Business Finance Lecture 5

Lecture 5: Discounted Cash Flow Valuation Future and present values of multiple cash ßows Present value with multiple cash ßows ! Example 1 Suppose you receive £1,000 in one year and £2,000 in 2 years. If you can earn 9% on your money, how much do you have to deposit today to exactly cover these amounts in the future? In other words, what is the present value of the two cash ßows at 9%? This method involves calculating the present values individually and adding them up. ! Payment of £1,000 in one year: ! PV = FV / (1+r)t ! = £1,000 / 1.091! = £1,000 / 1.09 ! = £917.43 ! Payment of £2,000 in two years: ! PV = FV / (1+r)t! = £2,000 / 1.092! = £2,000 / 1.1881! = £1,683.36 ! ! Therefore, £917.43 + £1,683.36! = £2,555.79 ! Answer: you need to deposit £2,555.79 now to cover these two payments. ! Example 2 Suppose you receive £1,000 a year for Þve years. If you can earn 6% on your money, how much do you have to deposit today to exactly cover these amounts in the future? In other words, what is the present value of the two cash ßows at 6%? This method involves discounting back to the present, one period at a time. ! !   This is very time consuming for any longer periods of time. Fortunately, there is a shortcut: we can use ANNUITY and PERPETUITY formulas where we have the same cash ßow every period for a number of periods (annuity) or forever (perpetuity). ! Annuity Ñ a level stream of cash ßows for a Þxed period of time. ! Perpetuity Ñ a level stream of cash ßows forever.! Annuity: present values The present value of an annuity of £C (or any other currency) per period for t period when the rate of return or interest rate is r given by: ! Present value of annuity = C x [1 Ñ present value factor]! r! C x {1Ñ[1/(1+r)t] }! r! C x {1Ñ 1 }! r r(1+r)t ! C x Annuity factor (r%, t) ! Annuity tables ! You can Þnd annuity factors in annuity tables. However, these do not give very accurate results, and so are often not appropriate in practice. It is important to learn how to use the formulas instead and, for example, build these into EXCEL spreadsheets. ! Example 3! LetÕs do example 2 again, but this time use the annuity formula. Suppose you receive £1,000 a year for Þve years. If you can earn 6% on your money, how much do you have to deposit today to exactly cover these amounts in the future? In other words, what is the present value of the Þve cash ßows at 6%? (It is annuity because it is the same value every period and it has a Þnite life). ! Reminder: present value of an annuity = C x {1Ñ[1/(1+r)t] }! r! = C x {1Ñ [ 1 ]}! r (1+r)t! = C x {1Ñ [1+r]-t}! r ! !   We know that: ! C = £1,000 ! r = 0.06 ! t = 5 years ! So, PV (annuity) = (£1,000 / 0.06) x (1 Ñ 0.06-5) ! = (£16,666.67) x (1 Ñ 0.747258)! = £16,666.67 x 0.25274! = £4,212.36! We therefore arrived at the same answer, but this time in one step using the PV (annuity) formula. ! Perpetuity A perpetuity is a series of cash ßows, C, paid indeÞnitely into the future. An example of a perpetuity: dividends payable on preference shares that do not have a redemption date. ! Present value of a perpetuity = C/r ! Example 4 ! Now letÕs assume that the £1,000 annual payments continue forever (this means it is a perpetuity, not an annuity). What is the present value of such a perpetuity at a constant interest rate of 6%? What is the present value now?! Present value of a perpetuity = C/r ! We know that: ! C = £1,000 ! r = 0.06 ! So, the present value of the perpetuity = £1,000 / 0.06 ! = £16,666.67! Discussion point ! ! Why is the answer to example 4 (£1,000 in perpetuity) £16,666.67 rather than inÞnity? Note that £1,000 in perpetuity is worth £16,666.67 today and Þve annual payments of £1,000 are worth £4,212.36. ! The increase in present value decreases each period until it is negligible as this graph illustrates. The curve merges to the horizontal. Hence, the reason why the end value is not inÞnity is because eventually when t = a great value, it makes the total value much smaller, almost equal to zero (because t is a power). !   Future value with multiple cash ßows ! Example 5 Now, letÕs look at the future value of multiple cash ßows. Suppose you receive £15,000 a year for four years. If you can earn 6% on your money, how much will your investment be worth in four yearsÕ time? In other words, what is the future value of the four cash ßows at 6%? ! Receive £15,000 at time 1, time 2, time 3 and time 4. ! FV (payment at time 1) = £15,000 x 1.063 = £17,865.24 ! FV (payment at time 2) = £15,000 x 1.062 = £16,854.00 ! FV (payment at time 3) = £15,000 x 1.06 = £15,900.00 ! FV (payment at time 4) = £15,000.00 ! Total = £17,865.24 + £16,854 + £15,900 + £15,000! = £65,619.24 ! Annuity: future values Alternatively, we could use the future value of an annuity formula to solve example 5. The future value formula is similar to the present value formula, but this time uses the future value factor as its base: ! Future value of annuity = C x {Future value factor Ñ 1}! r! = C x {(1+r)t Ñ 1}! r! = C x [(1+r)t Ñ 1]! r Example 5 ! Now, letÕs rework example 5 using the annuity formula. ! FV (annuity) = C x {(1+r)t Ñ 1}! r! Where:! C = £15,000! t = 4! r = 6% (0.06) ! What is the FV of the four cash ßows? ! FV = (£15,000 / 0.06) x (1.064 Ñ 1) ! = £250,000 x 0.262477 ! = £65,619.24 ! Check: yes, this is the same answer as the calculation done with the other, more complicated method. !   