Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014

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1 1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases

2 Learig objectives 2 Uderstad the time-value of moey Draw a timelie illustratig a give set of cash flows List ad describe the three rules of time travel Fid the et preset value of a stream of cash flows Calculate the preset value ad future value of perpetuities ad auities

3 Timelie of cash flows 3 Cocept A timelie is a liear represetatio of the timig of potetial cash flows Drawig a timelie of the cash flows will help you visualize the fiacial problem Assume that you made a loa to a fried. You will be repaid i two paymets, oe at the ed of each year over the ext two years

4 Timelie of cash flows 4 Differetiate betwee two types of cash flows Iflows are positive cash flows Outflows are egative cash flows, which are idicated with a (mius) sig Assume that you are ledig $10,000 today ad that the loa will be repaid i two aual $6,000 paymets The first cash flow at date 0 (today) is represeted as a egative sum because it is a outflow Timelies ca represet cash flows that take place at the ed of ay time period a moth, a week, a day, etc

5 Timelie of cash flows 5 Example

6 Three rules of time travels 6 Fiacial decisios ofte require combiig cash flows or comparig values. Three rules gover these processes

7 Three rules of time travels 7 Explaatio of the first rule A dollar today ad a dollar i oe year are ot equivalet It is oly possible to compare or combie values at the same poit i time Which would you prefer: A gift of $1,000 today or $1,210 at a later date? To aswer this, you will have to compare the alteratives to decide which is worth more. Oe factor to cosider: How log is later?

8 Three rules of time travels 8 Explaatio of the secod rule To move a cash flow forward i time, you must compoud it Suppose you have a choice betwee receivig $1,000 today or $1,210 i two years. You believe you ca ear 10% o the $1,000 today, but wat to kow what the $1,000 will be worth i two years. The time lie looks like this: Future value of a cash flow FV C (1 r) (1 r) (1 r) C (1 r) times

9 Three rules of time travels 9 Explaatio of the secod rule Example 1 (costat cash flows): Suppose you have a choice betwee receivig $5,000 today or $10,000 i five years. You believe you ca ear 10% o the $5,000 today, but wat to kow what the $5,000 will be worth i five years Timelie $5,000 x 1.10 $5, 500 x 1.10 $6,050 x 1.10 $6,655 x 1.10 $7,321 x 1.10 $8,053 I five years, the $5,000 will grow to: $5,000 (1.10) 5 = $8,053 The future value of $5,000 at 10% for five years is $8,053 You would be better off forgoig the gift of $5,000 today ad takig the $10,000 i five years

10 Three rules of time travels 10 Explaatio of the secod rule Example 2 (icreasig cash flows): Suppose we pla to save $1000 today, ad $1000 at the ed of each of the ext two years. If we ca ear a fixed 10% iterest rate o our savigs, how much will we have three years from today?

11 Three rules of time travels 11 Explaatio of the secod rule Compositio of iterest over time: origial ivestmet ($1,000), iterest rate (10%), ivestmet horizo (20 years)

12 Three rules of time travels 12 Explaatio of the third rule To move a cash flow backward i time, we must discout it Preset value of a cash flow PV C (1 r ) Example C (1 r )

13 NPV of a stream of cash flows 13 Valuig a stream of cash flows Based o the first rule of time travel, we ca derive a geeral formula for valuig a stream of cash flows If we wat to fid the preset value of a stream of cash flows, we simply add up the preset values of each PV formula PV N PV ( C ) 0 0 N C (1 r)

14 NPV of a stream of cash flows 14 Calculatig the NPV of a stream of cash flows Calculatig the NPV of future cash flows allows us to evaluate a ivestmet decisio. Net Preset Value compares the preset value of cash iflows (beefits) to the preset value of cash outflows (costs). NPV formula Example NPV I 0 N C 0(1 r)

15 Perpetuities, auities ad special cases 15 Perpetuities Whe a costat cash flow will occur at regular itervals forever it is called a perpetuity Preset value of a perpetuity Example ( i perpetuity) C PV C r

16 Perpetuities, auities ad special cases 16 Preset value of auities Whe a costat cash flow will occur at regular itervals for a fiite umber of N periods, it is called a auity Preset value of a auity C C C C N C 1 (1 r) PV... C 2 3 N r r r r ( 1 ) (1 ) (1 ) (1 ) 1(1 r) r Example: What amout must you ivest today at 6% compouded aually so that you ca withdraw $5,000 at the ed of each year for the ext 5 years? N

17 Perpetuities, auities ad special cases 17 Growig perpetuities Assume you expect the amout of your perpetual paymet to icrease at a costat rate, g Preset value of the growig perpetuities PV (growig perpetuity) Example: what s the preset value of a ivestmet that brigs a cash flow of $500 today, ad cotiues to geerate aual cash flows forever with a growig rate of 5% if the iterest rate is 8%? r C g

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