Summary
Samenvatting - Investments P5
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Summary of all lectures (book + slides) in the Investments course that will be given at VU Amsterdam in period 5 of the third year of study Economics and Business Economics.
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Week 1Lecture 1:
Refresher slides:
Basic terms:
- Investment portfolio: collection of investment assets
- Securities: invest in specific items across different asset classes, e.g. deposits, bonds, shares
- Risk-return trade-off (or no free lunch) determines choice of securities: how risk-averse are you?
- Passive versus active management
- Financial intermediaries: banks, investment funds…
- Primary versus secondary markets
Expected return, variance, and standard deviation:
- Expected return: 𝐸[𝑅] = ∑ 𝑃𝑅 × 𝑅
2 2
- [ ]
Variance: 𝑉𝑎𝑟(𝑅) = 𝐸 (𝑅 − 𝐸[𝑅] = ∑ 𝑃𝑅 × (𝑅 − 𝐸[𝑅])
- Standard deviation: 𝑆𝐷(𝑅) = 𝑉𝑎𝑟(𝑅)
Historical returns of stocks and bonds:
Historical returns are the same as realized returns. This is the return that actually occurs over a particular time
period:
𝐷𝑖𝑣𝑡+1 𝑃𝑡+1−𝑃𝑡
𝑅𝑡+1 = 𝑃𝑡
+ 𝑃𝑡
= Dividend yield + Capital gain rate
Average annual return:
𝑇
1 1
𝑅= 𝑇
(𝑅1 + 𝑅2 + . . . + 𝑅𝑇) = 𝑇
∑ 𝑅𝑡
𝑡=1
- where 𝑅𝑡 is the realized return of a security in year t, for the years 1 through T.
The variance and volatility of returns:
Variance estimate using realized returns:
𝑇
1 2
𝑉𝑎𝑟(𝑅) = 𝑇−1
∑ (𝑅𝑡 − 𝑅)
𝑡=1
The estimate of the standard is the square root of the variance.
Common versus independent risk:
Common Risk
> Risk that is perfectly correlated
> Risk that affects all securities
Independent Risk
> Risk that is uncorrelated
> Risk that affects a particular security
Diversification
> The averaging out of independent risks in a large portfolio
No arbitrage and the risk premium:
The risk premium for diversifiable risk is zero, so investors are not compensated for holding firm-specific risk.
,The risk premium of a security is determined by its systematic risk and does not depend on its diversifiable risk.
Measuring systematic risk:
Efficient Portfolio: A portfolio that contains only systematic risk. There is no way to reduce the volatility of the
portfolio without lowering its expected return.
Market Portfolio: An efficient portfolio that contains all shares and securities in the market.
The expected return of a portfolio:
The return on the portfolio, Rp, is the weighted average of the returns on the investments in the portfolio,
where the weights correspond to portfolio weights.
𝑅𝑃 = ∑ 𝑥𝑖𝑅𝑖
𝑖=1
The volatility of a two-stock portfolio:
Combining Risks: By combining stocks into a portfolio, we reduce risk through diversification. The amount of
risk that is eliminated in a portfolio depends on the degree to which the stocks face common risks and their
prices move together.
Determining covariance and correlation:
Covariance: The expected product of the deviations of two returns from their means
[ [ ]
- Covariance between Returns Ri and Rj: 𝐶𝑜𝑣(𝑅𝑖,𝑅𝑗) = 𝐸 (𝑅𝑖 − 𝐸 𝑅𝑖 )(𝑅𝑗 − 𝐸 𝑅𝑗 ) [ ]]
1
- Estimate of the Covariance from Historical Data: 𝐶𝑜𝑣(𝑅𝑖, 𝑅𝑗) = 𝑇−1
∑(𝑅𝑖,𝑡 − 𝑅𝑖)(𝑅𝑗,𝑡 − 𝑅𝑗)
𝑡
If the covariance is positive, the two returns tend to move together.
