# Effective Annual rate of Return

Hey everybody, can someone help me solve this problem?

Assuming a property tax rate of 1% per year, and and opportunity cost of 9% per year, what is the effective annual rate of return on the investment in a parcel of land purchased for \$100,000 cash and sold for \$200,000 net cash ten years later?

A. 0%
B. 10%
C. 50%
D. 100%

I believe the answer is 10%, but that is just a guess. I do not know how to work this problem… Any help is greatly appreciated.

Forgetting the property tax rate and opportunity cost lets look at just the nominal return on the investment. Investing \$100k cash today and receiving \$200k in cash ten years later. This would imply a 100% return over the course of ten years. Simply dividing 100 by 10 does not give you the annualized return thus 10% is incorrect. You have to ask yourself at what annual rate does \$100k grow into \$200k over ten years. The actual answer is about 7%. If you have a financial calculator just plug \$100k into the present value, \$200k into the future value and 10 years into the term and solve for the interest rate. Now figure in the 1% annual property tax expense and the 9% annual opportunity cost and you end up with a real rate of return that is negative. Perhaps it would be wiser to chose the investment yielding 9%?

I’m reading your post to mean you invest \$100k in year one, and in year ten you recieve \$300k (presumably from property disposition) and after returning the initial 100k investment, your net cash is \$200k. If that is the case, then you’re yearly return is 12.98%.

If you meant year ten brough you \$200k, from which your initial investment needed to be returned, really netting you \$100k profit, then your return would be closer to 8.0% per year.

That factors in the opportunity cost an reinvestment rates.

Using your opportunity cost as the discount rate the present value of the first investment is 126k, so you’re purchase at 100k is justified. The PV of the second scenario is only 84k, indicating a purchase price of 100k will yeild a lower rate of return.

Hope that helps.

Good luck,

Craig

To get a real answer, you also need to know the tax savings that you had over the 10 year period, various annual cash outlays, etc. but that gets back to grad. school economics…I’d stick with basic numbers to calculate it, but you DO want to figure in your tax savings (which will then come back in year 10 for depreciation)…

I don’t think you need any more information to answer the question the way it was presented.

Using the rule of 72’s or the rule or 78’s, we can just eyeball that the return for an investment that doubles from \$100K to \$200K is earning somewhere around 7%.

We also already know that the taxes and the opportunity cost are 10%, meaning that this invesment is a loser, just as the original responder said.

There’s no reason to overcomplicate this.

You’re probably right Paul,

why bother “complicating” our investments with things like analyzing numbers, appraisals, or engineering reports?

No reason to get snippy. I’ll put my analysis skills up against anyone’s, including yours.

The question, as asked, was simple, and dealt only with a time value of money issue. It merits only a simple answer.