Advanced Financial Modeling. Unit 2

Transcription

1 Advanced Financial Modeling Unit 2

2 Financial Modeling for Risk Management A Portfolio with 2 assets A portfolio with 3 assets Risk Modeling in a multi asset portfolio Monte Carlo Simulation

3 Two Asset Portfolio If there is a portfolio with 2 assets, we already know the returns and standard deviation of this portfolio. Returns are weighted average, while portfolio variance is given by the following formula Portfolio Variance = w 2 A*σ 2 (R A ) + w 2 B*σ 2 (R B ) + 2*(w A )*(w B )*Cov(R A, R B )

4 Questions 1. What is the covariance of an asset s returns with itself? 2. In a two asset portfolio, how many terms will be there in the variance covariance matrix?

5 Advanced Financial Modeling Unit 2

6 Financial Modeling for Risk Management A Portfolio with 2 assets A portfolio with 3 assets Risk Modeling in a multi asset portfolio Monte Carlo Simulation

7 Three Asset Portfolio But what happens if there are three assets in the portfolio. Returns are still weighted average, while portfolio variance is given by the following formula Portfolio Variance = w 2 A*σ 2 (R A ) + w 2 B*σ 2 (R B ) + w 2 C*σ 2 (R C ) + 2*(w A )*(w B )*Cov(R A, R B ) + 2*(w A )*(w C )*Cov(R A, R C ) + 2*(w B )*(w C )*Cov(R B, R C )

8 Three Asset Portfolio Example Assume a portfolio with 3 assets. The weights are 25%, 25% and 50% respectively. Below are the details of Standard Deviations and Correlations Portfolio Variance = w 2 A*σ 2 (R A ) + w 2 B*σ 2 (R B ) + w 2 C*σ 2 (R C ) + 2*(w A )*(w B )*Cor(R A, R B )* σ(r A )*σ(r B ) + 2*(w A )*(w C )*Cov(R A, R C )* σ(r A )*σ(r C ) + 2*(w B )*(w C )*Cov(R B, R C ) * σ(r B )*σ(r C )

9 Advanced Financial Modeling Unit 2

10 Financial Modeling for Risk Management A Portfolio with 2 assets A portfolio with 3 assets Risk Modeling in a multi asset portfolio Monte Carlo Simulation

11 Normal Distribution In Finance, returns are expected to be distributed normally. Let us understand the normal distribution first.

12 Normal Distribution Confidence Intervals If R is normally distributed, then 68% of observations fall within +/ 1.00std. deviations from mean 90% of observations fall within +/ 1.65std. deviations from mean 95% of observations fall within +/ 1.96std. deviations from mean 99% of observations fall within +/ 2.58std. deviations from mean

13 Normal Distribution Assume a stock with expected return 15%, and standard deviation of 20%. How would the normal curve look like?

14 Questions 1. Explain what you mean by Normal Distribution? 2. If average return is 2%, and SD is 1.5%, what is the probability of a return less than a) 3.5% b) -0.5%

15 Advanced Financial Modeling Unit 2

16 Financial Modeling for Risk Management A Portfolio with 2 assets A portfolio with 3 assets Risk Modeling in a multi asset portfolio Monte Carlo Simulation

17 Three Asset Portfolio Example Calculate the Portfolio Standard Deviation Case 1 Then, find the portfolio, subject to some conditions such as given below 1. A client wants to create a 3 stock portfolio 2. Maximize Return 3. The probability of a loss of more than 20% should not be more than 10%

18 Three Asset Portfolio Example Calculate the Portfolio Standard Deviation Case 2 Then, find the portfolio, subject to some conditions such as given below 1. A client wants to create a 3 stock portfolio 2. Minimize Risk 3. The probability of a loss of more than 20% should not be more than 10%

19 Advanced Financial Modeling Unit 2

20 Financial Modeling for Risk Management A Portfolio with 2 assets A portfolio with 3 assets Risk Modeling in a multi asset portfolio Monte Carlo Simulation

21 Monte Carlo Simulation Sometimes, we may need to run a simulation to find out if the results that are expected are feasible or not. Take for example an investment product, that says that if you invest Rs 100,000 today in an asset with mean of 20% and SD of 18%, you will be given 10 lakhs, if the total investment value crosses 8 lakhs at the end of 10 years, else you will be given Rs 3 lakhs. What can we do to evaluate this kind of a product? We will simulate this 10 year return, over 100 times, to arrive at an expected value.

22 Monte Carlo Simulation Sometimes, we may need to run a simulation to find out if the results that are expected are feasible or not. Take for example a trading scenario. One of my friends claims, that by trading Rs every day in Nifty, I can end the month with a likely return of 4% at the end of the month. We find that Mean Return is 0.05% on a daily basis, while annualized SD is 16% for Nifty. What can we do to evaluate this claim?