First principles of valuation: the time value of money Chapter five First principles of valuation: the time value of money

Learning objectives LO5.1 Understand how to determine the future value of an investment made today. LO5.2 Understand how to determine the present value of cash to be received at a future date. LO5.3 Understand how to find the return on an investment. LO5.4 Understand how long it takes for an investment to reach a desired value. continued

Learning objectives LO5.5 Understand how to determine the future and present value of investments with multiple cash flows. LO5.6 Understand how loan payments are calculated and how to find the interest rate on a loan. LO5.7 Understand how interest rates are quoted (and misquoted). LO5.8 Understand how loans are amortised or paid off.

Chapter organisation Future value and compounding Present value and discounting More on present and future values Present and future values of multiple cash flows Valuing equal cash flows: annuities and perpetuities Comparing rates: the effect of compounding periods Loan types and loan amortisation Summary and conclusions

Time value terminology Future value (FV) is the amount an investment is worth after one or more periods. Present value (PV) is the amount that corresponds to today’s value of a promised future sum. The number of time periods between the present value and the future value is represented by ‘t’. continued

Time value terminology The rate of interest for discounting or compounding is called ‘r’. All time value questions involve four values: PV, FV, r and t. Given three of them, it is always possible to calculate the fourth. Compounding is the process of accumulating interest in an investment over time, to earn more interest. Interest on interest is earned on the reinvestment of previous interest payments. continued

Time value terminology Discount rate is the interest rate that reduces a given future value to an equivalent present value. Compound interest is calculated each period on the principal amount, and on any interest earned on the investment up to that point. Simple interest is the method of calculating interest in which, during the entire term of the loan, interest is computed on the original sum borrowed.

Future value of a lump sum You invest $100 in a savings account that earns 10% interest per annum (compounded) for five years. After one year: $100  (1 + 0.10) = $110 After two years: $110  (1 + 0.10) = $121 After three years: $121  (1 + 0.10) = $133.10 After four years: $133.10  (1 + 0.10) = $146.41 After five years: $146.41  (1 + 0.10) = $161.05

Future values of $100 at 10% This graph illustrates the economic benefit of the time-value-of-money principle. Wealth increases by an increasing amount each year: how is this possible if the interest rate is fixed at 10%? It is possible because each new year’s cycle of investment yields a 10% return from a higher base value: Year 2 earns 10% from a base value of $110; Year 3 earns 10% from a base value of $121 and so on. The critical assumption here is that none of the money invested is withdrawn. It is preserved in the investment fund to be reinvested each year. (A good example are the compulsory superannuation contributions that Australian employees make to their super fund. The vesting rules prevent people from withdrawing their money until they retire.)

Future value of a lump sum The accumulated value of this investment at the end of five years can be split into two components: original principal $100.00 interest earned $61.05 Using simple interest, the total interest earned would only have been $50. The other $11.05 is from compounding. The key difference between compound interest and simple interest is that compound interest is a fixed interest amount earned on an increasing base level. Simple interest is a fixed interest amount earned on a fixed base level. continued

Future value of a lump sum In general, the future value, FVt, of $1 invested today at r per cent for t periods is: FVt = $1 × (1 +r)t The expression (1 + r)t is the future value interest factor (FVIF). Refer to Table A.1.

Example—Future value of a lump sum What will $1000 amount to in five years time if interest is 6$ per annum, compounded annually? FV = $1000(1 + 0.06)5 = $1000 × 1.3382 = $1338.22 - All else equal, higher compounding frequency leads to higher future value. This is a necessary mathematical function of geometric progression. continued

Example—Future value of a lump sum From the example, now assume interest is 6% per annum, compounded monthly. Always remember that t is the number of compounding periods, not the number of years. FV = $1000(1 + 0.005)60 = $1000 × 1.3489 = $1348.90 - All else equal, higher compounding frequency leads to higher future value. This is a necessary mathematical function of geometric progression.

Future value of $1 for different periods and rates - This example highlights the positive relationship between future value and ‘r’ and ‘t’. The 'future value interest factor' derives from the interplay between interest rate (r) and time (t).

