British Columbia Institute of Technology Fundamentals of Corporate Finance Second Canadian Edition prepared by: Carol Edwards BA, MBA, CFA Instructor, Finance British Columbia Institute of Technology copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Chapter 3 The Time Value of Money Chapter Outline Future Values and Compound Interest Present Values Multiple Cash Flows Level Cash Flows: Perpetuities and Annuities Inflation and the Time Value of Money Effective Annual Interest Rates copyright © 2003 McGraw Hill Ryerson Limited

How should you tackle this kind of problem? Introduction Money Problems … Assume interest rates are 4.3884%. You have just won a lottery and must choose between the following two options: Receive a cheque for $150,000 today. Receive $10,000 a year for the next 25 years. KEY QUESTIONS FOR YOU: Which option gives you the biggest “winnings” How should you tackle this kind of problem? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Introduction Money Problems … As a financial manager you will often have to compare cash payments which occur at different dates: Cash flows now, versus … … cash flows later. To make optimal decisions, you must understand the relationship between a dollar received (paid) today and a dollar received (paid) in the future. copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Introduction Money Problems … As a financial manager you will face two basic types of cash flow problems: Present Value (PV) problems. Future Value (FV) problems. copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Introduction Present Value (PV) Problems PV problems involve calculating the value today of future cash flow(s). For example: Interest rates are 7%. If I need to have $100,000 saved in 10 years, how much money must I put aside today to create that cash flow? Interest rates are 12%. If I need to create an income of $5,000 per year for 10 years, how much money must I put aside today to create that cash flow? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Introduction Future Value (FV) Problems FV problems involve calculating the value an investment will grow to after earning interest. For example: Interest rates are 5%. If I invest $1,000 today, how much will it be worth in 8 years? Interest rates are 10%. If I open an account and invest $2,500 per year, how much will it be worth in 12 years? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Future Values Compound Interest vs Simple Interest Future value is the amount to which an investment will grow after earning interest. There are two types of interest you may receive: Compound interest. Simple interest. copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Future Values Simple Interest Simple interest means that interest is earned only on your original investment: No interest is earned on the interest. Example: Assume interest rates are 6%. You invest $100 in an account paying simple interest. How much will the account be worth in 5 years? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Future Values Simple Interest You earn interest only on the amount invested. Therefore you would earn: $100 x 6% = $6.00 per year for 5 years. Answer – you would have $130 after 5 years: Balance in your account: $100 $106 $112 $118 $124 $130 Period (t) 1 2 3 4 5 copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Future Values Compound Interest Most financial problems you will deal with will involve compound interest. Compound interest means that interest is earned on interest. The result: the income you earn would be higher than it would be with simple interest. Can you see why? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Future Values Compound Interest Your income would be higher than it would be with simple interest because you earn interest on both the original investment and the interest earned in previous years. Try the example again using compound interest: Interest rates are 6%. You invest $100 in an account paying compound interest. How much will the account be worth in 5 years? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Future Values Compound Interest You earn interest on your interest: $100 x 6% = $6.00 the first year. $106 x 6% = $6.36 the second year. $112.36 x 6% = $6.74 the third year … etc. After 5 years you would have $133.82 : Balance in your account: $100 $106 $112.36 $119.10 $126.25 $133.82 Period (t) 1 2 3 4 5 copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Future Values Formula for Calculating FV FV = Investment x (1 + r)t Try the example again using the formula above: Interest rates are 6%. You invest $100 in an account paying compound interest. How much will the account be worth in 5 years? FV = $100 x (1 + 0.06)5 = $100 x 1.3382 = $133.82 copyright © 2003 McGraw Hill Ryerson Limited

How should you tackle this kind of problem? Present Value More Money Problems … Assume interest rates are 10%. You have just won a lottery and must choose between the following two options: Receive $1,000,000 today. Receive $1,000,000 five years from now. KEY QUESTIONS FOR YOU: Which option gives you the biggest “winnings” How should you tackle this kind of problem? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Present Value More Money Problems … This is an example of a present value problem. You shouldn’t even have to do a calculation to get the correct answer. Obviously the first option is the better choice! You would want to take the money today so that you could immediately start earning interest on your winnings. copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Present Value More Money Problems … The above example demonstrates a basic financial principle: A dollar received today is worth more than a dollar received tomorrow. The key question is: How much less valuable is a dollar received tomorrow as versus a dollar received today? That question is answered by using the interest rate (also known as the discount rate) to calculate the PV of the second option. copyright © 2003 McGraw Hill Ryerson Limited

