Time Value of Money Lectures 3 & 4 PDF

Topic 2 Time Value of Money Lectures 3 & 4 1 Aims and Objectives Introduction Compound interest and future values Discounting and present values Valuing multi-period cash flows Special cash flow patterns - annuities and perpetuities Compounding periods and effective annual rates Non-constant interes...

Topic 2 Time Value of Money Lectures 3 & 4 1 Aims and Objectives Introduction Compound interest and future values Discounting and present values Valuing multi-period cash flows Special cash flow patterns - annuities and perpetuities Compounding periods and effective annual rates Non-constant interest rates Real versus nominal interest rates 2 1.Introduction £100 now is worth more than £100 in one year’s time Eg. If you can invest £100 at 10% per year, £100 now grows to £110 in one year  at 10%, £110 in one year has the same value as £100 now {Why is the interest rate 10% (or whatever it is)?} For most of this topic, assume period = 1 year cash flows occur at the end of the year 3 2. Compounding and Future Values Future value is the amount a sum, of money grows to as, it earns interest over a period Example: Single Period Invest £1000 at 12% per year. What is the future value at the end of one year? FV = £1000 x (1 + 0.12) = £1120  principal of £1000 plus interest of £120 NB. 12%  0.12; ie. 0.12  12% In calculations we use 0.12 not 12 Interest rate = r per year  investment grows by, 1 + r per £ invested 4 Example: Compound interest What is the future value of £1,000 invested for 2 years at 12% per year? After one year you have £1,120 This is invested for a second year at 12% FV = £1120 x 1.12 = £1000 x 1.12 x 1.12 = £1254.4 The second period increase is £1254.4 - £1120 = £134.4 This comprises, 12% on original £1000 120.0 12% on 1st period interest of £120 14.4 ------£134.4 5 Compound interest each period’s interest earns interest in future periods If you invest at an interest rate of r per year for t years, the future value per £ invested is, (1 +r) x (1 + r) x….x (1 + r) = (1 +r)t Example: Compound interest - t periods The future value of £1000 invested at 0.12 for 6 years is, FV = £1000 x (1.12) 6 = £1973.8 simple interest - only the original principal (£1,000) earns interest 6 Note the effect of compound interest r 10% 20% Interest t= 1 100 200 2x on £1000 t= 5 610.5 1,488.0 2.4x after t years (£) t= 10 t= 20 1,593.7 5,727.5 5,191.7 37,337.6 3.3x 6.5x 20% FV 10% t 7 Discounting and Present Values What is the value now of £2,000 in 5 years, at 11% per year? Equivalent, how much do we need to invest now at 11% to give £2,000 after 5 years? Example:Single Period Discounting How much do we need to invest now at 11% to give £2,000 after one year? This is the present value (PV) of £2,000 in one year’s time at 11% We know, PV x 1.11 = £2000 and PV = £2000 = £1801.8 1.11 Example: Discounting – t periods Say you want £2,000 in 5 years’ time. We have PV x (1.11)5 = £2000  PV = £2000 = £1186.9 (1.11)5 8 4. Solving for r and t FV = PV (1 + r)t Four variables: FV, PV, r, t, given any 3 you can solve for the 4th! r = [FV/PV]1/t – 1; t = lnFV – lnPV = ln (FV/PV) ln(1 + r) ln(1 + r) Example: 1. If your investment doubles in 5 years what is r? r= (2/1)1/5 – 1 = 0.1487 = 14.87% 2. At r =30% how long to double your money? t= ln(2/1) / ln(1.3) = 2.64 years 9 5. Multiple cash flows Example: r = 0.09 with cash flows at the end of year 1 4 6 £2500 £900 £3600 What is the PV of the series and what is the FV after seven years? PV = 2500 + 900 + 3600 = 5077.7 1.091 1.094 1.096 FV = 2500(1.09)6 + 900(1.09)3 + 3600(1.09)1 = 9282.3 = PV (1 + r)t =5077.7 (1.09)7 = 9282.3 10 6. Special Patterns: Annuities and Perpetuities Annuity: a series of regular and constant cash flows for a period! What is the PV of a series of cash flows C at the end of each of the next t years at r per year? (a t year annuity of C at r)? PV = C + C + C + …+ C + (1 +r)1 (1 +r)2 (1 +r)3 (1 +r)t-1 C (1 +r)t (1) Multiply throughout by (1+r) (1+r)PV = C + C + C + C + …+ C (1 +r)1 (1 +r)2 (1 +r)3 (1 +r)t-1 subtract (1) from (2) rPV = C - C (1+r)t  PV = C x 1 – {1/(1+r)t} r (2) 11 Example: Annuities 1. What is the PV of a six year annuity of £4000 at r = 8%? PV = C x 1 – {1/(1+r)t} = 4000 x 1 – {1/(1.08)6} r 0.08 = £18,491.5 2. You borrow £30,000 now at an interest rate of 12% and plan to repay the loan using a ten-year annuity. What is the annual repayment? 