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.
- Adjust market cap for each share to excludes strategic holdings(SH)
- SH are held by LT investors for strategic purposes such as
holding companies or pension funds hedging their long-term liabilities.
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
m.i) the uses of investment indices
List (four) further uses of government bond indices.
Use of indices
- Benchmarking: A standard againt which yields on other fixed interest investments can be assessed.
- Yield curve Analysis: provide the general yield structures of fixed interest investments (summarise the yield curve). Provides information about market expectations for interest movements and economic conditions.
- approximate valuation of fixed interest pfs (without reference to all the individual bond prices)
- allow for comparison with yields on ordinary shares as a meausure of the yield gap between bonds and equities (as they summarise the yield curve).
- Portfolio construction: Helps with diversification by allocating investments across different segments of bond market
- Risk assessment: used to measure risk. High yields –> high credit risk and volatility vice versa. Also used to assess risk vs reward.
- Income planning: Investors seeking income-focused strategies use yield indices to identify segments of the bond market that offer attractive yield levels.
m.ii) principal indices in the SA and int stock and bond markets
Describe how the FTSE equity indices are calculated and list six figures, in addition to the capital value index, which are provided in respect of each FTSE index.
FTSE UK index series
- They are calculated on a weighted arithmetic average basis with market capilitisation as weights.
- weights based on free floats (which are rounded to a whole # according to the next higher band of 20%, 30%, 40%, 50% and 75% and 100%)
Other six figures:
1. actual dividend yield
2. price earnings ratio
3. total return index
4. ex-dividend adjustment
5. average dividend cover
6. Euro value index
m.ii) principal indices in the SA and int stock and bond markets
FTSE UK index series
m.ii) principal indices in the SA and int stock and bond markets
Outline the coverage of the following indices:
- FTSE 100
- FTSE 250
- FTSE 350 Supersectors
FTSE UK index series
FTSE 100:
- Consists the 100 largest quoted companies in the UK by market cap
- accounts for about 80% of the total equity market cap
- Main indicator ST market movements in the UK
- used as a basis for investment products (derivatives and EFT)
- for continuity and admnistrative reasons constituents changed once a quarter
FTSE 250:
- Consits of the 250 largest quoted companies ranking below the 100 companies by market cap
- Accounts for about 17% of total equity market cap
- Also a basis for stock derivatives
FTSE 350:
- Industry sector indexes derived from companies in the 100 and 250 indices.
- accounts for about 95% of total UK equity market
- sub-indices also calculated for high-yielding and low-yielding stocks
m.ii) principal indices in the SA and int stock and bond markets
Outline the coverage of the following indices:
- FTSE SmallCap
- FTSE All-Share
- FTSE Fledging
- FTSE AIM
FTSE UK index series
FTSE SmallCap:
- Covers all companies below the 350 with market cap above a certain limit and are actively traded
- about 350 constituents
- represents about 2% of the UK equity market cap
- index calculated at close of each day
FTSE All-Share
- Comprises of 100, 250 and SmallCap indices
- accounts for about 98% - 999% of the over total market cap
FTSE Fledging
- Consists of the remaining, sufficiently marketable, quoted copmanies that are too small to be included in the SmallCap index
FTSE AIM
- Covers some 1000 companies traded in Alternative investment Market.
- These companies are too small or too new to apply for full listing.
m.ii) principal indices in the SA and int stock and bond markets
List the main South African equity indices that comprise the FTSE/JSE Africa Headline Indices.
And list other indices published by the JSE.
FTSE UK index series
The most important equity indices in SA is the FTSE/JSE Headline Indices consisting of:
- FTSE/JSE ALSI (All share Index) consisting of 99% of all listed companies
- TOP40: Consists of the 40 largest stocks, constituting around 84% of the ALSI
- Mid-Cap index, made up of stocks from position 41 to 100, and around 14% of the ALSI
- Small cap index, made up of stocks from position 101, and about 2% of the ALSI
- Fledging index consisting of the 1% of listed companies not included in the ALSI
In addition of the headlines indices, other indices published by the JSE Classification system:
- Sector and subsector indices that are consistent FTSE Global Classification System.
- Secondary Market Indices: Development capital (small - medium sized with limited profit history); venture capital (CIS holding a pf of venture capital projects/single venture companies) and Alternative Index (smaller companies not yet listed on main board
Specialised Indices
- SWIX (shareholder weighted Index) - weights adjusted for foreign holding - freefloats reduced to reflect locally held shares
- ‘Style’ Indices - growth or value
- RAFI All-Share Index: Weights are by fundamental factors (sales, cashflow, bookvalue, dividends)
m.ii) principal indices in the SA and int stock and bond markets
Describe the FTSE Gilts Index series
Fixed Income indices
- Cover conventional and index-linked gilts
- Both price and yield indices are published…
- …with price indices subdivided according to term and yield indices subdivided according to term and coupon.
- Index numbers calculated using dirty prices, ie accrued interest included
- Accured interest and XD adjustment published for price index series
- price indices are constructed as weighted arithmetic indices, the weights being the market cap of the stocks
- indices are chain-linked to allow for new issues, redemptions and movements of stocks
m.ii) principal indices in the SA and int stock and bond markets
Describe the FTSE Gilts yield Index series (conventional gilts and index-linked gilts)
Conventional gilts:
- each yield index is constructed by fitting a curve to the gross redemption yields of the stocks in the particular category
- All irredeemable stocks are included in each coupon band to give stability to the long end of the curves.
- where stocks have optional redemption dates, whichever gives the lower redemption yield is used
Index-linkded gilts
- each yield index represents the average yields the stocks in that category
m.ii) principal indices in the SA and int stock and bond markets
List other figures published with the FTSE Gilt index
Fixed Income indices
Price indices:
- Index value
- accrued interest
- ex-dividend adjustment for the year to date
- day’s change
- total return index figure
- the weighting that each price index is given in the make-up of the all stocks index
- the gross redemption yield
- duration of the index
Yield indices:
- yesterday’s yield
- the yield the day before
- yield one year ago
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Chapter 2 and 318
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Chapter 2324
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Chapter 4: Specialist asset classes36
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Chapter 5: Specialist asset classes (2)48
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Chapter 12: Valuation of investments (1)19
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Chapter 13: Valuation of investments (2)24
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Chapter 6: Economic influences22
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Chapter 11: Fundamental Analysis22
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Chapter 7: Behavioural Finance32
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Chapter 8: Regulation of financial services13
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Chapter 9: Applications of the legislative and regulatory framework (1)19
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Chapter 10: Applications of the legislative and regulatory framework (2)19
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Chapter 14: Industry classification15
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Chapter 15: Indices36
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Chapter 16: Performance measurement (1)17
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Chapter 17: Performance measurement (2)14
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Chapter 18: Risk Control21
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Chapter 19: Actuarial techniques (1)13
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Chapter 20: Actuarial techniques (2)12
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Chapter 21: Portfolio Management (1)24
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Chapter 22: Portfolio Management (2)25
