# Time Value of Money. Background

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1 Time Value of Money (Text reference: Chapter 4) Topics Background One period case - single cash flow Multi-period case - single cash flow Multi-period case - compounding periods Multi-period case - multiple cash flows Perpetuities Annuities Mortgages Amortization schedules AFM Time Value of Money Slide 1 Background the economic value of a cash flow depends on when it occurs (i.e. the timing of the cash flow) \$1 today > \$1 tomorrow > \$1 a year from now >... present value calculations allow us to determine the value today of a stream of cash flows to be rec d/paid in the future, by taking into account the time value of money this is an important concept which is widely used both in corporate and in personal financial decision-making AFM Time Value of Money Slide 2

2 notation: PV = present value = value today of a stream of cash flows FV = future value = value at some future time of a stream of cash flows C = cash flow r = interest rate (a.k.a. discount rate) T = number of years m = number of periods in a year n = total number of periods (n = m T) we will often apply subscripts to indicate time, e.g. C t is a cash flow occurring at time t AFM Time Value of Money Slide 3 One Period Case - Single Cash Flow cash flow assumptions: timing and size are given in all our examples, and for now there is no risk as long as we measure investment alternatives at the same point in time, we can rank them consistently example: you have \$1,. An investment costs C = \$1, today, and returns C 1 = \$14, in a year. A bank offers a 5% annual interest rate on deposits. Should you make the investment (option A) or put your money in the bank (option B)? FV analysis: PV analysis: PV = C 1 /(1 + r) AFM Time Value of Money Slide 4

3 net present value: NPV = (PV of future cash flows) - (cost of investment today) NPV < don t invest (since the cost of the investment today exceeds the value today of all its future cash flows) calculate NPV for options A and B: observations: PV analysis and FV analysis both yield the same conclusion; the difference is only the time at which the cash flows are compared NPV also gives the same conclusion (A and B cost the same, so we are really just comparing their PVs) AFM Time Value of Money Slide 5 another example: a piece of land costs \$2, and will be worth \$21,5 a year from now. The bank pays a 4% annual interest rate. Should you invest? PV analysis: FV analysis: NPV analysis: AFM Time Value of Money Slide 6

4 Multi-Period Case - Single Cash Flow two types of interest: simple interest: interest earned only on original principal FV after 1 yr = C +C r FV after 2 yrs = C +C r +C r. n terms { }} { FV after n yrs = C + C r +C r + +C r = C (1 + n r) AFM Time Value of Money Slide 7 compound interest: interest earned on original principal and on previously earned interest FV after 1 yr = C (1 + r) FV after 2 yrs = C (1 + r) (1 + r). n terms { }} { FV after n yrs = C (1 + r) (1 + r) (1 + r) = C (1 + r) n AFM Time Value of Money Slide 8

5 the power of compounding (\$1 deposited at 6% and 1%) Time (years) % simple % compound % simple % compound other examples: text p. 85: Julius Caesar lent one penny to someone; assuming 6% annual interest, what would be owed on this loan 2, years later? the island of Manhattan was purchased in 1626 for the equivalent of \$24; what would this amount be worth in 25, assuming 5% annual interest? AFM Time Value of Money Slide 9 discounting moves a cash flow that is expected to occur in the future back to today (i.e. finding PV) compounding moves a cash flow from present to future value amounts (i.e. finding FV) cash flows at different points in time cannot be compared or aggregated, unless first brought to the same point in time (via discounting and/or compounding) example: C = 12, C 5 = 13. Adding these up to 25 is meaningless (like adding 12 + \$13), but we can find the value of the combined CFs at any point in time by discounting and/or compounding (like using exchange rates for currency conversion). AFM Time Value of Money Slide 1

