Risk and return (w2) Flashcards
What is the difference between nominal and real interest rate?
Nominal = growth rate of money
Real = growth rate of purchising power
How is real interest rate (rr) calculated?
Real interest rate = (rn - i) / (1 + i)
What is the Fisher Equation?
rr = rn + E(i)
Denotes the relationship between demanded nominal rates and inflation
As expected inflation E(i) increases, so does the expectation for the nominal rate (rn) to minimise the decrease in real rate (rr)
To ensure the value of money does not decrease overall
How to calculate a risk free rate of return on a zero coupon bond?
Risk free return at time t=1
rf = face value / price at maturity -1
rf = fv / P(t) - 1
Face value = par value –> price received at maturity
How is holding period return determined?
HPR = (Price in period t1 - Price in period t0 + Dividends in t1) / Price in period 0
HPR = (P1 - P0 + D1) / P0
Why are long term, low-risk investment unattractive?
Because of the negative correlation between inflation and real interest rate, the nominal rate often does not compensate for the soaring inflation, decreasing the value of the investment
How are expected returns of an investment calculated? Why is it important to estimate those?
Expected return = sum of (probability x return) for all states
E(r) = Σp(s) * r(s)
Important as when entering an investment, future return is unknown = must be estimated
Expected return = weighted mean
You’ll need E(r)
How to calculate variance of an investment?
Var = sum of [probability * (return in state - expected return) ]
Var = Σ p(s) * [ r(s) - E(r) ]^2
Note: return for each state is adjusted by the expected return to track deviation around the mean
Note: because it is a variance, r(s) - E(r) is SQUARED to ensure all values are positive
Risk = deviation around the mean
How to calculate variance when probabilities are unknown? How to convert it into SD?
VAR = 1/n * Σ r(s)
SD = sqrt of var
note: for SD, n becomes n-1
SD = sqrt of final VAR equation
What is risk premium?
The difference between the return on a risk free asset r(f) and the risky investment
Calculated as a difference between the HPR of both
What is Sharpe Ratio?
Sharpe Ratio = risk premium / SD of excess return
Note: since the risk free asset has a SD = 0, SD of excess return = SD of a risky asset
Sharpe Ratio = the slope of CAL
How does SD measures risk?
SD measures volatility
Therefore, the lower the SD the safer the stock
–> If SD is present, positive SD is preferred
If SD > E(r)
–> Expected returns do not compensate for the risk
–> The returns can be expected to be negative
How to estimate investment returns using historical data?
Since future states and their probabilities are unknown
Expected return = sum of historical returns / number of time periods
E(r) = 1/n * Σ r(s)
Probabilities of scenarios are replaced with historical outcomes which are observable
Why is normal distribution of the returns important?
3 reasons
- If returns follow a normal distribution, SD is an accurate measure of risk
- If all security returns are symmetrical, so will be the return of the portfolio
–> E(r) and SD can be used for estimating scenarios
What are the 2 consequences of not normally distributed returns?
- SD does not reflect risk accurately
- Sharpe ratio no longer gives an accurate measure of portfolio performance
What are the 2 reasons for negative skew in investment portfolio?
In terms of capitalisation: many small firms, a few giants
In terms of returns: small firms often outperform large firms (SMB)
What 3 properties do risk averse investors look out for in an investment?
- Risk free returns
- If risk is assumed, positive risk premium is required
- Mean variance principal
What is the equation for utility, using risk aversion? When will the risk aversion have a positive or negative value?
U = E(r) - (1/2 * A * var)
Risk aversion: A > 0, the aversion offsets the return
Risk seeking: A < 0, risk adds to the return in creating utility
Risk neutral: A = 0, the utility is solely based on E(r)
What is the mean-variance portfolio?
For portfolio A to dominate portfolio B
E(r) of A >= E(r) of B
SD of A <= SD of B
–> at least one must be an inequality to establish dominance
What characterises all portfolios on the same indifference curve?
- They yield the same utility
- They are equal according to the mean-variance criterion
However, higher indifference curve is preferred
How to calculate an optimal asset allocation in complete portfolio that minimises risk?
When expected returns, SD and risk aversion is given
y* = [ E(r) - r(f) ] / (A * Var)
Var = variance of the risky portfolio as risk free asset has a variance = 0
Derivative of r(f) + y * [E(r) - r(f) ] - [ 1/2 A * y^2 * var ]
What is the difference between risky and complete portfolio?
Risky = combination of risky investments, manipulated through weights
Complete = combination of risky and risk free asset, based on risk aversion
What is Capital Allocation Line?
Investment opportunity set
The graphical representation of the Sharpe Ratio
–> slope of CAL shows the marginal increase in E(r) for the unit increase of risk
Depicts all possible complete portfolios with different weights y in the risky portfolio
What causes the change in steepness of the CAL?
Different lending and borrowing rates in the investment opportunity set
How is the E(r) of the complete portfolio calculated?
E(r) = r(f) + y * [ E(r) - r(f)
Expected return of the complete porftolio = risk free return + risk premium adjusted by the weight of the risky portfolio in the optimal porfolio
How is the variance of the complete porfolio estimated?
Variance = y^2 * SD^2
SD = y * SD
What is a passive investment strategy? What are its advantages?
Diversification and market forced
Investment in a market index, which is diversified and adjusts based on supply and demand forces of the market
–> When using an active strategy, the cost of management must be considered, passive strategy gives the free rider benefit
–> Returns are comparable to those of an active investment strategy as it is difficult to outperform the market
What is the Capital Market Line?
Capital Market Line = graphical depict of feasable investment options in a complete portfolio, which uses risk free asset and market index instead of the risky portfolio
