Part 1 - CAPM
04 October 2025 11:12
CAPM under uncertainty
The CAPM model is based on the work of Sharpe (1964) and other authors
Assumptions and level of truth in reality of the model (also the assumptions of MPT)
- No transaction costs
○ Fairly realistic as there are TCs like brokerages and commissions but they are not usually too significant so can be considered as
non-existent
- All assets are infinitely divisible
○ Possible
- No personal taxes
○ Not true in reality
- Capital asset markets are purely competitive
○ High degree of competition in markets so not a bad assumption
- Investors choose portfolios based solely on ER and SD of such returns
○ Not true as there could be a range of factors which influences their decision like financial views (high dividend yields vs capital
gains), political, personal views etc.
- Short sales are permitted
○ Depends on when and where you are buying shares as some jurisdictions don't allow but not a bad assumption (typically short
selling is permitted in a bear market)
- Unlimited lending and borrowing at the risk-free rate is permitted
○ Nearly 1/2 true as you could invest virtually unlimited amounts in sovereign debt i.e. unlimited lending but borrowing is
impossible (big assumption, could say poor assumption)
- Investors have homogeneous expectations regarding asset returns, and variances and covariances of such returns
○ Not realistic assumption - some have different information and usually very different interpretations (of risk etc.)
CAPM derivation
Determining beta for asset j
Various ways to determine beta:
- As the slope of the OLS regression line when returns on asset j are regressed against returns on the market portfolio
○ Slope of the characteristic line of a share is beta (note the line is not the same as the CML)
- By formula
- Could also look it up
○ Betas are widely calculated and reported
○ Note alternative periods and data frequencies
i.e. a share's beta based on 5 years of monthly data will not be the same as the beta based on a year of weekly data
Proxies for M and Rf
- M often proxied by a broadly based market index
○ e.g. FT All Share Index, FTSE 100, FTSE 350, S&P 500
- Rf by the yield on short-dated low risk bonds
○ e.g. yield on 90-day gilts
Finance Page 1
, Stability of betas
- There is no such thing as a single correct beta for an individual asset
- Changing any of the following will change the calculation
○ Choice of M or the proxy for M
i.e. using a different index
○ Frequency of returns data used
Daily, monthly, quarterly etc.
○ Length of the period of returns data considered
Month, year, five years
○ The absolute period of data
Most recent period or a period from some time ago
- So formally you should be very specific about a beta
○ i.e. Beta of X plc against the FT All Share Index based on 2 years of monthly data, calendar years 2023 and 2024
To find Beta of M and of the risk free asset
- Start with CAPM and apply it to the market
- For Rf do the same with the Rf, setting μj to Rf
- In market equilibrium (CAPM equilibrium), every investment, every portfolio in the market will lie on the SML for that particular
market
○ Slope of SML is the market risk premium (NOT BETA)
Difference between sigma and beta
Beta of 0 - returns on an asset have no relation to the changes in return on the market
- For the asset with beta of -0.8 --> as returns on the market go up, returns on this asset go down
- For share/portfolio beta greater than 1, share returns will be 'sensitive to the market' or similar
- For share/portfolio beta = 1, the share returns will be 'of unit sensitivity to the market' or similar
- For share/portfolio beta less than 1 but non-negative, 'Low sensitivity to the market' returns or similar
- For share/portfolio beta negative, returns will be 'a hedge for the market' or similar
Finance Page 2
04 October 2025 11:12
CAPM under uncertainty
The CAPM model is based on the work of Sharpe (1964) and other authors
Assumptions and level of truth in reality of the model (also the assumptions of MPT)
- No transaction costs
○ Fairly realistic as there are TCs like brokerages and commissions but they are not usually too significant so can be considered as
non-existent
- All assets are infinitely divisible
○ Possible
- No personal taxes
○ Not true in reality
- Capital asset markets are purely competitive
○ High degree of competition in markets so not a bad assumption
- Investors choose portfolios based solely on ER and SD of such returns
○ Not true as there could be a range of factors which influences their decision like financial views (high dividend yields vs capital
gains), political, personal views etc.
- Short sales are permitted
○ Depends on when and where you are buying shares as some jurisdictions don't allow but not a bad assumption (typically short
selling is permitted in a bear market)
- Unlimited lending and borrowing at the risk-free rate is permitted
○ Nearly 1/2 true as you could invest virtually unlimited amounts in sovereign debt i.e. unlimited lending but borrowing is
impossible (big assumption, could say poor assumption)
- Investors have homogeneous expectations regarding asset returns, and variances and covariances of such returns
○ Not realistic assumption - some have different information and usually very different interpretations (of risk etc.)
CAPM derivation
Determining beta for asset j
Various ways to determine beta:
- As the slope of the OLS regression line when returns on asset j are regressed against returns on the market portfolio
○ Slope of the characteristic line of a share is beta (note the line is not the same as the CML)
- By formula
- Could also look it up
○ Betas are widely calculated and reported
○ Note alternative periods and data frequencies
i.e. a share's beta based on 5 years of monthly data will not be the same as the beta based on a year of weekly data
Proxies for M and Rf
- M often proxied by a broadly based market index
○ e.g. FT All Share Index, FTSE 100, FTSE 350, S&P 500
- Rf by the yield on short-dated low risk bonds
○ e.g. yield on 90-day gilts
Finance Page 1
, Stability of betas
- There is no such thing as a single correct beta for an individual asset
- Changing any of the following will change the calculation
○ Choice of M or the proxy for M
i.e. using a different index
○ Frequency of returns data used
Daily, monthly, quarterly etc.
○ Length of the period of returns data considered
Month, year, five years
○ The absolute period of data
Most recent period or a period from some time ago
- So formally you should be very specific about a beta
○ i.e. Beta of X plc against the FT All Share Index based on 2 years of monthly data, calendar years 2023 and 2024
To find Beta of M and of the risk free asset
- Start with CAPM and apply it to the market
- For Rf do the same with the Rf, setting μj to Rf
- In market equilibrium (CAPM equilibrium), every investment, every portfolio in the market will lie on the SML for that particular
market
○ Slope of SML is the market risk premium (NOT BETA)
Difference between sigma and beta
Beta of 0 - returns on an asset have no relation to the changes in return on the market
- For the asset with beta of -0.8 --> as returns on the market go up, returns on this asset go down
- For share/portfolio beta greater than 1, share returns will be 'sensitive to the market' or similar
- For share/portfolio beta = 1, the share returns will be 'of unit sensitivity to the market' or similar
- For share/portfolio beta less than 1 but non-negative, 'Low sensitivity to the market' returns or similar
- For share/portfolio beta negative, returns will be 'a hedge for the market' or similar
Finance Page 2