PV = Future Value x 1/(1 + r)t Present Value Formula for Calculating PV PV = Future Value x 1/(1 + r)t You have been offered $1 million five years from now. Interest rates are 10%. What is that worth to you in today’s dollars? PV = $1.0 million x 1/ (1 + 0.10)5 = $1.0 million x 0.620921 = $620,921 copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Present Value More Money Problems … Thus, you could have $1 million today. Or you could have the second option, which equates to $620,921 in today’s dollars. $1 million now The equivalent of $620,921 now vs You knew before that the first option was better, but now you can calculate exactly how much better off you are: $379,079 better off! copyright © 2003 McGraw Hill Ryerson Limited

Present Value vs Future Value PV and FV are related! Have you noticed that $620,921 becomes $1 million (and that $1 million requires $620,921) if you have a time period of 5 years and a discount rate of 10%? PV at 10% $1,000,000 $620,921 FV at 10% copyright © 2003 McGraw Hill Ryerson Limited

Present Value vs Future Value PV and FV are related! $620,921 invested for 5 years at 10% grows to $1 million. Or, working it in reverse, if rates are 10%, and you need $1 million in 5 years, you must put aside $620,921 right now. FV = PV x (1 + r)t = $620,921 x (1 + 0.10)5 = $620,921 x 1.61051 = $1 million PV = FV x 1/(1 + r)t = $1 million x 1/ (1 + 0.10)5 = $1 million x 0.620921 = $620,921 copyright © 2003 McGraw Hill Ryerson Limited

Present Value vs Future Value PV and FV are related! To calculate the FV of money which is available now (PV) to be invested for t years at an interest rate r, multiply the PV by (1+r)t. To calculate the PV of a future payment, run the process in reverse and divide the FV by (1+r)t. copyright © 2003 McGraw Hill Ryerson Limited

Present Value vs Future Value PV and FV are related! Note that: (1+r)t is called the future value factor. r is called the discount rate Finding the PV is often called discounting. copyright © 2003 McGraw Hill Ryerson Limited

Present Value vs Future Value Two Key Principles for Financial Calculations Think of the example in which we compared receiving $1 million today against $1 million received 5 years from now. You should see from that example that: A dollar received today is worth more than a dollar received tomorrow. Lesson: The value of cash flows received at different times can never be directly compared. You must first discount all cash flows to a common date and then compare them. copyright © 2003 McGraw Hill Ryerson Limited

Present Value vs Future Value Finding the Unknown … FV = PV x (1 + r)t PV = FV x 1/(1 + r)t The FV and PV formulas have many applications. Note that the variables used in these two equations are: FV PV r t Given any three variables in the equation, you can always solve for the remaining variable! copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Future Value Calculations So far, we have looked at problems involving only a single cash flow. This is unrealistic – most business investments will involve multiple cash flows over time. We need a method for coping with such streams of cash flows! copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Future Value Calculations EXAMPLE Assume interest rates are 8%. You make 3 deposits to your bank account: $1,200 today $1,400 one year later. $1,000 two years later. How much money will you have in your account 3 years from now? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Doing Future Value Calculations Calculate what each cash flow will be worth at the specified future date and add up these future values. 1 2 3 $1,200 $1,400 $1,000 FV in Year 3: $1,080.00 = $1,000 x 1.08 $1,632.96 = $1,400 x (1.08)2 $1,511.65 = $1,200 x (1.08)3 $4,224.61 copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Present Value Calculations Suppose we need to calculate the PV of a stream of future cash flows. We use basically the same procedure as for working with the FV of multiple cash flows: Calculate what each cash flow would be worth today, i.e. get its PV. Add up these present values. copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Present Value Calculations EXAMPLE Assume interest rates are 8%. You wish to buy a car making three installments: $8,000 today $4,000 one year later. $4,000 two years later. How much money would you have to place in an account today to generate this stream of cash flows? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Present Value Calculations You would need to place $15,133.06 in an account today to generate the desired cash flows: 1 2 -$8,000 -$4,000 PV today: $8,000.00 $4,000 / (1.08) = $3,703.30 $4,000 / (1.08)2 = $3,429.36 $15,133.06 copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Special Situations In the previous examples, we worked with multiple cash flows of different sizes. Sometimes we have a situation in which a series of equal cash flows is involved: What would you pay to own a guaranteed income of $1,000 per year to be received forever, if interest rates are 4%? Calculate what the value of your account would be if you were to deposit $2,500 per year for 5 years and interest rates are 7%? copyright © 2003 McGraw Hill Ryerson Limited

