Constructing an S&P 500 Benchmark with a Compounded Return Hurdle

Summary

The document considers how to build a benchmark that tracks the S&P 500 while earning an additional annual return of 150 basis points. One proposed approach annualizes each daily return, adds the hurdle, and converts back to a daily rate; the question notes that this method does not produce the expected cumulative excess over a long period.

The accepted response instead works in log-return space: convert daily simple returns to log returns, add the annual log hurdle spread across the average number of trading days, cumulatively sum the adjusted returns, and exponentiate to form an index level. It illustrates the result by comparing the adjusted and original series over the sample. Another response suggests using daily data including weekends, constructing a price series, calculating daily year-over-year returns, and adding the hurdle to those results. The alternatives rely on different calendar and compounding conventions; the document does not fully reconcile them or discuss benchmark risk matching, dividends, or implementation details.

Key ideas

• A benchmark with a fixed annual return hurdle needs a consistent compounding convention.
• The accepted method adds a daily share of the annual hurdle to log returns, then compounds the adjusted series.
• The example uses the average annual count of trading days to allocate the hurdle across sessions.
• A second suggestion uses daily calendar data and year-over-year returns, but its construction differs.
• The document does not establish that either method preserves risk matching or addresses every benchmark design choice.

Tags

# Custom benchmark construction (S&P500 + add-on)


# Custom benchmark construction (S&P500 + add-on)












If I have a strategy that has the same risk as S&P500 but also requires 150 bps on top of S&P500 Index, how would I construct such a benchmark?

I have the following approach, but it is not working out to the exact +150 bps after some time period:

- Calculate the daily S&P500 returns;

- Annualize the daily return series for each day and add the 150 bps;

- Convert the result from $Step$ $2$ back to daily rate. Essentially: $$r_{adj} = ((1+r_i)^{365}+0.015)^{1/365}$$

- Calculate the S&P500 + 150 bps based on the returns from $Step$ $3$

Any comments and thoughts would be appreciated. Thanks!

## Answer by demully (score 2, accepted)

https://quant.stackexchange.com/a/49362

So, you:

1- take your daily return series. I've used the SPY ETF including divis

2- take a log return series, ln(1)

3- add ln(1.015)/261 to 2, given 261 trading days on average each year

4- do a running sum series of 2 and 3

5- exp(4) to give you a price

Gives you:

The ratio between the two is 1.35 = 1.015^20, ie your 150bps compounded over the twenty years in the sample.

## Answer by AK88 (score 0)

https://quant.stackexchange.com/a/49361

There is another method that can be used:

- Get the daily S&P500 data, including weekends;

- Calculate the daily rate of return, including weekends;

- Create a price series using these data (base is 100, for example);

- Calculate the YoY daily returns using the price series;

- Add the 150 bps for each daily YoY return results.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.