Week 4 Reivew - Summary of key points from lecture notes on managed funds and the Treynor-Black PDF

Preview

Summary

Week 4 – Chapter 4, Chapter 6, 7.Key Points NotesManaged Funds Managed Funds Managed funds = financial intermediaries which collect money from many individuals and incest in a wide range of securities and other assets.ETFs & REITs Managed Funds and Listed Investment CompaniesHedge FundStrategy T...

Description

Week 4 – Chapter 4, Chapter 6.5, 7.2 Key Points

Notes Managed Funds Managed Funds Managed funds = financial intermediaries which collect money from many individuals and incest in a wide range of securities and other assets. ETFs & REITs Strategy

Typical full replication of benchmark

Regulation Fees Liquidity

HIGH Low Buy/sell in real time

Initial Price of 1 share Investment Evaluation Tracking error ¿ β

Managed funds can be classed as either

Managed Funds and Listed Investment Companies Mostly long positions – anything from benchmark tracking to active strategies to outperform benchmarks Medium: quarterly disclosure Medium End of day

$ 1000−$ 5000

Tracking error/ β /¿ Sharpe-Ratio/ α

Hedge Fund Non-directional bets – whatever they want Low: Annual Disclosure High Low: Lock-in periods

¿ $ 25000 α

Types of Managed Funds 1. Open-end a. All purchases and sales occur through the fund itself b. New shares are created when they are purchased and old shares are “destroyed” when they are sold back to the

open or closed ended. They are priced accordingly.

fund c. Often unlisted i. Unlisted funds = funds that are traded through the investment manager or a dealer. 2. Closed-end a. The fund does not redeem or issue share. All transactions must be with other investors b. Listed: funds that are traded on a public exchange (e.g. ETFs) c. Unlisted: (e.g. managed funds) Net Asset Value That value of the fund’s assets less its liabilities on a per share basis: market value of assets−liabilities NAV = shares outstanding Managed Fund Pricing 1. Open end a. Shares can be sold at NAV. b. Purchase price may be higher than NAV due to buy-sell spread and if the fund has a “contribution fee” 2. Closed-end a. Shares can sell at a premium or discount to NAV b. Exchange traded funds will usually trade close to their NAV c. Non exchange traded funds can deviate from their NAV significantly What do managed funds do? 1. Record keeping and administration a. Keep track of capital gains distributions, dividends, investments and redemptions 2. Diversification and divisibility a. Enable investors to hold fractional shares of many different securities. b. Can act as large investors where individual investors cannot. 3. Professional managers a. Staff who attempt to achieve superior investment results

b. Do they actually achieve much though? 4. Reduced transaction costs a. Can achieve substantial savings on brokerage fees and commissions because they trade large blocks of securities Fees for managed funds  Entry fee: paid upon first investing o Contribution fee: for additional investments after the first o Switching fee: for changing funds  Exit (withdrawal) fee: paid when disinvesting  Trailing (advisor service) fee: paid to the financial advisory  Management expense ratio  Performance fees (if the fund beats the benchmark) Implicit Fees  Soft dollars o Kick-backs to managers o Not cash, but service  Trading costs o Buy-sell spread from unlisted funds o Bid-ask spread for ETFs  Fund turnover Fund Fees and Performance  Fees are set by contract o Usually a percentage of the fund’s size  Not usually taken out up-front o accumulate and charged as the year progresses  Can affect net performance (can end up paying out the benefits of investing in the fund through the fees that you have to pay) Taxes and Tax Efficiency  Managed funds usually have pass through status

o The fund itself does not pay taxes, the investors do Taxes impact performance if not in a tax-exempt investment o Superannuation funds o Capital gains on all sales  High turnover: funds frequently selling stock can have large impact in terms of tax liability  You can be taxed on gains that occurred before you bought into the fun Investment Policies  Single asset funds o Invest only in specific asset classes  Multi-asset (diversified) funds 

Investment Philosophies  Passive management o Holding a well diversifies portfolio without attempting to search for security mispricing  Active management o Attempts to achieve portfolio returns more than commensurate with risk Index Models What is an index model? An index model is an Index model: atheoretical model, statistical model designed to estimate and distinguish the firm-specific and covariance risk.  Purpose is to segregate the risks that a security faces into diversifiable, firm-specific risk and undiversifiable, systematic atheoretical model used risk  An index model represents an assets returns for firm i as a function of firm specific ( e i ,t ) and K other asset or to distinguish portfolio covariance risks K between r i ,t −r f ,t =αi + ∑ β i ,k ( r k , t−r f , t ) + ei ,t firm k=1 specific and α i : a consistent return not explained by covariances or transitory firm-specific shocks. o covariance o This is not a risk measure. This term represents the expected return on the share beyond any return induced risk. by movements in the relevant factor.