Timing of cash ßows ! A note about cash ßow timing In present and future value problems, cash ßow timing is critically important. In almost all such calculations, it is implicitly assumed that the cash ßows occur at the end of each period. All the formulas we have discussed, all the numbers in a standard present value or future value table, and (very important) al the preset (or default) settings in a spreadsheet assume that cash ßows occur at the end of each period. Unless you are explicitly told otherwise, you should always assume that this is what is meant!! Annuities due Annuities due Ñ an annuity for which the cash ßows occur at the beginning of the period.! PV of an annuity due = (1+r) x PV of an annuity ! FV of an annuity due = (1+r) x FV of an annuity ! Because each cash ßow occurs one period earlier than with an ordinary annuity. Almost any type of arrangement in which we have to prepay the same amount each period is an annuity due. Suppose an annuity due has Þve payments of £400 each, and the relevant discount rate is 10%. The timeline looks like this: ! Example 6 ! Find the PV of a 3 year annuity that will make a series of $100 payments at the beginning of year for next three years. The discount rate is 10%. ! PV of an annuity = C x {[1 Ñ (1/1.103)]/0.10}! = 100 x 2.487! = 248.7 ! PV of annuity due = 248.7 x (1+0.10)! = $273.6! Summary ! We have looked at:! 1. How to calculate the present value or future value of multiple cash ßows, whether they are different or the same.! 2. How to use annuity formulae when the cash ßow is the same in every period.! 3. How to use present value (perpetuity) formula when the cash ßows continue forever.! A reminder of the key formulas For individual cash ßows that are each different! PV = C ! (1+r)t!   FV = C x (1+r)t! For cash ßows that are of the same value every period! PV of an annuity = C x [1 Ñ ( 1 )]! r (1+r)t! PV of a perpetuity = C (where cash ßows continue forever) ! r! FV of an annuity = C x [(1+r)t Ñ1]! r ! Quick link between annuity and annuity due values ! PV of annuity due = (1+r) x PV of annuity ! FV of annuity due = (1+r) x FV of annuity ! Compounding more than once a year This subject causes a fair amount of confusion, because rates are quoted in many different ways. Sometimes the way a rate is quoted is the result of tradition, and sometimes itÕs the result of legislation. Unfortunately, at times, rates are quoted in deliberately deceptive ways to mislead borrowers and investors. ! What if interest compounds monthly, quarterly, or semi-annually rather than one a year? All the formulas for PV of a single cash ßow, FV of a single cash ßow, PV of an annuity, FV of an annuity, PV of a perpetuity are still valid. However, we need to restate ÔrÕ and ÕtÕ to give: ! • The interest rate per period (for ÔrÕ), and ! • The number of periods (for ÕtÕ) ! Calculating ÔrÕ for a period less than a year ! Remember that ÔrÕ is the interest rate for a PERIOD. Also, market convention is that the quoted rate (QR) is always an annual rate unless stated otherwise. So: ! r = QR / m ! Where, QR = quoted rate ! m = the number of times the interest rate is compounded in one year ! Example 7 (a) Interest compounds semi-annually. That is, 2 times in one year. So:! m = 2! r = 15% / m! = 15% / 2! = 7.50% !   In time value of money calculations, we therefore use r = 0.075. ! (b) Interest compounds monthly. That is, 12 times in one year. So: ! m = 12 ! r = 15% / 12 ! = 1.25% ! In time value of money calculations, we therefore use r = 0.0125.! Calculating ÔtÕ where compounding is for periods of less than a year ! In time value of money formulas, ÔtÕ is the number of PERIODS. So:! t = years x m ! Where: m = the number of times the interest rate is compounded in one year. ! Example 8 You have invested £1,000 for 5 years in a deposit account that pays annual interest of 4%, compounded quarterly. What is the value of your investment at the end of 5 years? ! r = 4% / 4 ! = 1% = 0.0100! where m = 4 t = years x m ! = 5 x 4! = 20 periods FV = PV x (1+r)t ! = £1000 x (1.01)20 ! = £1,000 x 1.22019 ! = £1,220.19 ! Comparing rates using EAR Rates! Nominal Interest Rate Ñ the interest rate expressed in terms of the interest payment made each period. Also known as the state or quoted interest rate. ! Effective Annual Percentage Rate (EAR) Ñ the interest rate expressed as if it were compounded once per year. ! EAR = [1+ (quoted rate/m)]m Ñ 1! Example 9 Which of these is the best if you are thinking of opening a savings account? Which of these is best if they represent loan rates?!   