# Present value of a payment

Suppose I've just won 1'000'000 dollars in a game show. At the end of the program they tell me that they will pay me the prize as following: they will deposit in my bank 50'000 dollars every year for twenty years with a constant annual interest rate of 6% (this to prevent me from wasting too quickly the one million dollars I've just won). The present value of my prize is given by:$$V_t=z\frac{1-\left [ \frac{1}{(1+i)^n} \right ]}{1-\left [ \frac{1}{(1+i)} \right ]}$$

with $z=50'000$, $i=0.06$ and $n=20$ I get that: $$V_t\approx 50'000 \left (\frac{0.688}{0.566} \right )\approx 608'000$$

My book says that is a really great prize, but I'm not a millionaire at this point. I would have been a millionaire if they'd paid me the one million dollars right at the end of the program. But I can't understand why it looks like I'm less rich. Suppose I won't spend money they'll give me, in twenty years I'll have: $$1'000'000\cdot(1+0.06)=1'060'000$$ Which is obviously greater than 1'000'000. Could you please explain me what's the reasoning that there is behind the present value of 608'000 dollars? Thanks for any help you can provide.

• Is there something missing here? Why is $z=\$50,000$? – BKay Jan 20 '15 at 13:49 • @BKay Ops, sorry. My fault. The annual payment is 50'000 dollars every year for twenty year. – ChicagoCubs Jan 20 '15 at 13:52 ## 1 Answer You got the last sum wrong. In twenty years if you invest the million at 6 percent you'll have:$1,000,000 \cdot (1 + .06) ^ {20} = 3,200,000$I think the easiest way to to understand this result is with a table. This table asks what is the present value of each payment, the value of each year of payment in the year of the award (t=0). The far right column sums those payments to calculate the net present value of payments from time = 0 to time = t. You can see that the row 19, far right column number is the desired net present value of \$608k.

+------+-------------+-------------+-----------+--------------+------------------+
| Year | Value of $| Value of$  | Payment   | Value at 0   | Cumulative value |
| (t)  | in year 0   | in year t   | in year t | of payment   | of payments      |
|      | at time t   | at time 0   |           | in year t    | through t at 0   |
+------+-------------+-------------+-----------+--------------+------------------+
| 0    |  1.00       |  1.00       |  50,000   |  50,000      |  50,000          |
+------+-------------+-------------+-----------+--------------+------------------+
| 1    |  1.06       |  0.94       |  50,000   |  47,170      |  97,170          |
+------+-------------+-------------+-----------+--------------+------------------+
| 2    |  1.12       |  0.89       |  50,000   |  44,500      |  141,670         |
+------+-------------+-------------+-----------+--------------+------------------+
| 3    |  1.19       |  0.84       |  50,000   |  41,981      |  183,651         |
+------+-------------+-------------+-----------+--------------+------------------+
| 4    |  1.26       |  0.79       |  50,000   |  39,605      |  223,255         |
+------+-------------+-------------+-----------+--------------+------------------+
| 5    |  1.34       |  0.75       |  50,000   |  37,363      |  260,618         |
+------+-------------+-------------+-----------+--------------+------------------+
| 6    |  1.42       |  0.70       |  50,000   |  35,248      |  295,866         |
+------+-------------+-------------+-----------+--------------+------------------+
| 7    |  1.50       |  0.67       |  50,000   |  33,253      |  329,119         |
+------+-------------+-------------+-----------+--------------+------------------+
| 8    |  1.59       |  0.63       |  50,000   |  31,371      |  360,490         |
+------+-------------+-------------+-----------+--------------+------------------+
| 9    |  1.69       |  0.59       |  50,000   |  29,595      |  390,085         |
+------+-------------+-------------+-----------+--------------+------------------+
| 10   |  1.79       |  0.56       |  50,000   |  27,920      |  418,004         |
+------+-------------+-------------+-----------+--------------+------------------+
| 11   |  1.90       |  0.53       |  50,000   |  26,339      |  444,344         |
+------+-------------+-------------+-----------+--------------+------------------+
| 12   |  2.01       |  0.50       |  50,000   |  24,848      |  469,192         |
+------+-------------+-------------+-----------+--------------+------------------+
| 13   |  2.13       |  0.47       |  50,000   |  23,442      |  492,634         |
+------+-------------+-------------+-----------+--------------+------------------+
| 14   |  2.26       |  0.44       |  50,000   |  22,115      |  514,749         |
+------+-------------+-------------+-----------+--------------+------------------+
| 15   |  2.40       |  0.42       |  50,000   |  20,863      |  535,612         |
+------+-------------+-------------+-----------+--------------+------------------+
| 16   |  2.54       |  0.39       |  50,000   |  19,682      |  555,295         |
+------+-------------+-------------+-----------+--------------+------------------+
| 17   |  2.69       |  0.37       |  50,000   |  18,568      |  573,863         |
+------+-------------+-------------+-----------+--------------+------------------+
| 18   |  2.85       |  0.35       |  50,000   |  17,517      |  591,380         |
+------+-------------+-------------+-----------+--------------+------------------+
| 19   |  3.03       |  0.33       |  50,000   |  16,526      |  607,906 (award NPV)|
+------+-------------+-------------+-----------+--------------+------------------+


