Present value of a payment

Question

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.

Answer 1

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.