Chapter 15: Indices Flashcards
m. Construction of indices
Explain what is meant by:
- chain-linking
- free float.
Weighted arithmetic indices
Chain-linking
- Process used to maintain continuity in index value when number of shares issued by constiuent company changes
- used to ensure that changes in the index value are due to changes in the underlying companies’ performance, rather than changes in the number of shares issued.
- changes in number of shares might be due to rights issue/share buybacks, New issue of shares, merger/takeover/breakup or changes in consituent companies.
Free float
- Percentage of shares freely available for purchase on open market.
- Excludes strategic holdings.
m. Construction of indices
List four circumstances in which chain-linking would be required.
Weighted arithmetic indice
- rights issue/share buybacks by a constituent company
- New issue of shares in the sector covered by the index, e.g.,
- newly-formed company
- Privatisation
- demutualisation
- Merger/takeover or breakup involving the a constituent company or companies
- a change in the constituent companies in the index, resulting in a change in market cap due to share price movements. e.g., Q 100th largest company, share price of P changes so that theits market cap exceeds Qs.
m. Construction of indices
State the formula for a weighted arithmetic average capital value index.
Weighted arithmetic indices
i(t) = K (sum(w_i(P_i(t)/P_i(0)))/sum(w_i)
i(t) - capital index at t
K - constant related to the starting value of the index at 0 –> fixed so that index starts at 100 or 1000
w_i - weight applied to the ith constituent (market cap at 0)
P_i(t) - price at t
P_i(0) - price at 0 –> the last time at which there was a capital change
Weights are updated each time the number of shares issued change.
m. Construction of indices
State the formula for a weighted arithmetic average capital value index obtained by chain-linking and free-float.
Weighted arithmetic indices
I(t) = (sum(N_i(t) x P_i(t))/B(t)
Where:
- N_i(t) is the number of shares issued for i-th constituent at time t
- P_i(t) is price of i-th constituent at time t
- B(t) is the basee value, or divisor, at time t
- B(t) is obtained from B(t-1) through chain-linking process.
The numerator represents the total market cap of the index consituents
The formula only take into account changes in capital values
m. Construction of indices
Explain with aid of a formula what the ex-dividend adjustment represents.
Outline the assumptions that need to be made to allow for the effect of investment income.
Total return indices
xd_i(t) = N_i (t) x D_i(t)/B(t - 1) –> XD at the current time t (not an accumulation)
XD_i(t) = Sum_t (N_i (t) x D_i(t)/B(t - 1)) –> XD adjustment of ith share representing total dividends declared to date
XD(t) = Sum_i(XD_i(t)) –> XD adjustment accumulated to date for all constituent companies
Where:
- D_i(t) is the dividend per share declared by the ith constituent company at time t (net or gross, as required)
- B(t -1) is the divisor at the close of business on the previous day after allowing for any capital changes.
XD is reacts to the ex-dividend date rather than the date of receipt of dividends.
It is normally reset to zero at the start of each year.
An assumption needs to be made about:
- time of reinivestment of income
- whether it is reinvested net or gross of tax
- expenses of reinvestment
to allow for the effect of investment income
m. Construction of indices
State the formula of a holding period return.
Total return indices
TR(t) = (I(t) + XD(t) - I(t-1) - XD(t-1))/I(t-1) *100 –> holding period return
Generally:
HPR = (P(1) + d)/P(0)
Where:
- P(1) and P(0) are the vaues of the investment at the beginning and end of the period
- d is the income gennerated by the investment over the period.
- HPR is sometimes used as an approximation to IRR
- However, it is inaccurate, it fails to allow for the fact that part of the TR comes from reinvestment of d
This assumes implicitly that:
- dividends are subject to the rate of tax (if any) assumed in the calculation of the index
- there are no expenses or losses incurred in reinvesting the dividends.
m. Construction of indices
State the formula of the total return index obtained by linking successive HPRs.
