# How do you calculate “excess returns”?

(Although this is perhaps a very simple question, math-wise, it still has me puzzled so it's okay to ask for help here I guess. I really need to understand this, so any help is appreciated)

First, assume the S&P100 market index has performed like this on the first 3 days of the month:

day   open      close     return (abs)    return (%)
----------------------------------------------------
1     574.19    562.51    -11.68          -2.03
2     564.56    570.72      6.16           1.09
3     571.99    572.93      0.94           0.16


In other words, "the market" went down $11.68 or -2.03% on day 1. Now asssume that I have bought and sold the following stocks on each of these 3 days: day stock buy sell return (abs) return (%) ------------------------------------------------------------ 1 HPQ 40.81 40.14 -0.67 -1.64 2 FDX 89.79 87.89 -1.90 -2.12 3 PEP 63.49 63.24 -0.25 -0.39  So on day 1, I bought and sold HPQ for a return of -$0.67 or -1.64%. I now want to calculate, in %, how my trades performed against the market:

day    market return (%)    my return (%)    excess return (%)
-------------------------------------------------------------------
1      -2.03                -1.64             0.39  (-1.64 - -2.03)
2       1.09                -2.12             3.21
3       0.16                -0.39            -0.55
-------------------------------------------------------------------
total  -0.78                -4.15            -3.37  (-4.15 - -0.78)


For instance, on day 1 the market went down 1.64% but the stock I bought only went down 1.64%. In other words, my trade "outperformed" the market by 0.39%. When totalling, the market lost 0.78% during the three days, but my trades lost -4.15%. The way I interpret this, my trades underperformed 3.37% against the market.

Alternatively, if I take the first (574.19) and last (572.93) price of the market indexfund, the market only declined ((572.93 - 574.19) / 574.19) * 100 = -0.22%. Based on this number, my trades underperformed the market by -3.37 - -0.22 = -3.15%

Question 1 How do the 2 market performances that I calculated (-0.78% and -0.22%) relate to each other?

Question 2 How do I correctly culculate the relative performance of my trades against the market? Is is -3.37% or -3.15%? And if it's neither, how do I calculate with how many % my trading outperformed (or underperformed) the market?

[edit after thinking about incorporating overnight returns]

day  stock  close@yesterday  open@today  close@today  return (abs)  return (%)
------------------------------------------------------------------------------
1    HPQ    40.00            40.81       40.14         0.14          0.35
2    FDX    90.00            89.79       87.89        -2.11         -2.34
3    PEP    64.00            63.49       63.24        -0.76         -1.19
---------------------------------------------------------------------------- +
-3.18


(today's opening price is just there for reference. I've calculated return (abs) as close@today - close@yesterday and return (%) as (return (abs) / close@yesterday x 100))

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Normally the market return of a given day is calculated from the previous day's close, not from that day's open, so the return on day 2 is $570.72-562.51=8.21$ or When you add the returns on the three days you miss the rises in the index that happened overnight. This is why you get different returns by adding the return of the three days from taking the whole span. If you want to cover day 1 open to day 3 close, the $-0.22\%$ is correct. Similarly, you should calculate your return from purchase (presumably at market open day 1) to sale (market close day 3), but it should be based on the whole portfolio. On the assumption you bought the same dollar value of each stock, the portfolio performance is the average of the performances of the three stocks, not the sum. If you didn't buy the same amount, you need to weight the change by how much of each stock you bought. If you bought the same amount of each stock, your return is $-1.38\%$. Your excess return is then $-1.16\%$
Thanks for your reply, it clarifies the difference in market returns for me. On my returns you say: "Similarly, you should calculate your return from purchase (presumably at market open day 1) to sale (market close day 3)". But the situation is slightly different: on day 1, I bought and sold HPQ for a return of -1.64%. Then on day 2 and 3, I respectively bought and sold FDX and PEP. So I only held each stock for 1 day. I believe this is a different situation from where I would have bought 1 stock and held it for the 3 days. Shouldn't return be the sum here instead of average? –  Pr0no Aug 24 '12 at 23:52