Quantitative and Economics Flashcards
Use linear equation to forecast results one quarter ahead
Salest=Bo + B1t + et
- Analysis done over 15 years
- Intercept coefficient 10
- Trend coefficient 16
- Time variable (quarter #) t
Yt = 10 + 16(61) = 986
- Remember you are forecasting out an additional quarter plus the 60 quarters already studied.
Sample covariance (cov)
∑(x-xbar)(y-ybar) / n-1
Sample correlation coefficient (r)
Cov(x,y) / (Sx)(Sy)S = Standard Deviation
Limitations of Correlation Analysis
- Linear relationships (not quadratic)2. Outliers (news vs. noise)3. Not causation4. Spurious (Chance, mixed third variables)
Two-tailed “t-test” with n-2 degrees of freedom to tell if population correlation is 0 or not 0
t = r√n-2 / √1-r^2
Decision rule for two-tailed test
Reject H0 if1. t-stat > +ve tc (Positive critical value)2. t-stat < -ve tc (Negative critical value)3. t-stat > |Tc||Tc| = T critical value
Dependent Variable (which axis and description)
- Y-axis2. Seeking to explain or predict the Y variable
Independent Variable (which axis and description)
- X-axis2. Used to explain or predict the Y or dependent variable
Regression Model Equation
Yi = b0 + b1Xi+Ei1. i = 1,…,n2. This uses the estimates of Y (includes a carot above the Y and b)
Some of Squared Errors (SSE)
∑(Yi - Yi~ or (b0+b1Xi))^2
Calculation for b1
b1 = Cov(X,Y) / Var (X)
Calculation for b0
b0 = Yaverage - b1(Xaverage)
The midpoint of a regression series
= (Xbar, Ybar)1. will be plotted on the regression line
Standard Deviation
NAME?
Steps for calculating correlation coefficient
- calculate sum and mean of each variable2. Calculate cross product of (xi-xbar)(yi-ybar) for each variable (by row)3. Calculate squared deviation for each variable (by row)
Regression Line Equation
Yi~ = b0~+b1~Xi~ = ‘hat’ or estimate of these variables
Find correlation of two stocks using market model
Calcuate total risk (standard deviation) for both securities being compared. V’ is the common macro factor variance:Security 1 σ² = (β² x V’) + unsystematic risk²Calculate covariance:Beta(a) X Beta(b) X common macro factor varianceCalculate correlation: cov/totalriskA X total risk Y
Calculate a Cross Rate
Cross Rate = (F/USD) / (D/USD)
- Imagine that “usd” is a common denominator
Relative Purchasing Power Parity
ES1 = So X [(1+inflationF)^2 / (1+inflationD)^2]
International Fisher Relation (determine if projected inflation rates are consistent)
(1+rF%USD)/(1+rF%F) = 1+expectedinfUSD/1+infF = should equal exp infF
(1+rF%D)/(1+rF%USD) = 1+infD / 1+expinfUSD = should equal exp infD
Uncovered Interest Rate Parity Forecast
So[(1+rF%D)/(1+rF%F)] = Uncovered Interest Rate Parity Forecast
This is also how to calculate a no-arbitrage forward rate value.
DON’T FORGET TO BREAK THE RISK FREE RATE OUT BY THE TIME/360
Mundell Flemming Model (expansionary Monetary Policy and Fiscal Policy)
Expansionary Monetary Policy
- Flexible exchange rates: lower interest rates and currency drops
- Fixed exchange rates: expansionary policy is futile and limited to FX reserves
Expansionary Fiscal Policy
- Flexible exchange rates: increases inflation&interest rates driving currency higher
- Fixed exchange rates: To prevent currency appreciation currency will be sold and increase money supply
Average Cov of port approaches total average Cov when…
The number of assets in the portfolio are large.
Minimum Variance Portfolio
Portfolio that has the smallest variance among portfolios with identical expected returns.
Global Minimum Variance Portfolio
The portfolio that sits to the furthest left on the efficient frontier. All portfolios below are inefficient and all portfolios above may be efficient on the CAL.