If the covariance is negative, the two returns tend to move in opposite directions.
Correlation: a measure of the common risk shared by stocks that does not depend on their volatility
𝐶𝑜𝑣(𝑅𝑖,𝑅𝑗)
𝐶𝑜𝑟𝑟(𝑅𝑖, 𝑅𝑗) = 𝑆𝐷(𝑅𝑖) × 𝑆𝐷(𝑅𝑗)
Computing a portfolio’s variance and volatility:
For a two security portfolio:
2 2
𝑉𝑎𝑟(𝑅𝑃) = 𝑥1 𝑉𝑎𝑟(𝑅1) + 𝑥2 𝑉𝑎𝑟(𝑅2) + 2 𝑥1𝑥2𝐶𝑜𝑣(𝑅1, 𝑅2)
The volatility of a large portfolio:
More generally, the variance of a portfolio is equal to the weighted average covariance of each stock with the
portfolio: 𝑉𝑎𝑟(𝑅𝑃) = ∑ 𝑥𝑖 𝐶𝑜𝑣(𝑅𝑖, 𝑅𝑃) = ∑ ∑ 𝑥𝑖𝑥𝑗 𝐶𝑜𝑣(𝑅𝑖, 𝑅𝑗)
𝑖 𝑖 𝑗
Risk versus return: choosing an efficient portfolio:
Efficient Portfolios with Two Stocks: In an inefficient portfolio, it is possible to find another portfolio that is
better in terms of both expected return and volatility.
In the figure the portfolio consisting of 100% Coca-Cola and 0% Intel is inefficient because
we can make a portfolio with a higher expected return for the same level of volatility
(standard deviation). For example 40% Intel and 60% Coca-Cola.
Risk-free saving and borrowing:
Risk can also be reduced by investing a portion of a portfolio in a risk-free investment, like
T-Bills. However, doing so will likely reduce the expected return. On the other hand, an
,aggressive investor who is seeking high expected returns might decide to borrow money to invest even more in
the stock market.
Investing in risk-free securities:
Consider an arbitrary risky portfolio and the effect on risk and return of putting a fraction of the money in the
portfolio, while leaving the remaining fraction in risk-free Treasury bills.
The expected return would be:
[ ]
𝐸 𝑅𝑥𝑃 = (1 − 𝑥)𝑟𝑓 + 𝑥𝐸 𝑅𝑃 [ ]
[ ]
= 𝑟𝑓 + 𝑥 (𝐸 𝑅𝑃 − 𝑟𝑓)
The standard deviation of the portfolio would be calculated as:
2
[ ]
𝑆𝐷 𝑅𝑥𝑃 = 𝑥 𝑉𝑎𝑟(𝑅𝑃) = 𝑥 𝑆𝐷(𝑅𝑃)
Identifying the efficient portfolio:
To earn the highest possible expected return for any level of volatility we must find the portfolio with highest
Sharpe Ratio:
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝐸𝑥𝑐𝑒𝑠𝑠 𝑟𝑒𝑡𝑢𝑟𝑛 [ ]
𝐸 𝑅𝑃 −𝑟𝑓
𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 = 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦
= 𝑆𝐷(𝑅𝑃)
Combinations of the risk-free asset and the efficient portfolio provide the best risk and return tradeoff available
to an investor.
An investor’s preferences will determine only how much to invest in the tangent portfolio versus the risk-free
investment.
