Hello,
I got the following question recently in an interview with an investment bank in UK. Need some help with the answer:

There are two scenarios for investment.

First case - I invest a fixed sum, say \$1 million, in an index fund on 1Jan2001 as a lumpsum investment. Let's say fund was priced at \$100 per unit, so I receive 10,000 units on purchase date. I wait for 15 long years. On 31Dec2015, the fund was quoting at a price of \$500 per unit, and I sell all 10,000 fund units. Net profit = 10,000 * (\$500-\$100) = 4 million.

Second case - I invest a fixed amount, say \$66,666.67, on the first day of each year (like on 1Jan2001, 1Jan2002, 1Jan2003, .... upto 1Jan2015). In this case, each day's fund price will be different so I'll get different no. of units. After 15 years of such regular annual investment (from 1Jan01 to 1Jan15), I encash all my 'X' accumulated units of 31Dec2015 at the price of \$500 per unit. Net profit will be X accumulated units * (\$500-average buy price) = Whatever figure.

Question is - how do you compare the returns on the two investments?

My answer was CAGR was the right method for the first case, while DCF analysis was the right method for the second case. However, the intereviewer was not convinced and said that DCF needs risk-free rate (WACC) which changes every now and then. Additionally, he asked how CAGR and DCF results can be compared?

Can the experts help in providing some guidance?
1) What is the best method to compared the two scenarios?
2) If DCF, how to account for the changing risk-free rate? If CAGR, then how to account for each year's investment in second scenario?

Thanks

For the DCF, you can use the forward risk free rates or, almost equivalently, you can use a Monte Carlo simulation using some interest rate model. I'm assuming you want to use some CAPM type model dependent on the index fund for the discount rate.

For the CAGR model, you can split up the individual returns from each investment and discount each back to the present by either the same CAPM discount model, or you can just look at a non-risk adjusted CAGR and only discount the denominator by the risk free rate.

I believe that the risk adjusted (total return % discounter by CAPM) should be equivalent or roughly equivalent to the NPV DCF

For a simple example, make it \$100 investment at the start of 2 years and a price of \$100/unit the first year, \$150/ unit the second, and a selling price of \$200/unit. Assume CAPM gives a 10% constant discount rate and the risk free rate is 2%

NPV of this project (via s DCF) is (-100) + (-100)/1.02+ 332/1.1^2 = -100 - 98 + 274 = \$76

CAGR method would be

((332/(100+100/1.02))^1/2 -1 = 29.5%

These are just my thoughts but I'm happy to hear other ideas... I would say that a non risk-adjusted CAGR is inappropriate here though because it doesn't say anything about the index fund risk.... if it's risk-adjusted CAGR I think either method should be fine for comparison

Thanks for the detailed reply with example.

However, the 2 questions still remain unanswered.
The question is very clear as taken from a real-life scenario. CAGR gives a percentage, while DCF gives an absolute number.

Any thoughts on which is the best way to compare the two investments? 