Multiple Cash Flows Special Situations Any sequence of equally spaced, level cash flows is called an annuity. If the payment stream lasts forever, it is called a perpetuity. copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Perpetuities The PV of a perpetuity is calculated by dividing the cash payment by the interest rate: C r Cash Payment Interest rate PV of a perpetuity = = The interest rate on a perpetuity is calculated by dividing the cash payment by the PV: C PV Cash Payment Present Value Interest rate on a perpetuity = = copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Perpetuities We are now ready to answer a question we asked earlier: What would you pay to own a guaranteed income of $1,000 per year to be received forever, if interest rates are 4%? C r Cash Payment Interest rate PV of a perpetuity = = $1,000 4% = = $25,000 copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows PV of an Annuity: the Long Method In previous examples, we have worked with multiple cash flows of different sizes. Suppose we now need to calculate the PV of a stream of level future cash flows. We could use the same procedure as before: Calculate what each cash flow would be worth today, i.e. get its PV. Add up these present values. copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows PV of an Annuity: the Long Method EXAMPLE Assume interest rates are 10%. You wish to buy a car making three installments: $4,000 a year from now. $4,000 one year later. $4,000 two years later. How much money would you have to place in an account today to generate this stream of cash flows? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows PV of an Annuity: the Long Method You would need to place $9,947.41 in an account today to generate the desired cash flows: 1 2 3 -$4,000 -4,000 PV today: $4,000 / (1.10) = $3,636.36 $4,000 / (1.10)2 = $3,305.79 $4,000 / (1.10)3 = $3,005.26 $9,947.41 copyright © 2003 McGraw Hill Ryerson Limited

Multiple Cash Flows PV of an Annuity: the Short Cut! Yes! We have calculated that we need to put aside $9,947.41 to fund the following cash flows: $4,000 a year from now. $4,000 one year later. $4,000 two years later. However, is there an easier way to reach this answer? Yes! When you have level cash flows there is a short cut you can use … copyright © 2003 McGraw Hill Ryerson Limited

Multiple Cash Flows PV of an Annuity: the Short Cut! PV of an annuity = C x [ 1/r – 1/(r(1 + r)t)] (Where C = Cash Payment) PVannuity = $4,000 x [1/0.10 – 1/(0.10 (1 + 0.10)3)] = $4,000 x 2.48685 = $9,947.41 Using the PV of an annuity calculation, we get the same answer as before: Put aside $9,947.41 to fund the cash flows. copyright © 2003 McGraw Hill Ryerson Limited

Can you see the problem with using this method? Multiple Cash Flows Calculating the FV of an Annuity Suppose interest rates are 10% and you decide to save $4,000 per year for 20 years. How much will you have saved for your retirement? This is a FV problem. We could use the same procedure as we used for multiple cash flows of different sizes: Calculate what each cash flow would be worth in, 20 years, i.e. get its FV. Add up these future values. Can you see the problem with using this method? copyright © 2003 McGraw Hill Ryerson Limited

 Multiple Cash Flows Calculating the FV of an Annuity Yes! Calculating the FV this way would mean working out the FV for 20 separate cash flows ...  Is there an easier way? Yes! When you have level cash flows there is a short cut you can use … copyright © 2003 McGraw Hill Ryerson Limited

Multiple Cash Flows FV of an Annuity: the Short Cut! FV of an annuity = C x [ ((1 + r)t – 1)/r ] (Where C = Cash Payment) FVannuity = $4,000 x [ ((1 + 0.10)20 – 1) / 0.10 ] = $4,000 x 57.27499949 = $229,100 Using the FV of an annuity calculation, we see that you will have $229,100 in your account when you retire in 20 years. copyright © 2003 McGraw Hill Ryerson Limited

Which option gives you the biggest “winnings”? Multiple Cash Flows Our First Question … You now have all the tools necessary to answer the very first question we asked! Give it a try: Assume interest rates are 4.3884%. You have just won a lottery and must choose between the following two options: Receive a cheque for $150,000 today. Receive $10,000 a year for the next 25 years. Which option gives you the biggest “winnings”? copyright © 2003 McGraw Hill Ryerson Limited

PV of an annuity = C x [ 1/r – 1/(r(1 + r)t )] Multiple Cash Flows Our First Question … Option 1 is worth $150,000. To value Option 2, find the PV of $10,000 per year for 25 years at 4.3884%: PV of an annuity = C x [ 1/r – 1/(r(1 + r)t )] PV = $10,000 x [1/0.043884 – 1/0.043884 (1 + .043884)25] = $10,000 x 15.000 = $150,000 Both options are worth $150,000! copyright © 2003 McGraw Hill Ryerson Limited

