There are three types of index return: price return, gross total return, and net total return. The price return is ordinarily calculated without regard to cash dividends on Index Securities. The gross total return Index reinvests cash dividends in the Index on the ex-date. The dividend is reinvested in all the Index Securities proportionally to their respective Index weights. The net total return reinvests cash dividends after deduction of non-resident withholding tax rate who do not benefit from double taxation treaties. An Index Security’s withholding tax rate is based on the general tax rate of the Index Security’s country of incorporation. All three of these Index versions reflect extraordinary cash distributions.
To further understand how indices are calculated in real time as well as in historical backtesting, we refer to S&P Dow Jone’s description of index mathematics methodology. The key caveat is to understand that an index is not exactly the same as a portfolio. When a stock is added to or deleted from an index, the index level should not jump up or drop down; while a portfolio’s value would usually change as stocks are swapped in and out. To assure that the index’s value does not vary when stocks are added or deleted, a “base-weighted aggregative” method, with a divisor in the denominator to calculate index level is applied as is in the below formula.
Pi = Price of each constituent in the index
Qi = Quantity of each constituent in the index
The divisor provides a continuous measure of market valuation when changes such as stock splits, mergers, and spin-offs happen. So how is this divisor calculated? Let’s assume a corporate action occurred at t-1, stock r would be delisted, and replaced by stock s effective next day t when the market opens. To maintain the index level stable the following equation stands:
Divisor is equivalent to Divisor New, so the following derived formula also stands:
The benefits of applying this Divisor calculation mechanism is that it allows an incremental and continuous adjustment made to Divisor value, rather than to calculate index value from scratch every time an event takes place. Let’s use an extreme case such as “special dividend issuing” to illustrate. When the stock price declines significantly from dayt-1 to dayt accompanied by the special dividend. Using this “base-weighted aggregative” method, the indexer/indexer calculation agent employ this DivisorNew equation to ensure the index value level is same.
Lastly, I elaborate a bit more on the calculation involved with foreign currency denominated securities and currency hedging, which is increasingly demanded when more global indexes are created, and currency risk is in great need to be hedged.
The Foreign Exchange Rate usually is sourced from the WM Company. The Closing Spot Rates at 16:00:00 UK time is used in the calculation of the closing Index Values. SIX Financial Information Intraday Spot Rates are applied to the real-time Index calculations during the trading day. It’s just a matter of extending another multiplication factor of currency conversion rate into the above index level formula.
On the other hand, if the hedging mechanism is introduced, the calculation can be more complex, but maybe easier to understand if we break it down into the components as follows.
First, we know that to achieve a currency hedging objective, futures, forward are traded, MSCI uses the mid values of the 1‐month, 1‐week and TN (tomorrow next) Forward exchange rates published by WM/Reuters at 16:00:00 UK time to calculate the hedged index return/value. Other indexer/index calculation agents follow similar practices. The hedging can be done on a monthly basis or daily basis.
We use monthly hedging, symbols, and formulas referenced from Solactive’s hedging mechanism to illustrate the calculation here.
Next, the hedged index return can be dissected into two components: the performance of the unhedged index in the home currency and the Hedge Impact (aimed to represent the gain or loss on the Forward contracts) in the home currency.
The Hedge Impact, expressed in percent, is calculated as follows (all exchange rates are expressed as the amount of foreign currency for one unit of hedged currency):
AFRT = Adjustment factor on the last rebalancing date, calculated as follows:
= Hedged index on selection day (ST)
= Hedged index on rebalance day (RT)
Wi, ST = Weight of each index constituent’s currency on the selection date (ST)
Si, ST = Mid Spot rate of each index constituent’s currency on the selection date (ST)
Fi, RT = Mid Forward rate of each index constituent’s currency on the rebalancing date (RT)
IFi, t = Interpolated forward rate of each index constituent’s currency on day (t), calculated as follows:
D = Number of calendar days between the last and the next rebalancing data
d = Number of calendar days between t and the last rebalancing date
HIMt = Hedged impact on the index on day (t)
The hedged index is calculated as follows:
UIt = Unhedged index on day (t)
UIRT = Unhedged index on the last rebalancing date (RT)
Hit = Hedged index on day (t)
HIRT = Hedged index on the last rebalancing date (RT)