30,000 = C x 1 – {1/(1.12)10}  C = 30,000 0.12 5.65 = £5,309.5 12 Perpetuity: an annuity where the cash flows continue forever! PV = C + C + C + ……….. (3) (1 +r)1 (1 +r)2 (1 +r)3 multiply throughout by (1+r) (1+r)PV = C + C + C + C + …………. (1 +r)1 (1 +r)2 (1 +r)3 (4) subtract (3) from (4) rPV = C  PV = C r 13 7.Compounding Periods, Effective Annual Rates You invest £100 for 3 years at a flat rate of 10% per year with semiannual compounding. You get 5% interest every six months! Effective Annual Rate Investing £100 for a year at a flat rate of 10%, we get, with (a) annual compounding 100 x (1.1) = 110  £10 interest (b) semi-annual compounding 100 x (1 +0.1)2 = 110.25  £10.25 interest 2 (c) monthly compounding 100 x (1 +0.1)12 12 = 110.47  £10.47 interest 14 The effective annual rate corresponding to a flat rate of 10% is, Compounding period semiannual monthly weekly EAR 10.25% 10.47% 10.51% With compounding m times per year at a flat rate r, EAR = (1 + __r_ )m – 1 m r = m x [ (1 + EAR)1/m - 1 ] Investing C for t years at a flat rate of r per year with compounding m times per year, the future value is FV = C (1 +__r__ ) m mxt 15 Cash flows occurring during the year: Effective annual rates are easier to use for discounting and compounding for periods other than multiples of a year. Assume an EAR of 12.5%. The PV of £ 25000 at the end of 3 months is, ___25000 __= 24274.6 1/4 (1.125) The PV of £25,000 in 3½ years is, __25000 __= 16554.1 3.5 (1.125) Annual Percentage Rates: in the UK APR = EAR 16 A practical example www.wonga.com (now defunct) used to provide short term loans in the UK. At one point of time the advertised APR was around 4200%. Borrow £100 and make no repayments After 1 year Owe £4,200 After 2 years Owe almost £180,000 After 3 years Owe around £7,500,000 (£7 million) After 4 years Owe around £315,000,000 (£315 million) After 5 years Owe around £13,000,000,000 (£13 billion) After 6 years Owe around £557,000,000,000 (£557 billion) After 7 years Owe around £23,500,000,000,000 (£23.5 trillion) 17 8.Non-constant Interest Rates Example: You invest £100 and earn 11% during year 1, 9% during year 2, and 13% during year 3. FV after 3 years? FV = 100 x (1.11) x (1.09) x (1.13) = 136.72 What is the PV of £100 in 4 years’ time if interest rates are 8% (year 1), 12% (year 2), 6% (year 3), and 13% (year 4)? PV = __________100______________ = 69.02 (1.08) x (1.12) x (1.06) x (1.13) 18 9.Real v. Nominal Interest Rates Nominal interest rates = actual market rates Invest £100 for a year at 10%  £110 What if inflation over the year is 7%? You need £107 to be as well off at the end as you were at the start. What is your real return? If a Sunday paper costs £1 at the start, it costs £1.07 at the end. At the start, with £100, you could buy 100 copies. At the end, with “110, you can buy 110 = 102.8 copies 1.07 The real return is 2.8%. 19 1 + real rate = 1 + nominal rate 1 + inflation rate £110 is the nominal cash payoff real cash payoff is £110/1.07 = £102.8 To find the PV 1. discount nominal cash flows at a nominal rate 110= £100 1.1 2. discount real cash flows at a real rate 102.8= £100 1.028 real rate  nominal rate - inflation rate 20 Summary FV is the amount a sum of money grows to over a period of time. The process of calculating the FV is called compounding. PV is the reverse of FV and is calculated through discounting FV = PV (1 + r)t Annuities and perpetuities are special cash flow patterns. Increasing the compounding frequency results in a higher effective interest rate over the year called the EAR or the APR. Interest rates change frequently- non–constant interest rates. Real and nominal interest rates: real rate  nominal rate - inflation rate 21