What Was the Rate of Return on the Picasso Painting Sold in 1995?

[Jim01]

Homework Statement

In 1903, a Picasso painting was purchased for $600. The family of the original owner sold the painting in 1995 for $29,152,000. What rate of return (interest) did the family receive on the investment?

Homework Equations

Single Payment Compound Interest Formula:

F = P(1+i)ⁿ

where,

F= a future some of money (future value)
P= a present sum of money
n= number of interest periods
i = interest rate per interest period.

The Attempt at a Solution

solve for i:

F = P(1+i)ⁿ
F/P = (1+i)ⁿ
log(F/P) = n log (1+i)

This is as far as I get. I know that n log (1+i) does not equal n log 1 + n log i, but I don't know what to do to isolate the i. I can't just divide both sides by n log because there is no such thing as n log. I thought about moving n log (1+i) to the left side and setting the equation to zero, but I didn't get anywhere with that either.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[eumyang]

What is the base of "log"? When I look at it, I assume common log (base 10), but IIRC in other places the natural log (base e) is assumed. Whatever it is, you can raise a number to both sides. In other words,
$$x = y \rightarrow a^x = a^y$$
The number you choose for a should probably be the base of the logarithm you were using earlier.

 

[Jim01]

What is the base of "log"? When I look at it, I assume common log (base 10), but IIRC in other places the natural log (base e) is assumed. Whatever it is, you can raise a number to both sides. In other words,
$$x = y \rightarrow a^x = a^y$$
The number you choose for a should probably be the base of the logarithm you were using earlier.


69

The problem says nothing about the base, so I can only assume it is base 10. So then, if I understand you correctly, I would end up with:

10^log(F/P) = 10^{n log(1+i)}

Please excuse my ignorance, but I've always had a problem grasping logs.

so, if the above is correct, then I should be able to use the a^{log u} = u property and get:

F/P = (1+i)

i = (F/P) - 1

Is this correct?

 

[Jim01]

nope. That's not correct. That gives me an answer of 48,585.67. Seems a bit high for an interest rate. Plus it doesn't work when I plug the numbers into the original formula.

 

[Mark44]

When you get to this step,
F/P = (1+i)ⁿ,
instead of taking the log of both sides, take the n-th root of both sides (n is known). That will isolate 1 + i.

 

[Jim01]

When you get to this step,
F/P = (1+i)ⁿ,
instead of taking the log of both sides, take the n-th root of both sides (n is known). That will isolate 1 + i.

You are right, n is known. Even so, if I am doing this right I still end up with i = F/P - 1

(F/P)⁹² = (1+i)⁹²

i = (F/P) - 1

= ($29,152,000/$600) - 1

= 48,586.67%

Ok, this works out if I take $600 x 48,586.67%, however, when I try to plug $48586.67% into the original F=P(1+i)ⁿ formula, I get an overflow error when I should get $29,152,000.

F = P(1+i)ⁿ

= $600(1+48,586.67%)⁹² = overflow error

 

[Jim01]

I just thought of something. I assumed that n = 1995 - 103 = 92 years. This might be wrong. The painting was not compounded yearly. It was only sold once. Therefore, n=1.

Now it works

F = $600(1+48,586.67%)¹
= $29,152,600.00

Thank you everyone for your help.

 

[Mark44]

You are right, n is known. Even so, if I am doing this right I still end up with i = F/P - 1

(F/P)⁹² = (1+i)⁹²

No, this isn't right.
Assuming that n = 92 is correct, if you start with F/P = (1 + i)⁹², and then take the 92nd root (not power) of each side, what do you get?

i = (F/P) - 1

= ($29,152,000/$600) - 1

= 48,586.67%

Ok, this works out if I take $600 x 48,586.67%, however, when I try to plug $48586.67% into the original F=P(1+i)ⁿ formula,

$48586.67% doesn't make any sense. Is it a dollar amount or a percentage? It can't be both. Note that 48,586.67% = 485.8667 as a decimal value.

I get an overflow error when I should get $29,152,000.

F = P(1+i)ⁿ

= $600(1+48,586.67%)⁹² = overflow error