a) Bank 1 offers 15% compounded daily ! EAR = [1+ (Quoted rate / m)]m Ð 1 ! = [1+ (0.15 / 365)]365 Ð 1 ! = 16.18% ! b) Bank 2 offers 15% compounded monthly ! EAR = [1+ (Quoted rate / m)]m Ð 1! = [1+ (0.15 / 12)]12 Ð 1! = 16.08% ! c) Bank 3 offers 15.5% compounded quarterly ! EAR = [1+ (Quoted rate / m)]m Ð 1! = [1+ (0.155 / 4)]4 Ð 1! = 16.42% ! d) Bank 4 offers 15.5% compounded semi-annually ! EAR = [1+ (Quoted rate / m)]m Ð 1! = [1+ (0.155 / 2)]2 Ð 1! = 16.10% ! e) Bank 5 offers 16% compounded annually ! EAR = 16%! Answer: depositor: c, a, d, b, e! borrower: e, b, d, a, c! Loan types and loan amortisation ' Loan types ! Whenever a lender extends a loan, some provision will be made for repayment of the principal (the original loan amount). A loan might be repaid in equal instalments, for example, or it might be repaid in a single lump sum. There are an unlimited number of possibilities: ! Pure discount loans Ñ where the borrower receives money today, and repays a single lump sum at some time in the future (e.g. some Þnancial instruments such as treasury bills and commercial paper). ! Interest-only loans Ñ where the borrower pays interest each period, and reaps the whole principal (the original loan amount) at some point in the future (e.g. bonds). Note that if there is only one period, a pure discount loan and an interest-only loan are the same. ! Amortised loans Ñ where the borrower repays parts of the loan amount over time (e.g. mortgages). The process of providing a loan to be paid off by making regular principal reductions is called amortising the loan. !   Example 10 A T-bill (treasury bill) is a promise by the US government to repay a Þxed amount at some time in the future Ñ for example, in 3 months or 12 months. If a T-bill promises to repay £10,000 in 12 months, and the market interest rate is 7%, how much will the bill sell for in the market? ! Current market price = Present value! = £10,000/1.07! = £9,345.79 ! Amortised loans ! There are two approaches: ! • Fixed principal payment Ñ repay the principal in equal instalments plus all interest due on the outstanding amount. ! • Fixed total payment Ñ the borrower makes a single, Þxed payment every year. Most consumer loans (e.g. car loans) and mortgages are amortised following this approach. The payment includes repayment of principal and also interest. ! Example 11: an amortised loan with Þxed annual payment You borrow £10,000 for four years. The bank will lend you the money at 8% interest rate and will require that the loans are paid-off in four equal (end-of-year) instalment payments. ! a) What are the annual loan payments that the company will have to make to re-pay the loan? ! Present value of annuity = C x {1Ñ[1/(1+r)t] }! r! Where: ! r = 0.08 ! t = 4 periods ! PV (loan) = £10,000! So:! £10,000 = (C/0.08) x (1 Ñ 1/1.084) ! £10,000 = C x 3.31213 ! C = £10,000 / 3.31213 ! = £3,019.21 ! b) How much interest and principal will be repaid each year? ! Period Beginning*Balance Payment Interest 1 £10,000.00 £3,019.21 £800.00 2 £7,780.79 £3,019.21 £622.46 3 £5,384.04 £3,019.21 £430.72 4 £2,795.57 £3,019.21 £223.65 Total £12,077 £2,077 Principal Ending*Balance £2,219.21 £7,780.79 £2,396.75 £5,384.04 £2,588.49 £2,795.56 £2,795.56 £0.00 £10,000   Interest payment period 1 = 10,000 x 8% = £800 ! Principal payment period 1 = 3,019.21 Ñ 800 = £2,219.21 ! Ending balance period 1 = 10,000 Ñ 2,219.21 = £7,780.79! Example 12: an amortised loan with Þxed monthly payment Suppose we have a €100,000 commercial mortgage with an effective monthly interest rate of 1% and a 20 year (240-month) amortisation. What will the Þxed monthly payment be for the 20 year period? ! The monthly payment can be calculated based on an ordinary annuity with a present value of €100,000. ! Where:! r = 0.01! t = 240 periods ! PV (loan) = €100,000 ! Present value of annuity = C x {1Ñ[1/(1+r)t] }! r! So:! €100,000 = C x [(1 Ñ 1/1.01240)/.01]! €100,000 = C x 90.8194 ! C = €100,000 / 90.8194 ! = €1,101.09 ! Answer: the regular monthly payment needed on the loan is €1,101.09. ! Note: your amortisation table will have one period per line instead of one year per line. ! There are two ways to determine the balloon payment: 1. Hard way Ñ amortise the loan for 60 months to see what the balance is at that time. ! 2. Easy way Ñ recognise that after 60 months, we have a 240 x 60 = 180 month loan. Th payment is still €1,101.09 per month, and the interest is still 1% per month. ! The loan balance is the present value of the remaining payments: ! Loan balance: ! = €1,101.09 x [(1 Ñ 1/1.01180)/.01]! = €1,101.09 x 83.3217 ! = €91,744.69!  