Present value of a lump sum You need $1000 in five years time. If you can earn10 per cent per annum, how much do you need to invest now? Discount one year: $1000 (1 + 0.10) –1 = $909.09 Discount two years: $909.09 (1 + 0.10) –1 = $826.45 Discount three years: $826.45 (1 + 0.10) –1 = $751.32 Discount four years: $751.32 (1 + 0.10) –1 = $683.02 Discount five years: $683.02 (1 + 0.10) –1 = $620.93 continued

Present value of a lump sum In general, the present value of $1 received in t periods of time, earning r per cent interest is: PV = $1(1 + r)t = The expression (1 + r)–t is the present value interest factor (PVIF). Refer to Table A.2. $1 (1 + r)t

Example—Present value of a lump sum Your rich uncle promises to give you $100 000 in 10 years' time. If interest rates are 6% per annum, how much is that gift worth today? PV = $100 000 × (1 + 0.06)10 = $100 000 × 0.5584 = $55 840 The discount factor is the inverse of the compounding factor. Where FV = PV(1 + r)t then PV = FV / (1 + r)t or equivalently PV = FV (1 + r)-t . The PV formula is used whenever the price of an asset needs to be estimated.

Present value of $1 for different periods and rates This chart illustrates the situation for an investor who expects to receive cash in the future. Two principles are at play here: The longer the investor has to wait to receive that future cash flow, the lower will be its present value. As the discount rate rises, the present value declines.

Determining the discount rate You currently have $100 available for investment for a 8-year period. At what interest rate must you invest this amount in order for it to be worth $200 at maturity (in 8 years’ time)? In this example, you need to find the r for which the FVIF after 8 years is 2 (200/100). r can be solved in one of three ways: Remind the students that ln is the natural logarithm and can be found on the calculator. continued

Determining the discount rate Use a financial calculator: To determine the discount rate (r) in this example on a financial calculator r = 9.05% Enter: 5 −100 200 N I/Y PV FV PMT Solve for: 9.05 continued

Determining the discount rate Solve 200 = 100(1 + r)8 for r Remind the students that ln is the natural logarithm and can be found on the calculator. continued

Determining the discount rate Use the future value tables. If you look across the row corresponding to the 8th period in Table A1, the closest factor is 1.9926 in the 9% column, implying that r is 9%. Remind the students that ln is the natural logarithm and can be found on the calculator.

The rule of 72 The ‘rule of 72’ is a handy rule of thumb that states: If you earn r per cent per year, your money will double in about 72/r years For example, if you invest at 8 per cent, your money will double in about 9 years. This rule is only an approximate rule.

Finding the number of periods You have been saving up to buy a power generation company. The total cost will be $1 billion. You currently have $230 million. If you can earn 5% on your money, how long will you have to wait? This time you have solve 1000 = 230(1+ 0.05)n for n: Pointers for students: - To determine the number of periods without a financial calculator use the formula: ln(FV/PV)/ln(1+r)=t. - If using a financial calculator, they may need to input the values in brackets first, and then activate the ‘Ln’ function to obtain the log.

Finding the number of periods To determine the number of periods (t) in this example on a financial calculator t = 30.122 years Enter: 5 −230 1000 N I/Y PV FV PMT Solve for: 30.122 Pointers for students: - To determine the number of periods without a financial calculator use the formula: ln(FV/PV)/ln(1+r)=t. - If using a financial calculator, they may need to input the values in brackets first, and then activate the ‘Ln’ function to obtain the log.

Future value of multiple cash flows You deposit $1000 now, $1500 in one more year, then $2000 in two years and $2500 in three years in an account paying 10 per cent interest per annum. How much do you have in the account at the end of the third year? You can solve by either: compounding the accumulated balance forward one year at a time calculating the future value of each cash flow first, and then totalling them.

Solutions Solution 1 Solution 2 End of year 1: ($1 000  1.10) + $1 500 = $2 600 End of year 2: ($2 600  1.10) + $2 000 = $4 860 End of year 3: ($4 860  1.10) + $2 500 = $7 846 Solution 2 $1 000  (1.10)3 = $1 331 $1 500  (1.10)2 = $1 815 $2 000  (1.10)1 = $2 200 $2 500  1.00 = $2 500 Total = $7 846

Solutions on time lines Future value calculated by compounding forward one period at a time 1 2 3 Time (years) $0 1000 $1000 $1100 1500 $2600 $2860 2000 $4860 $5346 2500 $7846 x 1.1 x 1.1 x 1.1 Future value calculated by compounding each cash flow separately 1 2 3 Time (years) $1000 $1500 $2000 $2500 2200 1815 1331 $7846 x 1.1 x 1.12 x 1.13 Total future value

Present value of multiple cash flows You will deposit $1500 in one year’s time from now, $2000 in two years time and $2500 in three years time, in an account paying 10 per cent interest per annum. What is the present value of these cash flows? You can solve by either: discounting back one year at a time calculating the present value of each cash flow first, and then totalling them.