α> 0 → indicates an underpriced security. The security is earning returns more than what it should given its risk. o α< 0 → indicated an overpriced security. The security is earning returns less than what it should given its risk. COV ( r i , r k ) o β i ,k = : return due to covariances with other assets/portfolios VAR ( r k ) o The beta represents the influence of movements in a particular factor on the security’s returns. In this way, the beta is a measure of comparative sensitivity o e i ,t : firm specific or residual risk o This is the impact of unanticipated firm-specific events. The expected value of this is zero.

Using index models, we can decompose the risk of the firm into firmspecific and systematic risk.

o

Risk Decomposition K

r i ,t −r f ,t =αi + ∑ β i ,k ( r k , t−r f , t ) + ei ,t k=1

We can translate this form of the index model directly into risks (if we are looking at a single factor only): VAR ( r i −r f )=VAR ( α i +β i , k ( r k ,t −r f ,t )+ ei , t ) r k ,t −r f ,t β i ,k (¿) ¿ VAR ( r i −r f )=VAR ¿ VAR ( r i −r f )= β i2 σ 2k +σ e2 i

Therefore: total risk of a given asset is the sum of covariance/systematic risk and firm specific risk. σ2i =β2i σ 2k +σ e2 i



The first of these terms is the variance attributable to the uncertainty of the factor. This systematic risk is attributable to the risk in the factor risk premium. o The systematic risk of the share is dependent on both the sensitivity of the share to the factor and the volatility

of the factor The second of these terms is the variance attributable to firm-specific fluctuations. This is the variance in the share’s performance that is independent of the factor’s performance. Estimating Index Models with OLS  The index model equation may be interpreted as a single-variable regression equation of the share returns on the factor returns. o The excess returns (above the risk-free rate) of the security is the dependent variable that is to be explained by the regression. o Regression analysis allows us to use the sample of historical returns to estimate a relationship between the dependent variable (the security’s returns) and the explanatory variable (the factor returns)  Security Characteristic Line o Plot of a security’s expected excess return as a function of the excess return on the market o In this case we are using the market as the factor on which we are regressing the returns of the security. 

These statistics can be estimated using OLS regression. The R2 of the regression is an indication of the proportion of the total risk of the firm that is explained by the market (therefore, what risk is systematic vs not)



Features of the regression and of the SCL

o Regression intercept: the intercept of the SCL with the vertical axis  This represents the excess return on the security when there is zero excess return on the market. Therefore, this gives the excess return that the firm is expected to earn consistently when the market is not earning any excess returns.  Regression intercept = α i o Regression slope  This represents the sensitivity of the security’s returns to a unit change in the market’s excess return.  Regression slope = β o Residuals  The regression line does not represent actual returns. The line represents average tendencies.  Any deviation from the expected performance of the security (which is given by the regression) reflects unexpected, firm-specific shocks. o Regression R2  The residual variance, that is, the dispersion of the scatter of actual returns around the regression line, measures the effects of firm-specific events. Since the residuals represent firm specific shock, the dispersal of those residuals must be a representation of firm-specific events.  One way to measure the importance of systematic (explained by the regression) as opposed to firm-specific (unexplained = residuals) is the R2 measure. This is the ratio of systematic variance to total variance (that is, explained variance divided by total variance). systematic ( explained ) variance total variance β2i σ M2 2 R= 2 2 β i σ M +σ 2e 2

R=

i

We need to future adjust our

Predicting β  When calculating regressions, and therefore betas, we use historical data. But we are typically calculating this as a prediction tool. Therefore, we need figure out a way to make this representative of the future.

betas. We do this using that fact that they tend to mean revert.



Betas tend to mean revert o Over time, when the beta is below the mean, it will typically rise. When the beta is higher than the mean, it will typically fall. o Since the average beta is 1 (because the average share must by construction be the equivalent to the market index) we can calculate the following:  Note this is just one equation for a shrinkage estimator 1 2 Adjusted β= ∗historic β + ∗1 3 3 Portfolio Optimization: Using Index Models to Reduce Dimensionality  To create an efficient frontier, you need to calculate portfolio returns, variances and covariances (for every pair of assets in the portfolio). With 1000 assets, you need to calculate 500500 variances and covariances.  Instead, if we use an index model, then: Portfolio beta: β P , k =∑ wi β k , i Portfolio variance is: K

N

K

2 σ p=∑ ∑ β k , P βl , P cov ( r k , r l )+ ∑ w2i var ( e i ) i=1

k=1 l=1

Treynor Black Model How do we know whether to add an asset to the portfolio? Firstly, suppose you have an investment strategy that is bench-marked to K portfolios. The index model to explain your returns is: K

r i ,t −r f ,t =αi + ∑ β i ,k ( r k ,t −r f ,t )+e i ,t k=1

In reality, we will not know the actual efficient frontier, we only know the portfolios that we are benchmarking against. Given this, we must use the α of the asset to determine whether or not to add the asset to the portfolio.  α is statistically different from zero and is expected to persist → add the asset to your portfolio.

