Question

A coin sold at auction in 2017 for $3,158,000. The coin had a face value of $2 when it was issued in 1786 and had previously been sold for $420,000 in 1971. a. At what annual rate did the coin appreciate from its first minting to the 1971 sale? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) b. What annual rate did the 1971 buyer earn on his purchase? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) c. At what annual rate did the coin appreciate from its first minting to the 2017 sale? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Answer 1

A) future value = present value (1 + rate)ⁿ

• FV = $420,000
• PV = $2
• n = 1971 - 1786 = 185 years

420,000 = 2 (1 + r)¹⁸⁵

420,000 / 2 = (1 + r)¹⁸⁵

210,000 = (1 + r)¹⁸⁵

¹⁸⁵√210,000 = 1 + r

1.06848 = 1 + r

1.06848 - 1 = r

r = 0.06848 or 6.85%

B) future value = present value (1 + rate)ⁿ

• FV = $3,158,000
• PV = $420,000
• n = 2017 - 1971 = 46 years

3,158,000 = 420,000 (1 + r)⁴⁶

3,158,000 / 420,000 = (1 + r)⁴⁶

7.519 = (1 + r)⁴⁶

⁴⁶√7.519 = 1 + r

1.04483 = 1 + r

1.04483 - 1 = r

r = 0.04483 or 4.48%

C) future value = present value (1 + rate)ⁿ

• FV = $3,158,000
• PV = $2
• n = 2017 - 1786 = 231 years

3,158,000 = 2 (1 + r)²³¹

3,158,000 / 2 = (1 + r)²³¹

1,579,000 = (1 + r)²³¹

²³¹√1,579,000 = 1 + r

1.0637 = 1 + r

1.0637 - 1 = r

r = 0.0637 or 6.37%