Solutions Solution 1 Solution 2 End of year 2: ($2500  1.10–1) + $2000 = $4273 End of year 1: ($4273  1.10–1) + $1500 = $5385 Present value: ($5385  1.10–1) = $4895 Solution 2 $2500  (1.10)–3 = $1878 $2000  (1.10)–2 = $1653 $1500  (1.10)–1 = $1364 Total = $4895

Annuities An ordinary annuity is a series of equal cash flows that occur at the end of each period for some fixed number of periods. Examples include consumer loans and home mortgages. A perpetuity is an annuity in which the cash flows continue forever.

Present value of an annuity C = equal cash flow The discounting term is called the present value interest factor for annuities (PVIFA). Refer to Table A.3. 1{1/(1 + r)t} r PV = C × - Annuities can be solved in a single calculation because the cash flows are constant in their amount and in their flow pattern. The annuity formula should always be used to value this constant cash flow sequence.

Examples—Present value of an annuity You will receive $1000 at the end of each of the next ten years. The current interest rate is 6 per cent per annum. What is the present value of this series of cash flows? ×

Examples—Present value of an annuity You borrow $10 000 to buy a car and agree to repay the loan by way of equal monthly repayments over four years. The current interest rate is 12 per cent per annum, compounded monthly. What is the amount of each monthly repayment?

Finding the rate for an annuity You have a loan of $5000 repayable by instalments of $745.15 at the end of each year for 10 years. What rate is implicit in this 10-year annuity? To determine the discount rate (r) in this example, a financial calculator is used. r = 8% Enter: 10 5000 −745.15 N I/Y PV FV PMT Solve for: 8.00

Finding the number of payments for an annuity You have $2000 owing on your credit card. You can only afford to make the minimum payment of $40 per month. The interest rate is 1% per month. How long will it take you to pay off the $2000? continued

Finding the number of payments for an annuity To determine the number of payments (t), a financial calculator is used. t = 69.66 months ÷ 12 = 5.81 years Enter: 1 2000 −40 N I/Y PV FV PMT Solve for: 69.66

Future value of an annuity The compounding term is called the future value interest factor for annuities (FVIFA). Refer to Table A 4.

Example—Future value of an annuity What is the future value of $1 000 deposited at the end of every year for 20 years if the interest rate is 6% per annum?

Perpetuities The future value of a perpetuity cannot be calculated, as the cash flows are infinite. The present value of a perpetuity is calculated as follows: where C is cash flow and r is rate.

Comparing rates The nominal interest rate (NIR) is the interest rate expressed in terms of the interest payment made each period. The effective annual interest rate (EAR) is the interest rate expressed as if it was compounded once per year. When interest is compounded more frequently than annually, the EAR will be greater than the NIR.

Calculation of EAR m = number of times the interest is compounded

Comparing EARs Consider the following interest rates quoted by three banks: Bank A: 8.3% compounded daily Bank B: 8.4% compounded quarterly Bank C: 8.5% compounded annually continued

Comparing EARs continued

Comparing EARs Which is the best rate? For a saver, Bank B offers the best (highest) interest rate. For a borrower, Banks A and C offer the best (lowest) interest rates. The highest NIR is not necessarily the best. Compounding during the year can lead to a significant difference between the NIR and the EAR, especially for higher rates.

