# 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:

1. Calculate the daily S&P500 returns;
2. Annualize the daily return series for each day and add the 150 bps;
3. Convert the result from $$Step$$ $$2$$ back to daily rate. Essentially: $$r_{adj} = ((1+r_i)^{365}+0.015)^{1/365}$$
4. Calculate the S&P500 + 150 bps based on the returns from $$Step$$ $$3$$

Any comments and thoughts would be appreciated. Thanks!

• what do you mean 'requires 150 bps on top of'? are you trying to create a portfolio starting with the SP500, adjusting weights, that results in annual returns +150 bps? – Chris Oct 23 '19 at 17:12
• Hi Chris. The idea is to be able to beat S&P500 by 150 bps, given the same risk level as S&P500. While performance is reported on a monthly basis, I would like to construct this index on a daily basis to be aware of the strategy's performance on a daily basis. Does this make sense? – AK88 Oct 23 '19 at 17:18
• it does now, thanks...insofar as you'd like SP500 returns + 150 bps with the same vol, surely there's some kind of disconnect though? I mean, if active managers could simply take existing benchmark holdings, doctor positions and increase portfolio return without impact risk, everyone would beat their benchmark. there's not an obvious/easy way to dial up returns without also increasing risk. – Chris Oct 23 '19 at 17:41
• yeah, I do not think we are we are replicating S&P500 at all. The strategy can be based on bond ETFs (exaggerating a bit, but it is entirely up to the PM). I am comparing S&P500 vol vs the strategy vol. At the same time, I am comparing (annualized) S&P500 + 150 bps performance on a given day with the (annualized) strategy performance. – AK88 Oct 23 '19 at 18:09
• maybe I'm missing what you're trying to do...but if you have daily holdings-level returns and weights, you can simply lever up individual holdings such that you get a portfolio return equal to SP500 + buffer (ie, 150 bp / 252 trading days). – Chris Oct 23 '19 at 19:41

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.

• Thanks. I think this is the continuous version of the method in the OP. If you look at YoY return at any given point in time, will you be able to see exactly 150 bps add-on? – AK88 Oct 25 '19 at 17:30
• yes, the grey line is the ratio between new and old - always growing steady 150bps. Could be off by 1.5bps if # of trading days out by day or two, to correct for that do it by /365 and charge every Monday 3 days for the weekend. – demully Oct 25 '19 at 18:24
• accepted, upvoted. thanks! – AK88 Nov 1 '19 at 4:05

There is another method that can be used:

1. Get the daily S&P500 data, including weekends;
2. Calculate the daily rate of return, including weekends;
3. Create a price series using these data (base is 100, for example);
4. Calculate the YoY daily returns using the price series;
5. Add the 150 bps for each daily YoY return results.