Sharpe Ratio (formula and slope of what?)
- Sharpe ratio = (ErP - Rf%) / SdP
- Sharpe ratio = Slope of Capital Markets Line
Expected Return of Portfolio
ErP = W₁(Er₁) + W₂(Er₂)…
Market Portfolio has Highest Possible Sharpe Ratio, Therefore….
All other portfolios must have a sharpe ratio below the market’s sharpe ratio.
A Factor Portfolio (definition)
A portfolio with a beta of 1 to a single factor and a beta of zero to other factors. Will have more active factor risk than a tracking portfolio.
Y-Axis Intercept for Multifactor Model
= Expected value of multifactor model. The start calculation is the intercept because multifactor models are based on sensitivity to surprises.
Ex-post Information Ratio
IR = (α / standard error of α) / √n
- √n: The number of periods covered by the past α
- α: Estimated coefficient of alpha
- α / Standard error of α: the t-statistic of α
- Standard error of α = SDα / √n
Annualized Value Added for Portfolio
Anualized VA = α - (ƛ X ω²)
Step 1: Annualized α = α(n)
Step 2: Annualized ω = ω(√n)
- ω: Residual risk*
- α: Residual return*
Optimal Annual Residual Risk (for portfolio)
ω* = IR/2ƛ
- ω*: Optimal annual residual risk*
- IR: Information ratio*
Information Coefficient for Market Timing
IC = (2 X Winning%) - 1
Information Ratio for Market Timing
IR = IC X √BR
- BR = Breadth (number of trades, which is like n-periods for managers)*
- IC = Information coefficient*
- BR may be independent bets on individual stocks which would then be BR/#of independent stocks*
Combined Information Ratio
Step 1: IRcom² = IRx² + IRy²
Step 2: IRcom = √IRcom²
- IRcom: Information ratio for both strategies
Capital Deepening (economics)
- An increase in the capital to labor ratio.
- More foriegn investment can drive this.
- Moves economy further along the productivity curve
GDP Growth Rate attributable to Total Factor Productivity
GDP = ΔTFP + α(LT growth rate in capital) + [(1-α)(LT growth rate in labor)]
- α: 1 - (labor cost/total factor cost)*
- TFP is equal to the growth rate of technology*
Problem Predicted by Classical Economic Model (dismal)
Technology will not lead to individual wealth because it will increase population growth until only subsistence is possible.
Wrong because….
- Technology correlated with slowing population growth
- Technologies growth greater than population growth
Exogenous Economic Growth Models Predict…
Technological advances will benefit multiple sectors of the economy as new investments are made within and across companies.
Covered Interest Rate Parity / Covered Interest Arbitrage
Covered parity: F/S = (1+Rf%”A”)/(1+Rf%”B”)
Covered arbitrage:
- If F/S < rate calculation then short A
- If F/S > rate calculation then short B
- F/S: Forwardrate/Spotrate*
- A/B: Units of A per B ie: USD/GBP = 2*
Proper transformation for dependant variable in linear equation if there is a exponential rate of growth
Logarithmic transformation
- You may be able to multiply each side by its natural log.
Using an AR model of Order 1
- Find mean reverting value?
- Will next period be higher or lower than previous?
- Mean Reverting Value = Intercept coef / 1 - (Lag 1 coef)
A. If MRV is lower than prior value then next value expected to fall (revert)
B. If result of linear model is lower than prior value then next is expected to fall
P-Value
The smallest level of significance at which the null hypothesis can be rejected.
If p=.02 then null is rejected for .05 level of significance
Auto Regressive Heteroskedasticity
- What does it mean?
- Is it exhibited in the model?
- ARCH tests to see if variance of error terms is constant
- If lagged residual squared coefficient p-value is below significance level (<.05) then it is in the model
Sunset provisions in regulation
A law will cease unless other laws are implemented to extend it.
Regulatory Burden
the costs to the regulated entity from regulations.
r Squared
- how much of the movement of the dependent variable can be explained by the dependent variable.