Types of loans An interest-only loan requires the borrower to only pay interest each period, and to repay the entire principal at some point in the future. An amortised loan requires the borrower to repay parts of both the principal and interest over time. - Bonds issued by companies are an example of interest-only loans. The company has to make interest payments for the life of the loan, and repays the principal upon expiry of the bond. - Most home loans are amortised loans. The borrower has to make payments of principal and interest, and by the end of the loan all principal has been paid back. http://news.infochoice.com.au/home-loans/interest-calculator.aspx

Amortisation of a loan

Spreadsheet example Use the following formulas for TVM calculations: FV (rate, nper, pmt, pv) PV (rate, nper, pmt, fv) RATE (nper, pmt, pv, fv) NPER (rate, pmt, pv, fv) The formula icon is very useful when you can’t remember the exact formula Click on the Excel icon to open a spreadsheet containing four different examples. - Click on the tabs at the bottom of the worksheet to move between examples.

Quick quiz—Part I What is the difference between simple interest and compound interest? Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years. How much would you have at the end of 15 years using compound interest? How much would you have using simple interest? - Formula: 500(1.08)15 = 500(3.172 169) = 1 586.08 500 + 15(500)(.08) = 1 100 - You may wish to take this opportunity to remind students that, since compound growth rates are found using only the beginning and ending values of a series, they convey nothing about the values in between. For example, a firm may state that 'EPS has grown at a 10% annually compounded rate over the last decade' in an attempt to impress investors of the quality of earnings. However, this just depends on EPS in year 1 and year 11. For example, if EPS in year 1 = $1, then a '10% annually compounded rate' implies that EPS in year 11 is (1.10)10 = 2.5937. So, the firm could have earned $1 per share 10 years ago, suffered a string of losses, and then earned $2.59 per share this year. Clearly, this is not what is implied by management’s statement above.

Quick quiz—Part II What is the relationship between present value and future value? Suppose you need $15 000 in 3 years. If you can earn 6% annually, how much do you need to invest today? If you could invest the money at 8%, would you have to invest more or less than at 6%? How much? - Relationship: The mathematical relationship is PV = FV / (1 + r)t. One of the important things for them to take away from this discussion is that the present value is always less than the future value when we have positive rates of interest. N = 3; I/Y = 6; FV = 15 000; PV = 15 000 / (1.06)3 = 15 000(0.839 619 283) = 12 594.29 N = 3; I/Y = 8; FV = 15 000; PV = 15 000 / (1.08)3 = 15 000(0.793 832 241) = 11 907.48

Quick quiz—Part III What are some situations in which you might want to know the implied interest rate? You are offered the following investments: You can invest $500 today and receive $600 in 5 years. The investment is low-risk. You can invest the $500 in a bank account paying 4%. What is the implied interest rate for the first choice, and which investment should you choose? Implied rate: N = 5; PV = -500; FV = 600; r = (600 / 500)1/5 – 1 = 3.714% Choose the bank account because it pays a higher rate of interest (assuming tax rates and other issues are consistent across both investments). How would the decision be different if you were looking at borrowing $500 today, and either repaying at 4%, or repaying $600? In this case, you would choose to repay $600 because you would be paying a lower rate.

Quick quiz—Part IV When might you want to compute the number of periods? Suppose you want to buy some new furniture for your family room. You currently have $500, and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don’t add any additional money? Calculator: PV = –500; FV = 600; I/Y = 6; Formula: t = ln(600/500) / ln(1.06) = 3.13 years.

Comprehensive problem You have $10 000 to invest for five years. How much additional interest will you earn if the investment provides a 5% annual return, when compared to a 4.5% annual return? How long will it take your $10 000 to double in value if it earns 5% annually? What annual rate has been earned if $1000 grows into $ 4000 in 20 years? N = 5 PV = –10,000 At I/Y = 5, the FV = 12 762.82 At I/Y = 4.5, the FV = 12 461.82 The difference is attributable to interest. That difference is 12 762.82 – 12 461.82 = 301 To double the 10 000: I/Y = 5 PV = –10 000 FV = 20 000 Note, the rule of 72 indicates 72/5 = 14 years, approximately. N = 20 PV = –1 000 FV = 4 000

Summary and conclusions For a given rate of return, the value at some point in the future of an investment made today can be determined by calculating the future value of that investment. The current worth of a future cash flow or series of cash flows can be determined for a given rate of return by calculating the present value of the cash flow(s) involved. It is possible to find any one of the four components (PV, FV, r, t) given the other three. continued

Summary and conclusions A series of constant cash flows that arrive or are paid at the end of each period is called an ordinary annuity. For financial decisions, it is important that any rates are converted to effective rates before being compared.