# I. Why is there a time value to money (TVM)?

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1 Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios the tricks I. Why is there a time value to moey (TVM)? Which would you prefer: a cheque from me to you for \$,, dated today or a cheque from me to you for \$,, dated oe year from ow Why? What does the TVM mea?

2 Time Value alculatios Basics so far: Sigle cash flow over oe period PV ( + r) FV ( + r) Objective: Exted to multiple, or, periods II. The Basic Time-Value-of-Moey Relatioships for a Sigle ash Flow where FV t+ t ( + r) PV (+ r) t+ t r is the iterest rate per period is the duratio of the ivestmet, stated i the time uits of the period for r t is the cash flow at period t t+ is the cash flow at period t+ FV t+ is the future value at time period t+ PV t is the preset value at time period t 2

3 Ituitio for the Future Value formula Future Value of \$ ivested for 2 years at 8% per year compouded yearly Year 2 FV 2 (+r) 2 \$ (+.8) 2 \$.664 Note: simple iterest is just 6 over the two years but compoud iterest is 6.64 which icludes the simple iterest plus iterest o iterest. The Power of ompoudig Give a iterest rate of 8% per year ad a iitial \$, ivestmet, compare the compoud iterest i the 2 d year ad the 2 th year. What is the total compoud iterest over the 2 year ivestmet? Explai the power of compoudig 3

4 Ituitio for the Preset Value Formula Preset value of \$ received i 2 years give a discout rate of 8% per year. Year 2 PV 2 (+r) 2 \$ (+.8) 2 \$.8573 Practice exercises Fid the value i 3 years of \$, received i 8 years. Fid the value i years of \$25, ivested i 2 years. How may years was the moey ivested if \$, grew to \$,? If you ivested \$5 for 7 years ad ow have \$, how much did your ivestmet retur? 4

5 Housekeepig fuctios:. Set to 8 decimal places: TVM i your HP B alculator Yellow DISP 8 2. lear previous TVM data: Yellow ALL 3. Set paymet at Begiig/Ed of Period: Yellow MAR BEG/END Always keep set o Ed the display should ever say Begi 3. Set # of times iterest is calculated (compouded) per year to : Yellow PMT P/YR Usig the HP B TVM fuctios N umber of periods or umber of paymets I/YR the effective iterest rate or discout rate per period PV preset value or a preset cash flow PMT repeatig cash flow paymets (ot yet discussed) FV future value or a future cash flow 5

6 III. Groups of ash Flows osider the followig series of cash flows: Year 2 3 \$, \$,5 \$,2.5 \$,55.33 What is PV ; what is FV? We could apply our PV ad FV formulae to each idividual cash flow but that would be too paiful! Mathematics of Perpetuities ad Auities Fortuately, math provides a simplified way osider a growig perpetuity: Goes o forever 2 3 t \$, \$,5 \$,2.5 2 (+g) 3 (+g) 2 t (+g) t- 6

7 Sum of a ifiite series PV of the growig perpetuity is mathematically equivalet to the sum of a ifiite geometric series. The sum is defied ad is fiite as log as the PV of each subsequet cash flow is a fractio (less tha ) of the PV of the previous cash flow. I.e., as log as r > g PV of a growig perpetuity This sums the PV s of each idividual cash flow i the growig perpetuity. is the first cash flow PV r g PV is the PV oe period before the first cash flow This formula is oly correct whe r > g. If r g or if r < g, the the PV is ifiite. 7

8 Growig Perpetuity Example To service the curret atioal debt, the govermet plas to make the followig series of paymets begiig i oe year ad cotiuig i perpetuity: \$8 billio iitially ad the growig by 4% each year. The iterest rate o log-term debt is 6%. What is the PV of these paymets? Note: the PV of all future debt paymets is equal to the pricipal amout curretly outstadig. PV of a Growig Auity A auity is a fiite series of cash flows i.e., a series that has a ed assume the ed as at time. We ca determie the PV of the growig auity by subtractig off the latter part from a growig perpetuity (+g) (+g) - + (+g) 8

9 PV of a Growig Auity so the PV of the growig auity is just the PV of the whole growig perpetuity mius the PV of the latter part of the growig perpetuity. The latter part of the growig perpetuity is just aother growig perpetuity that starts at a later time with a differet iitial cash flow. PV of a Growig Auity PV r g r g + ( + r) Subtract off the PV of the latter part of the growig perpetuity PV r g ( + g) r g ( + r) PV of the whole growig perpetuity PV ( + g) r g ( + r) PV is the PV oe period before the first cash flow 9

10 Growig Auity Example I 2 years you pla o retirig ad you would like icome each year that grows at the expected iflatio rate of 3%. You desire your year 2 icome to be \$5, ad you expect to eed 3 years of retiremet icome. If you are cofidet your savigs will ear 8% per year, how much do you eed saved by year 2? FV of a Growig Auity If PV discouts all the cash flows to time zero ad sums up the discouted amouts the FV, the future value of all the cash flows take to time, is just PV (+r) ( + g) FV ( + r) r g ( + r) [( + r) ( g ] FV ) r g + I effect, this takes all cash flows of the growig auity, icludig the last cash flow, forward to the time period of the last cash flow

11 FV of Growig Auity Example Give your retiremet plas of the previous example, how much do you eed to save each year begiig i oe year ad edig with year 9? Assume your savigs will ear 8% ad you icrease your cotributios by 6% each year. Simple (o-growig) series of cash flows For costat auities ad costat perpetuities, the time value formulas are simplified by settig g. We ca use the PMT butto o the fiacial calculator for the auity cash flows, PV r PV FV r (+ r) r regular perpetuity [(+ r) ] regular auity

12 Regular Auities & Perpetuities Examples Your father loaed you \$2, as your dow paymet o your ew house. If you repay him i equal amouts of \$2,6 each of the ext years, what rate of iterest are you, i effect, payig him? Regular Auities & Perpetuities Examples You have wo the lottery ad are offered cash paymets of \$ millio per year for the ext 2 years (first paymet is oe year from today). If you could ivest at a rate of %, how much as a sigle lump sum would you be willig to receive today i exchage for the 2 yearly cash flows? 2

13 Regular Auities & Perpetuities Examples You have just doated to the Uiversity of Maitoba ad your doatio stipulates that the Uiversity must sped the icome eared from your doatio each year. If your doatio is \$ millio ad it ears a 6% rate of retur, how much ca be spet each year ad for how log ca this cotiue? IV. Some fial warigs Eve though the time value calculatios look easy there are may potetial pitfalls you may experiece Be careful of the followig: PV of auities or perpetuities that do ot begi i period ; remember the PV formulas give always discout to exactly oe period before the first cash flow. If the cash flows begi at period t, the you must divide the PV from our formula by (+r) t- to get PV. Note: this works eve if t is a fractio. 3

14 Be careful of auity paymets out the umber of paymets i a auity. If the first paymet is i period ad the last is i period 2, there are obviously 2 paymets. How may paymets are there if the st paymet is i period 2 ad the last paymet is i period 2 (aswer is use your figers). How about if the st paymet is ow (period ) ad the last paymet is i period 5 (aswer is 6 paymets). If the first cash flow is at period t ad the last cash flow is at period T, the there are T-t+ cash flows i the auity. Be careful of wordig A cash flow occurs at the ed of the third period. A cash flow occurs at time period three. A cash flow occurs at the begiig of the fourth period Each of the above statemets refers to the same poit i time! If i doubt, draw a time lie. 4

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