6 some examples: how much must be invested today in order to receive \$1, in 5 years if interest is compounded at 7% per annum? what is the annual compound interest rate equivalent to a simple interest rate of 6% for a five year investment? suppose a bank s annual compound interest rate is 4%. What is the PV of \$1 invested for 5 years at 4% simple interest? AFM Time Value of Money Slide 11 Multi-Period Case - Compounding Periods stated annual rate: (SAR) not the whole story unless the compound frequency is given effective annual rate: (EAR) if compounding occurs m times per year, then EAR = (1 + SAR/m) m 1 example: SAR = 1%, C = \$1 annual compounding FV 1 = \$1 (1 +.1) = \$1.1 EAR = 1% semi-annual compounding FV 1 = \$1 (1 +.1/2) 2 = \$1.125 EAR = 1.25% quarterly compounding FV 1 = \$1 (1 +.1/4) 4 = \$1.138 EAR = 1.38% AFM Time Value of Money Slide 12

7 we can also have monthly, weekly, daily compounding, etc. example: Mastercard statement says that the annual interest rate is 18.4%, the daily interest rate is.541%, and interest is compounded daily. What is EAR? ) m T in the limit: lim m (1 + SAR m = e SAR T this is called continuous compounding: FV 1 = \$1 e.1 = \$1.152 EAR = 1.52% why does EAR increase as compounding period decreases (i.e. as compounding frequency increases)? AFM Time Value of Money Slide 13 converting between compounding frequencies: e.g. 1% compounded semi-annually is equivalent to what rate compounded weekly? compounding over several years at non-annual compounding frequencies: FV T = C (1 + SAR/m) m T FV of \$1 invested for 6 years at 5% compounded monthly: continuous compounding over many years: FV T = C e SAR T C 7 = \$1,, what is PV today if interest rate is 5.5% compounded continuously? AFM Time Value of Money Slide 14

8 Multi-Period Case - Multiple Cash Flows time line: t = t = 1 t = 2 t = 3... t = k... t = n C C 1 C 2 C 3... C k... C n t = k implies end of year k, beginning of year k + 1 PV = C + C 1 1+r + C 2 (1+r) 2 + C 3 (1+r) C n (1+r) n e.g. if r = 6%, calculate PV of the following cash flows: t = t = 1 t = 2 t = 3 -\$1 \$15 -\$8 \$3 AFM Time Value of Money Slide 15 Summarizing to here: simple interest: FV = C (1 + n r) discrete compounding/discounting: ( FV T = C 1 + SAR ) mt PV = m C T ( 1 + SAR m = C (1 + r) n = C T (1 + r) n continuous compounding/discounting: FV T = C e rt, PV = C T e rt ) mt with multiple CFs, treat each one separately and add (as in the example on slide 15). There are no convenient formulas if cash flows vary in general, but there are for nice cash flows. AFM Time Value of Money Slide 16

9 Perpetuities a perpetuity is a stream of equal cash flows that occur every period forever, 1st payment one period from now: t = t = 1 t = 2 t = 3 t = 4 t = 5 C = C 1 = C C 2 = C C 3 = C C 4 = C C 5 = C PV of a perpetuity is PV = = C r C (1 + r) + C (1 + r) 2 + C (1 + r) 3 + e.g. if r = 3%, find PV of a perpetuity paying \$2/year: AFM Time Value of Money Slide 17 a growing perpetuity is a stream of cash flows that occurs every period forever, has 1st payment one period from now, and grows at a rate of g per period: C C(1 + g) C(1 + g) 2 C(1 + g) 3 PV of a growing perpetuity is PV = C (1 + r) = C (r g) + C(1 + g) (1 + r) C(1 + g)2 C(1 + g) (1 + r) 3 (1 + r) 4 + AFM Time Value of Money Slide 18

10 the formula above is valid only if g < r e.g. you wish to purchase a preferred share of ABC Co. The share is expected to pay a dividend of \$2.5 next year, thereafter growing at a rate of 3%. Assume an interest rate of 8%. How much should you be prepared to pay for the share? (Assume that you will never receive your initial capital back.) note that we could have g < (a decreasing perpetuity) - reconsider the example above but assume dividends will decline at a rate of 2% per year forever: AFM Time Value of Money Slide 19 Annuities ordinary annuity (a.k.a. annuity in arrears): a stream of equal cash flows that occur every period, for a specified number of periods, at the end of each period. Examples include car leases, mortgages, pensions, etc. Our PV formula below (slide 22) assumes that the 1st cash flow is one period from now. e.g. find PV of an annuity paying \$1, per year for 4 years, r = 7%: , 1, 1, 1, AFM Time Value of Money Slide 2