Total return indices
TRI(t) = TRI(t-1)[I(t)/(I(t) - income(t, t-1)]
where:
- TRI(t) is the total return index;
- income(t, t-1) is the income received from t - 1 to t (net or gross as required)
Alternatively, following formula can be used:
TRI(t) = TRI(t-1)[(I(t+1) + income(t, t-1))/I(t-1)]
The above is used more often
Total return between time a and b (b>a) is then given as:
TRI (b)/TRI(a) -1
m. Construction of indices
When do you assume the dividends are reinvested?
1.2 Total return indices
Usual assumption is to use the ex-dividend date. However, this may lead to problems if the index is used by index tracking funds, since they will not be able to reinvest the dividends until they actually receive it. The index fund might underperform the TRI due to the missed opportunity to earn returns on the immediate reinvestment assumed by the formula.
m. Construction of indices
Give two different ways of estimaiting the income received over the time period from t -1 to t from the index constituents.
1.2 Total return indices
- income(t-1, t) = XD(t) - XD(t-1)
where XD(t) is the ex-dividend adjustment at time t
- income(t-1,t) = I(t)*y(t)/n
where I(t) is the capital value index and y(t) dividend yield, both at time t and n is the number of time periods per annum.
m. Construction of indices
State the formula for an unweighted arithmetic index of capital values.
Unweighted (price) arithmetic indices
I(t) = Ksum_i(P_i(t)/P_i(0)
where:
- P_i(t) is price of the ith consituent at time t
- K is a constant
m. Construction of indices
Explain the main problems with such an index.
Unweighted (price) arithmetic indices
- Unsuitable for performance measurement:
This is unsuitable for performance measurement since actual performance reflects weights held, whereas this give equal weight to each share.
- Sensitivity to Stock Selection:
The index value is heavily influenced by the choice of stocks included. A single high-priced stock can significantly impact the index value compared to a low-priced stock, even if the high-priced stock’s performance isn’t representative of the broader market.
- Ignores Company Size:
The index doesn’t consider the market capitalization of companies. A small company with a high stock price can have the same weight as a large, established company with a lower stock price. This can misrepresent the overall market performance.
- Limited Diversification:
An unweighted index may not be well-diversified across sectors or industries. This can expose investors to higher risk if a particular sector or industry underperforms
- Potential for Manipulation:
Since the index is heavily influenced by the selection of stocks, there’s a potential for manipulation if the index composition isn’t carefully chosen and monitored.
6.* Difficulty with Reinvestment:*
The formula doesn’t explicitly account for dividend reinvestment. If dividends are not considered, the index might not accurately reflect the total return an investor would experience.
m. Construction of indices
State the formula for an unweighted geometric index of capital values.
Explain the main problem with this.
Geometric indice
I(t) = K[(multiplication function_i P_i(t)/P_i((0)]^(1/n)
m. Construction of indices
Describe three advantages and disadvantages of an unweighted geometric index relative to a weighted arithmetic index as a measure of price changes.
Geometric indice
Three advantages:
- It does not require weights – which might not be available in some circumstances;
- It is simpler to calculate and understand/explain (especially if it ignores corporate changes);
- It can be used to give an indication of short-term price movements;
- It gives a better representation of the broader market trend than an arithmetic index (due to the geometric index change being closer to the median of price changes).
Three disadvantages:
- The index goes to zero if one of the components goes to zero;
- Being unweighted makes it less relevant for performance measurement;
- The geometric index undershoots the arithmetic index in a rising market, and overshoots in a falling market
m. Construction of indices
List factors to consider when constructing an index.
- Purpose of index
- Consituents and basis for inclusion/exlcusion
- Type of index (weighted, freefloats)
- frequency of calculation of index values (& updating index constituents and weights)
- base date and value
- how to deal with income (XD adjustment, total return index)
- price data used (mid-market prices?)
- how to deal with capital gains changes.
- Sources and availability of data
- (costs of constructing the index)
m.i) the uses of investment indices
List the main uses of indices
Use of indices
- Portfolios
- as benchmark of Investment performance of pfs
- valuing a Notional pf
- to provide basis for the creation of Derivative instruments relating the market or sub-section of the market
- basis for Index tracker funds
- market movements
- Charting long-term history of market movements and levels
- Estimating future market movements based on past trends, ie for technical analysis
- Measure Short-term market movements
5. analysing Sub-sectors of the market
INDICES’S