This leads us to the Treynor-Black model. This model requires:  N estimates of the securities’ α i (non-market risk premia)  N estimates of the β i 2  N estimates of the firm-specific variances: σ e  One estimate of the market risk premium: r m−r f 2  One estimate of the market variance: σ m i

Treynor Black Optimization Procedure: 1. Compute the initial position in the “active” portfolio αi 0 wi = 2 σe 2. Rescale the initial weights such that they sum to 1 0 w w i= N i i

∑ w0i i=1

3. Using the rescaled weights, calculate the α A =∑ w i α i β A =∑ w i β i

α and β

4. Compute the residual variance of the active portfolio N

a.

σ2e =∑ w 2i σ 2e A

i=1

of the portfolio.

The index model should fully explain covariances so we need not calculate any correlation term.

i

5. Compute the initial weight of the active portfolio in the overall risky portfolio

αA σ e2

0

w A=

A

E [ r M ]− r f 2

σM 6. Adjust the initial weight allocated in the active portfolio 0 wA ¿ w A= 0 1+w A ( 1−β A ) 7. Calculate the weight in the passive, benchmark portfolio ¿ ¿ w M =1−w A Using these final weights, we can calculate the optimized risk premium on the portfolio: α A +β A (E [r M ]− r f ) E [ r O ]− r f =w M ( E [ r m ] −r f ) + w A ¿ E [ r O ]− r f =w A¿ α A +( w M +w ¿A β A )( E [ r M ]−r f ) The variance of the portfolio is then: 2 ¿ ¿ σ2A=( w M + w A β A ) σ M2 +( w A σ A ) The Sharpe Ratio of the new optimized portfolio is: α A 2 E [ r O] −r f ¿ = S O= S M + σO σ eA

√

Summary

( )

Managed funds: market value of assets−liabilities NAV = shares outstanding

Index Models: K

r i ,t −r f ,t =αi + ∑ β i ,k ( r k , t−r f , t) + ei ,t k=1

E ( e i ,t ) =0

β i ,k =

COV ( r i , r k ) VAR ( r k )

2 2 2 2 Risk Decomposition: total risk = sum of firm specific and systematic risk → σ i =β i σ k +σ e

i

2

2

R:

2

ratio of the variance explained by the index model and the total variance → R =

1 2 Future adjusted beta (based on mean reversion): Adjusted β= ∗historic β + ∗1 3 3 Portfolio beta: β P , k =∑ wi β k , i K

K

N

Portfolio variance: σ p =∑ ∑ β k , P βl , P cov ( r k , r l) + ∑ w2i var ( e i ) 2

k=1 l=1

i=1

Treynor-Black Optimization: Treynor Black Optimization Procedure: 1. Compute the initial position in the “active” portfolio

2

βi σ M 2 2 β σ M +σ e 2 i

i

αi

0

wi =

σ2e 2. Rescale the initial weights such that they sum to 1 0 w w i= N i i

wi0 ∑ i=1

3. Using the rescaled weights, calculate the α α A=∑ wi α i β A=∑ wi β i

and β

of the portfolio.

4. Compute the residual variance of the active portfolio N

a.

σ e =∑ w 2i σ 2e 2

A

i=1

i

5. Compute the initial weight of the active portfolio in the overall risky portfolio αA 2

w 0A=

σe

A

E [ r M ]−r f

σ 2M 6. Adjust the initial weight allocated in the active portfolio 0 wA w ¿A= 0 1+w A ( 1−β A ) 7. Calculate the weight in the passive, benchmark portfolio ¿ ¿ w M =1−w A Using these final weights, we can calculate the optimized risk premium on the portfolio:

α A +β A (E [r M ]− r f ) E [ r O ]−r f =w M ( E [ r m ] −r f ) +w A ¿ E [ r O ]−r f =w A¿ α A +( w M +w¿A β A ) ( E [ r M ] −r f ) The variance of the portfolio is then: 2 ¿ ¿ 2 2 σ A =( w M + w A β A ) σ M +( w A σ A ) The Sharpe Ratio of the new optimized portfolio is: 2 E [ r O]−r f α S O= S M¿ + A = σ eA σO

√

( )...