11 e.g. find FV at the end of year 4 of an annuity paying \$1, per year for 4 years, r = 7%: , 1, 1, 1, AFM Time Value of Money Slide 21 a simplifying formula for the PV of an annuity - find a formula for the PV of an ordinary annuity paying C for a total of n periods, assuming an interest rate of r per period: PV of annuity = C 1 (1 + r) n r AFM Time Value of Money Slide 22

12 a simplifying formula for the FV of an annuity - find a formula for the FV (at the end of n periods) of an ordinary annuity paying C for a total of n periods, assuming an interest rate of r per period: FV of annuity after n periods = C (1 + r)n 1 r exercise: use the formulas on slides 22 and 23 to verify the calculations on slides 2 and 21 AFM Time Value of Money Slide 23 note that our PV formulas for annuities and perpetuities all assume that the 1st payment is one period from now and that the rate of interest per period is r ordinary annuity examples: you have an outstanding balance on your Visa account of \$2,4. If you can only afford to repay at the rate of \$15 per month, how long will it take you to repay the entire amount if the interest rate on the outstanding balance is 24% compounded monthly? AFM Time Value of Money Slide 24

13 redo the previous example, but assume that the interest rate is 24% compounded semi-annually: you are 25 years old and wish to have savings of \$2, by the time you retire on your 55th birthday. You are willing to put money aside for next 2 years. Assuming an interest rate of 6% until your retirement, how much must you set each year? AFM Time Value of Money Slide 25 annuity due (a.k.a. annuity in advance): payments are at the start of each period PV annuity due = (1 + r) PV ordinary annuity PV(FV) of annuity due > PV(FV) of ordinary annuity e.g. Sue has won a lottery, which pays \$25, per year for 8 years, starting today. Calculate PV today and FV (in 8 years) of the lottery payout (assume r = 5%). AFM Time Value of Money Slide 26

14 delayed annuity: payments start after a delay by a certain number of periods this involves a two step calculation; e.g. Bill will graduate in 4 years, and will thereafter earn an annual income of \$75, for 35 years (assume all income is received at end of a year). Calculate the PV of Bill s lifetime earnings (assume r = 4%). AFM Time Value of Money Slide 27 infrequent annuity: payments occur less often than once per year this also involves a two step calculation; e.g. you plan to purchase a new \$35, car every 5 years for next 3 years starting in 5 years. Calculate the PV of your car purchases (assume r = 6.5% per annum). AFM Time Value of Money Slide 28

15 growing annuity: like an ordinary annuity, but payments grow at a rate of g per period n n + 1 C C(1 + g) PV = C C(1 + g) 2 [ 1 (1+g) (1+r) r g C(1 + g) n 1 ] n e.g. XYZ Fund will pay out distributions over next 1 years. The first distribution, \$8 per unit, will be paid out a year from today. Subsequent distributions will grow at a rate of 8%. No further cash flow is expected once the ten distributions have been paid out. Find the PV of one fund unit (assume r = 11%). AFM Time Value of Money Slide 29 Mortgages many varieties, but these are basically annuities with monthly payments, semi-annual compounding, a 25 year maturity (usually), and a term (typically 5 years) less than the maturity e.g. find the monthly payment on a \$225, mortgage, assuming the interest rate on the initial 5 year term is 6% find the monthly payment after the initial 5 year term, if the interest rate changes to 8% AFM Time Value of Money Slide 3

16 Amortization Schedules applicable to loans being paid off over a number of periods, with constant monthly payments (e.g. mortgages, leases) the loan is said to be amortized over the loan period does not apply to debt where only interest is paid periodically, with principal repaid as a lump sum at maturity (e.g. corporate bonds) see spreadsheet handout (try to recreate it yourself) note that as time passes: principal balance decreases (eventually to zero when the last payment is made) interest portion of each payment becomes smaller principal portion of each payment becomes larger AFM Time Value of Money Slide 31

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