Calculus

MAT 111/Elementary Calculus   Fall 2020/70389

Project 2: Maximizing Proceeds from an Asset Sale Due Wednesday, November 25 at 11:59 pm Adapted from: Contemporary Calculus, by Bartkovich and Barrett The purpose of this project is to understand how an asset appreciates and optimize the best time to sell it and reinvest the proceeds.

INSTRUCTIONS: Answer all questions in Parts 1 and 2. Your paper must be typed. There is very little math notation needed, but any that you use should be inserted correctly. For example, for exponents, if you wish to write ?ଶ, you may use equation editor or similar (as I did) or the superscript button, t2 with correct parentheses if needed. I will not accept t2 or t^2 in place of ?ଶ. Viewing windows for graphs should be appropriate to the graph, and the correct domain should be indicated. Part 1 Background: A dealer in sports cards has a rare baseball card and is trying to decide when to sell it. She knows its value will grow over time, but she can sell the card and invest the money in a bank account, and that money would also grow over time. The question: When is the best time to sell the card? Based on experience, the value of the card like many collectibles will grow according to the proportional model: ?ሺ?ሻൌ??௞√௧Where ?൐0,?൐0 are constants and ?ሺ?ሻ is the value of the card at ? years after 2020.

Step 1: Graph ?ሺ?ሻ when ? ൌ2500,? ൌ0.5. Use a graphing utility (for example, https://www.desmos.com/calculator(it will only accept x as your variable, not t) or https://www.geogebra.org/graphing). YOU MUST CHOOSE an appropriate viewing window so that the features of the graph are easy to view and will not earn credit unless you choose a reasonable window. (In Desmos, click on the wrench tool in the upper right corner to choose your viewing window).

Step 2: What is the interpretation of ?? Suppose that the 25‐year old dealer decides to sell the card at time ? sometime in the next 40 years: 0൑?൑40. At that time, she’ll invest the money in a bank account at interest rate ? (compounded continuously) for the remaining years until she turns 65, at which point she’ll withdraw the whole amount and use it to fund her dream trip around the world. This means she holds onto the card for ? years, then sell the card and deposits the proceeds into a bank account for the next 40െ? years, as illustrated in the diagram: The dealer would like to answer the question: “When should she sell the card to maximize the amount in 40 years?”. Let’s assume ? ൌ2500,? ൌ0.5,? ൌ6%ൌ.06 and think about how much money she will have at each point. We’ll let ?ሺ?ሻൌAmount in 2060 if the card is sold after ? years ሺso in 2020൅?ሻ, for 0൑? ൑40What is ?ሺ10ሻ? This means after 10 years the card is worth ?ሺ10ሻ dollars (Where ?ሺ?ሻൌ2500?.ହ√௧), which is then compounded continuously for the remaining 30 years at 6% so: ?ሺ10ሻൌ?ሺ10ሻ?.଴଺∗ଷ଴ൌ2500?.ହ√ଵ଴?.଴଺∗ଷ଴ൌ2500?.ହ√ଵ଴ା.଴଺ሺଷ଴ሻൌ2500∗29.404ൌ73,510.60MAKE sure you understand the algebra above!

MAT 111/Elementary Calculus Fall 2020/70389 Project 2: Maximizing Proceeds from an Asset Sale Due Wednesday, November 25 at 11:59 pm

Step 3: How much will she have if she sells the card after 35 years? Will it be more or less than if she sells after 10 years? Show your work. Using the work we’ve done above, the general formula for ?ሺ?ሻ is: ?ሺ?ሻൌ?ሺ?ሻ?଴.଴଺ሺସ଴ି௧ሻൌ2500?.ହ√௧?଴.଴଺ሺସ଴ି௧ሻൌ2500?.ହ√௧ା଴.଴଺ሺସ଴ି௧ሻൌ2500?.ହ√௧ି.଴଺௧ାଶ.ସTo maximize ?ሺ?ሻ (find the best time to sell the baseball card), we set ?ᇱሺ?ሻൌ0 and solve for ?. First rewrite: ?ሺ?ሻൌ2500?.ହ√௧ା଴.଴଺ሺସ଴ି௧ሻൌ2500?.ହ√௧ି.଴଺௧ାଶ.ସ and since it’s an exponential function, it’s easy to differentiate using the chain rule: ?ᇱሺ?ሻൌ2500?.ହ√௧ି.଴଺௧ାଶ.ସ∗???൫.5√?െ.06?൅2.4൯ൌ2500?.ହ√௧ି.଴଺௧ାଶ.ସ∗൬.25√?െ.06൰Then ?ᇱሺ?ሻൌ0 when ଵସ √௧െ.06ൌ0.

Step 4: Solve the equation above. When should the dealer sell the baseball card? How much money will she have?

Step 5: Graph ?ሺ?ሻ (for 0൑?൑40) using an appropriate viewing window. Does the graph confirm the answer you found in step 4?

Step 6: We’re now going to look at ?ሺ?ሻ for different values of the interest rate, ?. Write out ?ሺ?ሻ for ? ൌ3.5% and for ? ൌ10% and then graph. From the graph, when should she sell the card in each case? What is the effect of having a low interest rate as opposed to having a higher one? Does your answer make sense to you? Explain Part 2 You will now choose a collectible and do a similar analysis.

Step 1: Choose a collectible, antique, artwork or similar that is expected to appreciate. You can find something at: http://www.greatestcollectibles.com/collectibles‐price‐guides/#.X0GNSzV7lQJ or on any other site that you wish. Your item should be selling now for over $100. What is the item and how much is it worth now? Include a screenshot or link. We will assume that it appreciates according to the formula, namely ??଴.ହ√௧ where ? is the current selling price.

Step 2: What would you like to use the money for in 45 years? Step 3: You will now choose an interest rate unique to you as indicated. The interest rate that you will be able to invest the money at after selling the item will depend on the number of letters in your first name and last digit of your telephone number. If you have x letters in your first name and the last digit of your telephone number is y, then your interest rate is x.y%. Since my first name is Lisa, and the last digit of my telephone number is 5, I can invest at 4.5%. If your number is less than 3.9%, add 2 to it What is your interest rate? Step 4: Let ?ሺ?ሻ be the amount of money at the end of your time period if you sell your item after ? years. Find ?ሺ?ሻ and simplify. What is the domain of ?ሺ?ሻ? Step 5: Graph ?ሺ?ሻ. Based on the graph, when do you think would be the best time to sell your item? How much money do you think you will have? Step 6: Use calculus to maximize ?ሺ?ሻ. Check your answer with your guesses in Step 5. Will you have enough money to do what you wanted it for in Step 2 (assuming 2020 prices, of course, not a reasonable assumption)