Now that we see how this \$608 number is calculated, how should we interpret it? The classic answer is to ask "what someone would pay you for your prize?" For simplicity, let's ignore risk or assume that the 6% number fully encapsulates the risk. Imagine an investor who is risk neutral, has deep pockets, but critically, has the same investment choices as the game show. What would they pay you for your prize? What if there were many such investors such that they were competing away all the profits to pay you exactly what they thought that investment was worth.? What's the absolute maximum they'd pay? They would pay \$607,906.

Why? Because say they invested \$607,906 at 6% per year with the plan of selling anything left over? What would happen to their balance over time? +---------------+--------------+--------------+--------------------+ | Starting | Interest (t) | Cash Out (t) | Ending Balance (t) | | Balance (t) | | | | +---------------+--------------+--------------+--------------------+ | 607,906 | 0 | 50,000 | 557,906 | +---------------+--------------+--------------+--------------------+ | 557,906 | 33,474 | 50,000 | 541,380 | +---------------+--------------+--------------+--------------------+ | 541,380 | 32,483 | 50,000 | 523,863 | +---------------+--------------+--------------+--------------------+ | 523,863 | 31,432 | 50,000 | 505,295 | +---------------+--------------+--------------+--------------------+ | 505,295 | 30,318 | 50,000 | 485,612 | +---------------+--------------+--------------+--------------------+ | 485,612 | 29,137 | 50,000 | 464,749 | +---------------+--------------+--------------+--------------------+ | 464,749 | 27,885 | 50,000 | 442,634 | +---------------+--------------+--------------+--------------------+ | 442,634 | 26,558 | 50,000 | 419,192 | +---------------+--------------+--------------+--------------------+ | 419,192 | 25,152 | 50,000 | 394,344 | +---------------+--------------+--------------+--------------------+ | 394,344 | 23,661 | 50,000 | 368,004 | +---------------+--------------+--------------+--------------------+ | 368,004 | 22,080 | 50,000 | 340,085 | +---------------+--------------+--------------+--------------------+ | 340,085 | 20,405 | 50,000 | 310,490 | +---------------+--------------+--------------+--------------------+ | 310,490 | 18,629 | 50,000 | 279,119 | +---------------+--------------+--------------+--------------------+ | 279,119 | 16,747 | 50,000 | 245,866 | +---------------+--------------+--------------+--------------------+ | 245,866 | 14,752 | 50,000 | 210,618 | +---------------+--------------+--------------+--------------------+ | 210,618 | 12,637 | 50,000 | 173,255 | +---------------+--------------+--------------+--------------------+ | 173,255 | 10,395 | 50,000 | 133,651 | +---------------+--------------+--------------+--------------------+ | 133,651 | 8,019 | 50,000 | 91,670 | +---------------+--------------+--------------+--------------------+ | 91,670 | 5,500 | 50,000 | 47,170 | +---------------+--------------+--------------+--------------------+ | 47,170 | 2,830 | 50,000 | 0 | +---------------+--------------+--------------+--------------------+  That is, they'd have exactly enough money to make the required \$50,000 payments in every period with nothing left over.

• Thanks for the explanation and for the table. But I still do not understand what's the interpretation of the result. Once I've found that the present value of the payment is 608'000 dollars, what can I say about it or what kind of information can I get from it? – ChicagoCubs Jan 20 '15 at 17:03
• The present value of 608,000 dollars means that you could replicate the payoff of 50k per year with only that 608k initially. For example, split the 608k into sizes matching the sizes of the 20 chunks listed in column "the value at 0 of..." Then invest all that money and cash out the first chunk (50k) on in the first period, take out ~47k in the second period, take out ~45 in the third period, etc.. The PV means you can do this with only ~608k in the first time period. Essentially, it means that money you have in hand now is worth more than having it in the future. – jmbejara Jan 20 '15 at 17:24
• @BKay So basically can I say that receiving 1'000'000 dollars in twenty years is the same as receiving 608'000 dollars today (with an interest rate of 6%)? – ChicagoCubs Jan 20 '15 at 19:04
• Receiving 608k today is the same as receiving 50k annually for 20 years. – jmbejara Jan 20 '15 at 21:44