Lista 1 – Entrega 19/02/2010.
1. Escolha um caso de perdas financeiras e faça um pequeno resumo sobre os acontecimentos (cerca de 2 ou 3 páginas). Não esqueça de contextualizar os fatos no ambiente econômico, relatar as falhas que levaram aos problemas, as medidas que poderiam ter sido tomadas para evitar as perdas e descrever os tipos de risco.
2. Ex. 14.4 pg. 342 Financial Risk Management Handbook – Jorion Exemplo 14.4: FRM EXAM 2004 – Questão 3
Let ht be the variance at t and r2t-1 the squared return at t-1. Which of the following GARCH models will take the shortest time to revert to its mean?
a. ht = 0.02 + 0,06 r2t-1 + 0,93 ht-1
b. ht = 0.03 + 0,04 r2t-1 + 0,94 ht-1
c. ht = 0.04 + 0,05 r2t-1 + 0,95 ht-1
d. ht = 0.05 + 0,01 r2t-1 + 0,96 ht-1
3. Ex. 14.5 pg. 344 Financial Risk Management Handbook - Jorion EXAMPLE 14.5: FRM EXAM 2002 – QUESTION 13
The GARCH model is useful for simulate asset returns. Which of the following statement about this model is false?
a. The Exponentially Weighted Moving Average (EWMA) approach of RiskMetrics is a particular case of a GARCH process.
b. The GARCH allows for time-varying volatility.
c. The GARCH can produce fat tails in the return distribution.
d. The GARCH imposes a positive conditional mean return.
4. Ex. 14.6 pg. 344 Financial Risk Management Handbook – Jorion EXAMPLE 14.6: FRM EXAM 1999 – QUESTION 103
The current estimate of daily volatility is 1.5 percent. The closing price of an asset yesterday was $30.00. The closing price of the asset today is $30.50. Using EWMA model with λ = 0.94, the updated estimate of volatility is:
e. 1.5096 f. 1.5085 g. 1.5092 h. 1.5083
5. Ex. 14.7 pg. 344 Financial Risk Management Handbook – Jorion EXAMPLE 14.7: FRM EXAM 1999 – QUESTION 72
Until January 1999 the historical volatility for the Brazilian real versus the U.S.
dollar had been very small for several years. On January 13, 1999, Brazil abandoned the defense of currency peg. Using data from the close of business on January 13th, which of the following methods for calculating volatility would have shown the greatest jump in measured historical volatility?
a. 250-day equal weight
b. Exponentially weighted with a daily decay factor of 0.94 c. 60-day equal weight
d. All of the above 6. Hull – 17.2 pg. 389
17.2 What is the difference between the exponentially weighted moving average model and the GARCH(1,1) model for updating volatilities?
7. Hull – 17.3 pg. 389
17.3 The most recent estimate of the daily volatility of an asset is 1.5% and the price of the asset at the close of trading yesterday was $30.00. The parameter λ in the EWMA is 0.94. Suppose that the price of the asset at the close of trading today is $30.50. How will this cause the volatility to be updated by the EWMA model?
8. Hull – 17.4 pg. 389
17.4 A company uses a EWMA model for forecasting volatility. It decides to change the parameter λ from 0.95 to 0.85. Explain the likely impact on the forecasts.
9. Hull – 17.6 pg. 389
17.6 A company uses GARCH(1,1) model for updating volatility. The three parameters are ω, α and β. Describe the impact of making a small increase in each of the parameters while keeping the others fixed.
10. Hull – 17.8 pg. 389
17.8 Assume that S&P 500 at close of trading yesterday was 1,040 and the daily volatility of the index was estimated as 1% per day at that time. The parameters in a GARCH(1,1) model are ω = 0.000002, α = 0.06 and β = 0.92. If the level of the index at close of trading today is 1.060, what is the new volatility estimate?
11. Hull – 17.9 pg. 390
17.9 Suppose that the current daily volatilities of asset A and asset B are 1.6% and 2.5%, respectively. The prices of the asset at close of trading yesterday were
$20 and $40 and the estimate of the coefficient of correlation between the returns on the two assets made at that time was 0.25. The parameter λ used in EWMA model is 0.95.
a. Calculate the current estimate of the covariance between the assets.
b. On the assumption that the prices of the assets at close of trading today are
$20.5 and $40.5, update the correlation estimate.
12. Hull – 17.11 pg. 390
17.11 Suppose that the current daily volatilities of asset X and asset Y are 1.0% and 1.2%, respectively. The prices of the assets at close of trading yesterday were
$30 and $50 and the estimate of the coefficient of correlation between the returns on the two assets made at this time was 0.50. Correlations and volatilities are updated using GARCH(1,1) model. The estimates of the model’s parameters are α = 0.04 and β = 0.94. For the correlation ω = 0.000001, and for the volatilities ω = 0.000003. If the prices of the two assets at close of trading today are $31 and $51, how is the correlation estimate updates?
13. Hull – 17.17 pg. 391 – (Se preferir troque a base por IBOVESPA – 3 anos de dados ou uma ação qualquer líquida).
17.7 An Excel spreadsheet containing 500 days of daily data on a number of different exchange rate and stock indices can be downloaded from the author’s Web site:
www.rotman.utoronto.ca/~hull
Choose one exchange rate and one stock index. Estimate the value of λ in the EWMA models that minimizes the value of:
Σ
(vi – βi )2where vi is the variance forecast made at the end of day i-1 and βi is the variance calculated from data between day i and day i + 25. Use the Solver tool in Excel. Set the variance forecast at the end of the first day equal to the square of the return on that day to start the EWMA calculations.
14. Use dados diários do Ibovespa entre 2001 e 2008.
a. Teste a existência de heterocedasticidade
b. Estime um modelo GARCH(1,1) para o Ibovespa, supondo que os retornos diários são normais com média nula.
