Expert answer:When to Sell the Basketball Signed by Kobe Bryant?
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When to Sell the Basketball Signed by Kobe Bryant?
Submission Deadline: 2 pm, December 13, 2015
A person is selling a rare basketball on eBay with the following description:
I’m selling Kobe Bryant signed official game ball with inscription ”5x Champ” and the ball is
authenticated by PSA/DNA. The ball with come with PSA/DNA COA and Panini authentic COA.
This ball is 100% guaranteed genuine. This is a very rare ball only 24 balls signed by Kobe is like
this. And this is one of the ball. You will not find this on eBay at all. Plus Kobe Bryant is about
to retire and he’s still a 5 time champ. The display case is not included with the item. Email me
for any questions. Thanks.
Now the seller is trying to decide when to sell it, and currently his asking price is 1,600 USD. He
knows its value will grow over time especially after Kobe’s retirement next year, but he could sell
it and invest the money in a bank account, and the value of the money would also grow over time
due to interest. The question is: when should the seller sell the basketball? Experience suggests
to the seller that over time, the value of the basketball, like many other collectibles, will grow in a
way consistent with the following model:
V (t) = Aeθ
√
t
where A and θ are constants, and V (t) is the value of the basketball in dollars at the time t years
after the present time.
• 1. Plot this function against t when A = 1, 600 and θ = 0.5.
• 2. What is the interpretation of A?
• 3. Plot the function V (t) for several different values of θ. What effect does θ have on the
value of the basketball over time? For your convenience, you can choose the values of θ from
the set {0.1, 0.3, 0.5, 0.7, 0.9}.
Suppose that the seller, who is 35 years old, decides to sell this basketball at time t, sometime
in the next 30 years: 0 ≤ t ≤ 30. At that time t, he will invest the money he gets from the sale in a
bank account that earns an interest rate of r, compounded continuously, which means that after t
years, an initial investment of B USD will be worth Bert USD. When he turns 65, he will take the
money in his bank account for his retirement. Let M (t) be the amount of money in his account
when he turns 65, where t is the time at which he sells his basketball.
• 4. Write down the closed-formed expression of M (t).
• 5. Plot your function M (t) against t when A = 1, 600, θ = 0.5, and r = 0.05.
• 6. If those values of the constants were accurate, then when should the seller sell the basketball to maximize the amount in his retirement account when he turns 65?
• 7. Plot the function M (t) for several different values of θ, while holding r constant. What
does a larger value of θ imply about the value of the basketball over time? (Refer back to
question 3.) And now, what does a larger value of θ imply about the best time to sell the
basketball? Do these two facts seem consistent with one another?
• 8. Plot the function M (t) for several different values of r, while holding θ constant. What
does a larger value of r imply about the best time to sell the basketball? Is that consistent
with the meaning of r? For your convenience, you can choose the values of r from the set
{0.01, 0.03, 0.05, 0.07, 0.09}.
• 9. Let to be the optimal time to sell the basketball, i.e., the time that will maximize M (t).
Try to find to in the general model. Note that your solution should be a function of the
constant variables A, θ and r.
• 10. Plot M (t) against t for different combinations of A, θ and r, and verify that your expression for to does accurately predict when the best time will be to sell the basketball.
• 11. Are the properties of to (as it relates to θ and r) consistent with what you found in step
7 and step 8?
• 12. There is another way to decide when to sell the basketball instead of thinking about
putting the money from the sale into a retirement account. Suppose that today (time = 0)
the seller puts an amount of money W into a bank account that earns interest at an annual
rate of r, compounded continuously, so that at time t the bank account√ will be worth W ert .
If at time t the basketball is sold for an amount equal to V (t) = Aeθ t , how much money
W would the seller have needed to invest initially in order for the bank account value and
the baseball card value to be equal at the time of the sale? That amount W is called the
present value of the basketball if it ends up being sold at time t. Model the present value of
the basketball as a function of the time t when it is sold. Find the time when selling the card
would maximize its present value. Is the answer consistent with the one you found earlier, in
step 9 above?
…
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