Stella section

Question 1

Define asset price

Answer 1

the amount paid for an asset, represents amount of value the market has been assigned to an asset

Question 2

Define log returns

Answer 2

The log return n is defined as the logarithmic price changes on an asset, with appropriate adjustments for any dividend
payments. Let pt and rt denote the price and the log-return at time t respectively. If ignoring dividends, then rt = log(pt/pt−1).

Question 3

Define gross return

Answer 3

Ratio of two prices

Question 4

Define simple returns

Answer 4

simple return Rt of an asset over time period t can be measured
by the sum of the change in its market price plus any income received
over the holding period divided by its price at the beginning of the
holding period

Rt = (pt − p_(t−1) )/ p_(t−1)

Question 5

Relate simple returns to log returns

Answer 5

rt = log(1 + Rt)

Question 6

Give the 11 stylised facts

Answer 6

AHGAIVSCLVA

• absence of autocorrelations
• heavy tails
• gain/loss symmetry
• aggregational gaussanity
• intermittency
• volatility clustering
• slow decay of autocorrelation in absolute terms
• conditional heavy tails
• leverage effect
• volume/volatility correlation
• asymmetry in time scales

Question 7

State the three most important stylised facts

Answer 7

• The first important stylized fact for returns is that their distribution is not normal.
• The second important stylized fact is that the sample autocorrelations of
  returns are generally close to zero, regardless of the time lag.
• The third major stylized fact is about the positive dependence between absolute returns or
  squared returns on nearby day

Question 8

Give the RWH for modelling returns

Answer 8

rt = µ + et, t = 0, ±1, ±2, . . . ,
where {et} is weakly stationary with E[et] = 0 and Var(et) < ∞

note et = σtεt

Question 9

What is the H0 for testing the RWH in modelling returns?

Answer 9

Cov(rt,rt+h) = 0 for h≠0, i.e., {et} ∼ WN(0, σ^2)

Question 10

Give the Q-test for the RWH

Answer 10

Qτ = n(n + 2)*sum(k=1 to τ) ρˆ2_k/(n − k) ≈
sum(k=1 to τ) nρˆ2_k

Under the RWH, Qτ→ χ^2_τ

Question 11

Give the variance-ratio test for modelling returns by the RWH

Answer 11

VR(N) = V(N) / NV(1) = 1 + 2/N * sum(τ=1 to N-1) (N − τ )ρτ

zN = (VR( ˆ N) − 1) / sqrt(v_N/n)∼ N(0, 1)

Question 12

When does the variance ratio test perform better than the q-test?

Answer 12

If the null hypothesis is heteroskedastic

Question 13

Give the sample skewness formula

Answer 13

β_2 = 1/n * sum(i=1 to n) (ri − rbar)^3 / S^3

Question 14

Give the sample kurtosis formula

Answer 14

κ_2 = 1 / )n−1) sum(i=1 to n) (ri − rbar)^4 / S^4

Question 15

Give the JB test statistic

Answer 15

JBn = n*(β^2/6 + (κ_2 − 3)^2 /24→ χ_2

Question 16

Define volatility

Answer 16

Volatility is a measure of price variability over some period of time. It is defined as the standard deviation of the change in the logarithm of a price
during a stated period of time

Question 17

Define conditional volatility

Answer 17

the standard deviation of a future log return that is conditional on known information such as the history of previous returns

Question 18

Define unconditional volatility

Answer 18

the standard deviation of a log return without

conditioning on the history of previous returns

Question 19

Define a GARCH model

Answer 19

rt = µt + et, et = σtεt
where
- εt is a white noise,
- µt is a trend process (modelling the conditional mean).
- σt is a volatility process (modelling the conditional variance)

Question 20

Give the formula for sharpe ratio

Answer 20

SR = E(rt - rf) / sqrt( Var(rt)

Question 21

Define sharpe ratio

Answer 21

The ratio of the expected excess return of an investment to

its return volatility or standard deviation

Question 22

Give the single factor model for asset returns

Answer 22

rit = αi + βirmt + εit, (1)
where:

• rit and rmt are the log returns of the ith stock and the common
  market index respectively
• αi is the i-th stock specific effect, independent of market
  performance
• εit represents a random error E[εit] = 0 and V[εit] = σ^2_ei,
• βi is a factor loading that measures the expected change in rit for a unit change in rmt
• εit’s uncorrelated with each other and uncorrelated with rmt

Question 23

Give expectation of a single factor model

Answer 23

E[rit] = αi + βiE[rmt]

Question 24

Give variance of a single factor model

Answer 24

V[rit] = βi^2 * σm^2 + σei^2

Question 25

Derive estimators of βi and αi

Answer 25

S = sum(i=1 to n) (rit − α − βi * rmt)^2

For β, dS/dβ = 0
For α, dS/dα = 0

βi =sum (t=1 to n) [(rit − rbar_i) * (rmt − rbar_m)] / sum(i=1 to n) (rmt − rbar_mt)^2

αi = rbar_i − βi*r¯bar_m

Question 26

Give the estimator for σei^2

Answer 26

1 /(n-2) * sum(t=1 to n) (rit − αˆi − βˆi*rmt)^2

Question 27

Give the estimator for σm^2

Answer 27

1 /(n−1) * sum (t=1 to n) (rmt − rbar_m)^2

Question 28

Give the estimator for σi^2

Answer 28

βiˆ2 * σm^2 + σε^2

Question 29

Give the estimator for σij

Answer 29

βiβjσm^2

Question 30

Define CAPM

Answer 30

E(Ra) = Rf + βa(E(Rm) − Rf)
where:
- Ra is the return on an asset
- Rf is the risk-free rate of return
- Rm is the return on the market

Question 31

Define CAPM in terms of excess returns

Answer 31

E(ra) = βaE(rm)

ra = Ra - Rf
rm = Rm - Rf

Question 32

in terms of CAPM give estimators for βa and αa

Answer 32

βa = Sam / Smm
αa = rbar_a - βa * rbar_m

where
Sam = sum(i=1 to n) (r_at - rbar_a) * (r_mt - rbar_m)

Smm = 1/n * sum(i=1 to n) (r_mt - rbar_m)^2

Question 33

Give an estimator for var(αa)

Answer 33

1/(n-2) * (Saaf - Sam^2/Smmf)

Question 34

Give the test statistic for testing CAPM

Answer 34

t = αa / sqrt ( Var(αa)hat)

we reject H0 if |t| > tn−2(η/2)

Question 35

Define value at risk

Answer 35

For a confidence level α, the Value-at-Risk, VaR(α), is chosen so that the probability of a loss (over a given time horizon) larger than the VaR(α) is equal to α

If L is the loss over the holding period, then VaR(α) is the α-th upper quantile of L

VaR(α) = inf{x : P(L > x) ≤ α}.