# Effective annual return rate and the annual percentage return rate

1. Nov 3, 2014

### beaf123

Hey guys. I have a very easy math problem which I cant seem to solve. Its been a while since I have had any math. A painting is bought in
1963 for 92,000 pounds, was sold in 2012 10.1 million pounds.
Calculate the effective annual return rate and the annual percentage return rate of this investment.

I found a formula and this is what I did. This is the effective annual interest rate. Does anyone know the difference between this and the annual percentage reurn rate?
The effective annual interest rate:
= 92000((1+ X/49) ^49 – 1) = 10 100 000 { /92000
(1+X/49)^49 = 111 } ln of both sides and properties of logs
49ln(1+X/49) = ln111
1n(1+X/49)=ln(111)/49
1+X/49 = e^0.096
X = (e^0.096)*49 –49 = 4.937%
Annual interest rate = 4,937 %

2. Nov 3, 2014

### Mentallic

1963 was 51 years ago ;)

That result is way off, and it should be obvious to you because 5000% interest (to multiply its value by 50 every year) for 50 years would return 92000*5050 which is astronomically large. Your calculation for the annual percentage return rate should be

$$92,000(1+X)^51 = 10,100,000$$

and the annual return rate is would depend on which year you're looking at. Essentially, for the nth year, the value of the painting would have increased by

$$I_n = P_n-P_{n-1}$$

where In is the interest return in the nth year and Pn stands for the price in the nth year. This value can be calculated by a similar formula to the above, except we don't have 51 in the exponent, but rather n because we want the price after n years, not 51 years.

$$P_n=92,000(1+X)^n$$

and so

$$I_n=92,000(1+X)^n - 92,000(1+X)^{n-1}$$

factoring out the largest common factor of both which is $92,000(1+X)^{n-1}$ yields
$$I_n =92,000(1+X)^{n-1}(1+X-1)$$

$$I_n=92,000X(1+X)^{n-1}$$

3. Nov 3, 2014

### SteamKing

Staff Emeritus
I think the OP simply made a typo when writing the Annual interest rate. He had 4.937% written in the line above.

4. Nov 3, 2014

### beaf123

Thank you for the reply. You are right I am way off. Should be around 14% I think, but over 1 year since I ve done math like this.

So the annual return rate is the change in value from one year to another? And why do you use 51 and not 49. Because you are including the starting and the finishing year? Any reason for that?

5. Nov 3, 2014

### SteamKing

Staff Emeritus
You would use 49 years, since the painting was bought in 1963 and sold in 2012. I think Mentallic missed that bit of information from the OP.

6. Nov 3, 2014

### jbriggs444

The calculation appearing in the original post is... strange.

In a more typical APR calculation, one might start with the actual annual appreciation, divide it according to the compounding interval (for instance, into 12 months) and figure out a monthly interest rate that would produce the actual annual appreciation. Multiply that by 12 and you would have a monthly interest rate expressed on an annual basis.

In original post, you are starting with an actual total appreciation, dividing it up into 49 intervals, figuring out the per-year rate of return that would produce the actual total appreciation and multiplying that by 49 to get an annual interest rate expressed on a 49 year basis. Whatever the result is, it is not an APR.

If what you really want is the per-year appreciation rate, that's simply the 49'th root of the total appreciation. The associated APR may vary from this depending on the nominal compounding interval.

Last edited: Nov 3, 2014
7. Nov 3, 2014

### Mentallic

Oh right, it did mention it was sold in 2012.

By the way, is my latex working for others?

edit: Never mind, my latex is working again.

Last edited: Nov 3, 2014