2.
Determine F Value that tests the hypothesis that all coefficients equal zero
F = Mean regression sum of squares / Mean error sum of squares
F = (R²/total dependent variables) / (ESS / Degrees of Freedom)
Conditional Heteroskedasticity
Variance of the regression errors are not constant and related to regression independent variables
If stock value based on CAPM is greater than forecasted expected returns…
The stock is overvalued.
Country most likely to benefit from capital deepening
The country with the highest α
α = Share of capital to GDP or cost of capital to Total Factor Cost
Factors that affect the growth rate of labor
- Increase in labor participation rate
- Increase in hours worked
- Higher population growth (or immigration)
In steady state (equilibrium) economy “rent” or marginal price of capital is?
r = αY/K
Y = GDPo
K = Capital base
Active Factor Risk
Deviations of a portfolio’s factor sensitivity from the benchmark’s (like the S&P 500) sensitivities.
Arbitrage Pricing Model
- Need to know factor sensitivities and the risk premium for factors.
- Risk premium equals Rf% - The return of a factor portfolio to each factor
- Re = Rf% + β1(ƛ1) + β2(ƛ2)
Calculate Standard Error from slope estimate
= slope estimate / t-stat
Calculate Confidence Interval from slope estimate
= Slope Estimate +/- (crit-t X standard error)
When using t-table with 99% confidence which column should be used?
- one tailed (ie greater than or equal to): .01
- two tailed (not zero): .005
Asset Market Approach focuses on?
- Fiscal not Monetary Policy
Fisher Effect (not Fisher relation)
Real interest rates are a combination of expected inflation and interest rates.
Covered Interest Rate Parity / Arbitrage (correct)
1+rf%D = (1+rf%F)F / Spot rate
Work through borrowing and paying back the currency expected to drop and profiting from the currency expected to rise (both currency values will go up with interest accumulation but the borrowed currency will be negative)
Alpha in the GDP Growth Equation
- 1 - (total labor cost/total factor cost)
- 1 - (GDP paid to labor/GDP)
Endogenous Growth Model (GDP)
Economic growth due to savings and productivity do not reach a steady state or equilibrium because technological growth externalities.
Neoclassical asserts that the economy only grows above equilibrium in brief spurts.
Relative Version of PPP (currency rates)
Determined solely by the difference in expected inflation rates.
So - ((InfA% - InfB%) X So) = Forward Rate
- Predicts that A will grow in value against B if B’s inflation rate is greater
Club Convergence (which economic Growth Model?)
- Neoclassical
- Lower per capital income members of a “club” will converge to the average for the club.
- Lower level club members will converge as they borrow or imitate technologies of higher level members.
Conditional Heteroskedasticity
The error term has a constant variance with the INDEPENDANT variable
Steady State rate of Economic Growth
Y/Y = (TFP% / 1-a) + n
- TFP%: Growth in total factor productivity
- 1-a (this is the % of labor cost to total factor productivity)
- n: growth rate in labor force
Growth rate in potential GDP
= Growth rate of labor + Growth rate of labor productivity
= n +
International Fischer Effect (real interest rates)
R% = N% - EXPECTEDInf%
Taylor Rule
i = rn + π + α(π-π*) + β(y-y*)
i = prescribed interest rate
rn = Neutral real policy rate
π = Current inflation
π* = Target inflation
y = log of current level of output
y* = log of economy’s potential/sustainable level of output
Bo and B1 if regression is a random walk
Bo will be close to 0 and B1 will be nearly 1.
Best forecast for a random walk variable into the next period
The current period’s value is the best forecast for the next period of a random walk variable.
Test whether regression series is ARCH by what?
B1 (or C1) will be significantly different from 0
When analyzing two time series in regression you need to ensure 2 things…
- Neither the dependent variable series nor the independent variable series has a unit root… or
- That both series have a unit root and are cointegrated.
Cointegrated variables
Two or more time series are cointegrated if they share a common stochastic drift. Like futures and stock indexes.
