms. This will be known if she calculates the Internal Rate of Return. Internal Rate of Return (IRR) is the amount of profit you get by investing in a certain project. It is expressed as a percentage. An IRR of 10% means you make 10% profit per year on the money invested in the project. IRR is a popular economic criteria for evaluating capital projects, since investors like Shirley Winters can easily identify with rates of return. IRR is calculated by finding the discount rate whereby the Net Investment amount equals the total present value of all cash inflows. The IRR is a discount rate that makes the Net Present Value = 0. This can be seen from the example below: Keep changing the IRR by small amounts till the right hand side equals the Left Hand Side. That happens at a rate of 18.5%. This becomes the IRR. This is how the IRR is arrived at. The table gives the details of investments and returns of two projects. The cost of capital for both the projects is 14% per annum and the deposit rate is 10%. Boutique Year Investment ($) Cash Flow Net Flow Discount factor @18.5% Present Value 1 250000 40000 -210000 0.8439 -177223 2 50000 50000 0.7122 35610 3 80000 80000 0.6010 48083 4 100000 100000 0.5072 50723 5 100000 100000 0.4281 42806 NPV -> 0 Restaurant Year Investment ($) Cash Flow Net Flow Discount factor @17.53% Present Value 1 300000 60000 -240000 0.8509 -204207 2 80000 80000 0.7240 57918 3 80000 80000 0.6160 49280 4 100000 100000 0.5241 52413 5 100000 100000 0.4460 44596 NPV -> 0 If the IRR is higher than 14%, then we would accept the project. In our example, the IRR 18.5% for Project-A and 17.53% for Project-B. Since the IRR is higher than the cost of capital, we can invest in both the projects, but Project-A is superior to Project-B. One of the problems with IRR is the so-called reinvestment rate assumption. This means the amount $35610/- which is surplus of the 2nd year is reinvested in the project at 18.5%, equivalent to the IRR. This assumption need not be true as the surplus may be deployed at a different rate. We will correct this distortion by modifying our IRR calculation (MIRR). The MIRR can be calculated using the Excel function: =MIRR(,,). In order to eliminate the reinvestment rate assumption, we will modify the IRR incorporate changes in the reinvestment rate. Accordingly, the MIRR for the Project-A is 15.47% and Project-B is 14.55%, at reinvestment rate of 10%, which are still higher than the cost of capital and hence economically viable. Payback Period Shirley is keen on getting rid of the loan as soon as possible and would like to know how long it will take to do so. The Payback period is the periods for the investment to be recovered from the net cash flow of the project. It is usually undiscounted and calculated by deducting the investment cost from each year’s net cash flow until the entire investment is recovered. The time taken to recover the investment is called the ‘payback period’. It is demonstrated for the two projects discussed earlier. Boutique Year Investment ($) Net Cash Flow Total Cash Flow Net Flow (3-2) Annual recovery of investment Discount factor @14% Present Value (5 x 6) Annual recovery (discounted) (1) (2) (3) (3a) (4) (5) (6) (7) (8) 1 250000 (b) 40000 40000 -210000 -210000 0.8772 -184211 -184211 2 50000 90000 50000 -160000 0.7695 38473.38 -145737 3 (a) 80000 170000(c) 80000 -80000 0.6750 53997.72 -91739.4 4 100000 270000 100000 (d) 20000 0.5921 59208.03 -32531.4 5 100000 370000 100000 120000 0.5194 51936.87 19405.47 It can be calculated in a straightforward manner using the formula. PBP = a + ( b - c ) / d Where , PBP is pay back period, a is the year last year of negative recovery b is the investment c is the total cash flow upto year ‘a’ d is the net flow in the year following year ‘a’ = 3 + (250000 - 170000) / 100000 = 3.9 Years (rounded to single decimal) Restaurant Year Investment ($) Net Cash Flow Total Cash Flow Net Flow (3-2) Annual recovery of investment Discount factor @14% Present Value (5 x 6) Annual recovery (discounted) (1) (2) (3) (3a) (4) (5) (6) (7) (8) 1 300000 60000 60000 -240000 -240000 0.8772 -210526 -210526 2 80000 140000 80000 -160000 0.7695 61557.4 -148969 3 80000 220000(c) 80000 -80000 0.6750 53997.72 -94971.2 4 100000 320000 100000 20000 0.5921 59208.03 -35763.2 5 100000 370000 100000 120000 0.5194 51936.87 16173.7 In projects A and B, the annual capital recovery is shown in column (5). With the data from the table, the payback for both the projects has been computed and found to be at 3.9 years. Both the projects have the same payback period. Profitability Index Profitability index is the ratio of the Present Worth of the net cash flows of the Project to the Initial Cash Outflow. This can be illustrated with the help of the Boutique example. The cash in-flow of the project is $40000, $50000, $80000, $100000 and $100000, respectively for the first to the fifth year. The cost of capital is 10%, which is taken as the discount rate and the calculations are shown. The Present Worth of the Net cash outflows is $268185 while the Initial cash outlay or the investment is $2,50,000. The Profitability Index is = 268185/ 250000= 1.07 Should the Boutique Project be accepted' Yes since the ratio is greater than one, the project is viable as it can cover the cost of borrowing. The measures of project worth help Shirley Winters evaluate and select the projects. She should continually try to spot potential difficulties so as to overcome them from time to time. Capital rationing occurs when the capital is constrained during a particular period. If Shirley Winters can raise only a part of her project cost, then the selection will have to based on the criteria that maximises shareholders value by investing the part of the capital. Finally a post completion audit has to be carried out to compare the actual costs and returns with those that were projected. This will help in identifying the weaknesses so that corrective action can be taken and better decisions taken in future.