Multiple Cash Flows Cash Flows Growing at a Constant Rate What if the cash flows in a financial problem are not equal, but are instead growing at a constant rate? For example: Assume the discount rate is 8%. You are thinking of buying a condo which generates $12,000 per year in net cash flow in perpetuity. These cash flows grow at 3% per year. What is the maximum price you should pay for this condo? copyright © 2003 McGraw Hill Ryerson Limited

Discount Rate – Growth Rate Multiple Cash Flows Valuing Growing Perpetuities The PV of a growing perpetuity is calculated by dividing the cash payment by the discount rate less the growth rate: C r-g Cash Payment Discount Rate – Growth Rate PV of a perpetuity = = $12,000 0.08 – 0.03 = = $240,000 copyright © 2003 McGraw Hill Ryerson Limited

Multiple Cash Flows Cash Flows Growing at a Constant Rate In the previous problem we assumed that the cash flows grew at a constant rate forever. It may be more reasonable to assume a constant growth rate for a limited time period. For example: Assume the condo in the previous problem generates $12,000 per year in net cash flow for 20 years. These cash flows grow at 3% per year. Now, what is the maximum price you should pay for this condo? copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Multiple Cash Flows Valuing Finite Growing Cash Flows The PV of cash flows which grow at a constant rate for a limited time period (T) is calculated by: (1- [ ]T) (1 + g) (1 + r) C r-g PV of cash flow = (1- [ ]20) (1 + 0.03) (1 + 0.08) $12,000 0.08 – 0.03 = = $240,000 * 0.61250 = $147,000 copyright © 2003 McGraw Hill Ryerson Limited

Effective Interest Rates EAR vs APR So far, we have used annual interest rates applied to annual cash flows. But interest can be applied daily, weekly, monthly, semi-annually – or for any other convenient time period. The simplest way to deal with this situation is to convert to annual rates. There are two ways you may do this: Calculate the Effective Annual Rate (EAR). Calculate the Annual Percentage Rate (APR). copyright © 2003 McGraw Hill Ryerson Limited

Effective Interest Rates Annual Percentage Rate (APR) The annual percentage rate (APR) is an interest rate that is annualized using simple interest. For example: Your credit card charges 1.5% per month. What is the annual charge? APR = Quoted rate x Number of Periods Per Year = 1.5% x 12 = 18% copyright © 2003 McGraw Hill Ryerson Limited

Effective Interest Rates Effective Annual Interest Rate (EAR) The effective annual interest rate (EAR) is an interest rate that is annualized using compound interest. For example: Your credit card charges 1.5% per month. What is the annual charge? EAR = (1 + Quoted rate) Number of Periods Per Year = (1 + 0.015)12 = 19.5% copyright © 2003 McGraw Hill Ryerson Limited

Effective Interest Rates Calculating the EAR Convert the APR to a period rate and then apply the equation: (1 + Period Rate) m (m = Number of periods per year) Quoted APR: 12% (a) (b) (c) Compounding Period Periods per Year (m) Period Rate (= a/b) EAR = (1+ c)m - 1 1 year 1 12.0000% Semiannually 2 6.0000% 12.3600% Quarterly 4 3.0000% 12.5509% Monthly 12 1.0000% 12.6825% Daily 365 0.0329% 12.7475% copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Summary of Chapter 3 Future value (FV) is the amount to which an investment will grow after earning interest. Find the FV of an investment by multiplying the investment by the future value factor of (1+r)t where t is the time period and r is the discount rate. The present value (PV) of a future cash payment is the amount you would need to invest today to create that future cash payment. Find the PV of an investment by multiplying the future cash payment by the discount factor of 1/(1+r)t. copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Summary of Chapter 3 Apply the formulas for FV and PV to calculate the value of multiple cash flows. A level stream of payments which continues forever is called a perpetuity. One which continues for a limited number of years is called an annuity. You can use the FV and PV formulas to calculate their value or you can use the shortcut formulas. Variations on the shortcut formulas allow you to calculate the value of multiple cash flows growing at a constant rate. copyright © 2003 McGraw Hill Ryerson Limited

copyright © 2003 McGraw Hill Ryerson Limited Summary of Chapter 3 Interest rates for periods of less than one year are often quoted as annual rates by converting to either an APR or an EAR. Annual percentage rates (APR) do not recognize the effect of compound interest, that is, they annualize assuming simple interest. Effective Annual Rates (EAR) annualize using compound interest. EAR equals the rate of interest per period compounded for the number of periods in a year. copyright © 2003 McGraw Hill Ryerson Limited