The Capital Asset Pricing Model (CAPM):
- The equilibrium model that underlies modern financial theory
- Derived using principles of diversification with simplified assumptions
Assumption behind the CAPM:
- Investors are price takers: individual trades do not affect prices
- Single-period investment horizon
- Investments are limited to traded assets
- No taxes or transaction costs
- All investors are rational mean-variance optimizers
- Information is costless and available to all
- Investors have homogeneous expectations
This results in the following equilibrium implications:
- All investors choose to hold the same portfolio: the market portfolio
- The proportion of each stock in the market portfolio is the market value of the stock expressed as a
percentage of total market value
- The market portfolio is the tangency portfolio: the Capital Market Line (CML) is the best
attainable CAL
→ The risk premium of the market portfolio M:
Recall that the optimal investment in the risky portfolio for the individual investor is:
𝐸(𝑟𝑀−𝑟𝑓)
𝑦= 2
𝐴σ 𝑀
In equilibrium net borrowing and lending across investors is 0 ➔ y=1 on average, so we
obtain:
2
[ ]
𝐸 𝑟𝑀 − 𝑟𝑓 = 𝐴σ 𝑀
, The mutual fund theorem:
Why do all investors hold the market portfolio?
- The passive strategy (= investing in the market index) is efficient
The mutual fund theorem (the separation property):
- All investors choose to hold a market index mutual fund
- The allocation between the mutual fund and the risk-free asset depends on individual investor’s risk
aversion
Matrix notation: portfolio variance:
2
The variance of the portfolio is: σ 𝑀
= 𝑤'∑ 𝑤
- Where w is a vector of portfolio weights
- and ∑ is the variance-covariance matrix
Asset i’s contribution to portfolio’s variance:
𝑛
𝑤𝑖 ∑ 𝑤𝑘 𝐶𝑜𝑣(𝑟𝑖, 𝑟𝑘)
𝑘=1
The expected returns on individual securities:
The contribution of asset i to the market portfolio’s variance is:
𝑤𝑖𝐶𝑜𝑣(𝑟𝑖, 𝑟𝑀)
The contribution to the market risk premium:
[ ]
𝑤𝑖(𝐸 𝑟𝑖 − 𝑟𝑓)
Then, the reward-to-risk ratio for asset i is:
𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚. [ ]
𝑤𝑖(𝐸 𝑟𝑖 −𝑟𝑓) [ ]
𝐸 𝑟𝑖 −𝑟𝑓
𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖 𝑡𝑜 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
= 𝑤𝑖𝐶𝑜𝑣(𝑟𝑖,𝑟𝑀)
= 𝐶𝑜𝑣(𝑟𝑖,𝑟𝑀)
And for the market portfolio:
𝑀𝑎𝑟𝑘𝑒𝑡 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 [ ]
𝐸 𝑟𝑀 −𝑟𝑓
𝑀𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
= 2 The market price of risk
σ 𝑀
In equilibrium the reward-to-risk ratios of the individual securities and the market should be equal, so that we
obtain:
𝐶𝑜𝑣(𝑟𝑖,𝑟𝑀) 𝐶𝑜𝑣(𝑟𝑖,𝑟𝑀)
[ ]
𝐸 𝑟𝑖 − 𝑟𝑓 =
σ
(𝐸[𝑟𝑀] − 𝑟𝑓)
2
𝑀
here
σ
2
𝑀
= β𝑖
𝐸[𝑟𝑖] = 𝑟𝑓 + β𝑖 (𝐸[𝑟𝑀] − 𝑟𝑓)
Bond pricing:
𝑇 𝑇
𝑃𝐵 = ∑ ( 𝐶
𝑡=1 (1+𝑟)
𝑡 ) +
𝐹
(1+𝑟)
𝑇 here ∑ (
𝑡=1 (1+𝑟)
𝐶
𝑡 ) is PV of coupon payments | 𝐹
(1+𝑟)
𝑇 is PV of par
× ⎡⎢1 − ⎤
𝐶 1 𝐹
= 𝑟 𝑇 ⎥+ 𝑇
⎣ (1+𝑟) ⎦ (1+𝑟)
- 𝑃𝐵 is the bond price
- 𝐹 is the face value
- 𝐶 is coupon payment per period = 𝐹 × 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑝𝑒𝑟𝑖𝑜𝑑
- 𝑟 is the periodic interest rate
- 𝑇 is the number of periods until maturity
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