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← Introduction to Present Value

A choice between money now and money later.

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Showing Revision 6 created 10/13/2011 by eko subagyo.

  1. We'll now learn about what is arguably the most useful concept in finance
  2. and that is called the present value.
  3. And if you know the present value
  4. then it's very easy to understand
  5. the net present value and the discounted cash flow
  6. and the internal rate of return
  7. and we'll eventually learn all of those things.
  8. But the present value, what does that mean?
  9. Present value.
  10. So let's do a little exercise.
  11. I could pay you a hundred dollars today.
  12. So let's say today
  13. I could pay you one hundred dollars.
  14. Or (and it's up to you) in one year, I will pay you
  15. I don't know, let's say in a year I agree to pay you $110.
  16. And my question to you
  17. and this is a fundamental question of finance
  18. everything will build upon this
  19. is which one would you prefer?
  20. and this is guaranteed.
  21. I guarantee you, I'm either going to pay you $100 today
  22. and there's no risk, even if I get hit by a truck or whatever.
  23. This is going to happen, if the Earth exists, I will pay you $110 in one year.
  24. It is guaranteed, so there's no risk here.
  25. So it's just a notion of
  26. You're definitely gonna get $100 today, in your hand
  27. or you're definitely gonna get $110 one year from now.
  28. So how do you compare the two?
  29. And this is where present value comes in.
  30. What if there were a way
  31. to say, well what is $110
  32. a guaranteed $110 in the future?
  33. What if there were a way to say
  34. How much is that worth today?
  35. How much is that worth in today's terms?
  36. So let's do a little thought experiment.
  37. Let's say that you could put money
  38. in some, let's say you could money in the bank.
  39. And these days, banks are kind a risky.
  40. But let's say you could put it in the safest bank in the world.
  41. Let's say you could put it in government treasuries
  42. which are considered risk free
  43. because the US government, the treasury
  44. can always indirectly print more money.
  45. We'll one day do a whole thing on the money supply.
  46. But at the end of the day
  47. the US government has the rights on the printing press, etc.
  48. It's more complicated than that, but for these purposes, we assume
  49. that the US treasury, which essentially is
  50. you lending money to the US government
  51. that it's risk free.
  52. So let's say that
  53. you could lend money
  54. Let's say today, I could give you $100
  55. and that you could invest it
  56. at 5% risk free.
  57. So you could invest it 5% risk free.
  58. And then a year from now, how much would that be worth?
  59. In a year.
  60. That would be worth $105 in one year.
  61. Actually let me write $110 over here.
  62. So this is a good way of thinking about it.
  63. You're like, okay. Instead of taking the money
  64. from Sal a year from now
  65. and getting $110 dollars,
  66. If I were to take $100 today and put it in something risk free
  67. in a year I would have $105.
  68. So assuming I don't have to spend the money today
  69. This is a better situation to be in. Right?
  70. If I take the money today and risk-free
  71. invest it at 5%, I'm gonna end up at
  72. $105 in a year.
  73. Instead, if you just tell me
  74. Sal, just give me the money in a year and give me $110
  75. you're gonna end up with more money in a year.
  76. You're gonna end up with $110.
  77. And that is actually the right way to think about it.
  78. And remember, everything is risk-free.
  79. Once you introduce risk,
  80. And we have to start introducing different interest rates and
  81. probabilities, and we'll get to that eventually.
  82. But I want to just give the purest example right now.
  83. So already you've made the decision.
  84. We still don't know what present value is.
  85. So to some degree
  86. when you took this $100 and you
  87. said, well if I lend it to the government
  88. or if I lend it to some risk-free bank at 5%
  89. in a year they'll give me $105
  90. This $105 is a way of saying, what is the one-year value of $100 today?
  91. So what if we wanted to go in the other direction?
  92. If we have a certain amount of money
  93. and we want to figure out today's value
  94. what could we do?
  95. Well to go from here to here, what did we do?
  96. We essentially took $100
  97. and we multiplied by 1+5%.
  98. So that's 1,05
  99. So to go the other way,
  100. to say how much money
  101. if I were to grow it by 5%
  102. would end up being $110, we'll just divide by 1,05
  103. And then we will get the present value
  104. And the notation is PV
  105. We'll get the present value of $110 a year from now.
  106. So $110 year from now.
  107. So the present value of $110 in 2009
  108. It's currently 2008
  109. I don't know what year you're watching this video in.
  110. Hopefully people will be watching this in the next millenia.
  111. But the present value of $110 in 2009
  112. — assuming right now is 2008— a year from now, is equal to $110
  113. divided by 1,05.
  114. Which is equal to— let's take out this calculator
  115. which is probably overkill for this problem— let me clear everything.
  116. OK, so I want to do 110 divided by 1,05
  117. is equal to 104 (let's just round) ,76.
  118. So it equals $104,76.
  119. So the present value of $110 a year from now
  120. if we assume that we could invest money risk-free at 5%— if we would get it today—
  121. the present value of that is— let me do it in a different color, just to fight the monotony—
  122. the present value is equal to $104,76.
  123. Another way to kind of just talk about this is to get
  124. the present value of $110 a year from now, we discount the value by a discount rate.
  125. And the discount rate is this.
  126. Here we grew the money by— you could say—
  127. our yield, a 5% yield, or our interest.
  128. Here we're discounting the money 'cause we're backwards in time—
  129. we're going from a year out to the present.
  130. And so this is our yield. To compound the amount of money we invest
  131. we multiply the amount we invest times 1 plus the yield.
  132. Then to discount money in the future to the present,
  133. we divide it by 1 plus the discount rate— so this is
  134. a 5% discount rate.
  135. To get its present value.
  136. So what does this tell us?
  137. This tells us if someone is willing to pay $110— assuming this 5%, remember
  138. this is a critical assumption— this tells us that if I tell you
  139. I'm willing to pay you $110 a year from now
  140. and you can get 5%, so you can kind of say
  141. that 5% is your discount rate, risk-free.
  142. Then you should be willing to take today's money if
  143. today I'm willing to give you more than the present value.
  144. So, if this compares in— let me clear all of this,
  145. let me just scroll down— so let's say
  146. that one year— so today, one year—
  147. so we figured out that $110 a year from now, its
  148. present value is equal to— so the present value of $110—
  149. is equal to $104,76.
  150. So— and that's 'cause I used a 5% discount rate (and that's the key assumption)—
  151. what this tells you is— this is a dollar sign, I know it's hard to read—
  152. what this tells you is, is that if your choice was between
  153. $110 a year from now and $100 today,
  154. you should take the $110 a year from now.
  155. Why is that?
  156. Because its present value is worth more than $100.
  157. However, if I were to offer you $110 a year from now or
  158. $105 today, this— the $105 today— would be the better choice,
  159. because its present value— right, $105 today
  160. you don't have to discount it, it's today— its present value
  161. is itself.
  162. $105 today is worth more than the present value of $110, which
  163. is $104.76.
  164. Another way to think about it is, I could take this $105 to the bank,
  165. get 5% on it, and then I would have— what would
  166. I end up with?— I would end up with 105 times 1,05, it's equal to $110,25.
  167. So a year from now, I'd be better off by a quarter.
  168. And I'd have the joy of being able to touch my money for a year,
  169. which is hard to quantify, so we leave it out of the equation.