Intensive Actuarial Training for Bulgaria

January 2007

Lecture 16 – Portfolio Optimization and Risk

ManagementBy Michael Sze, PhD, FSA, CFA

Financial Planning Process

• Aim: To match financial needs to resources on long term and short term bases

• Steps– Know your client: needs, income, – Planned events: education, major purchases, retirement– Contingent events: death, sickness, job loss, divorce– Projected cashflow until the younger spouse is 100

years old

• Graphs, what ifs: different investment returns, pay increases

Determinants of the Required Rate of Return

• Real risk free rate

• Expected inflation rate premium

• Risk premium

Nominal Risk-Free Rate

• Expected Inflation = 4%

• Real Rate = 2%

• Nominal Rate = 2% + 4% = 6%

• This is the approximation method. The correct method is:

• (1 + real risk free rate)(1 + inflation rate) – 1• [1.02 × 1.04] – 1 = 6.08%

Risk Premium Factors

• Business risk

• Financial risk

• Liquidity risk

• Exchange rate risk

• Country risk

Life Cycle of Wealth

• Accumulation

• Consolidation

• Spending

• Gifting

Portfolio Management

• Policy statement

• Investment strategy

• Implementation

• Monitor & rebalance

Policy Statement

• The roadmap that guides the investment process

• It forces investors to understand and articulate their needs and constraints

• Sets performance standards and specifies benchmarks

• Imposes investment discipline on the client and the portfolio manager

Objectives

• Risk tolerance

• Return objectives

Constraints

• Time horizon

• Liquidity needs

• Taxes

• Legal/regulatory

• Unique needs

Asset Allocation

• Which asset classes will be allowed in the portfolio? (policy)

• What is the normal weighting for each asset class? (policy)

• How far can we deviate from the stated policy? (timing)

• What specific securities will be purchased for the portfolio? (selection)

Asset Allocation

• Equities are riskier but need to be represented in a significant way in the portfolio to preserve capital value over long time periods

• Over long periods government securities are less volatile than equities. Asset allocation can help reduce the volatility of a portfolio’s returns.

Risk Aversion

• Prefers higher return for same level of risk

• Prefers lower risk for same level of return

Measures of Risk

• The most common measure of risk used in finance is variance and standard deviation which is the square root of the variance

• Alternative measures of risk include the range and semivariance

• You should be able to calculate variance with both expectational and historical data

E r = w 1 R 1 + w 2 R 2

w 1 + w 2 = 1

2 2 2 2 2p 1 2 1 21 1 2 2σ = w σ + w σ + 2 w w c o v

1 2 1 2 1 2c o v = σ σ ρ

1 1 2

1 2 12

2 2 2 2 2

w = 40%, R = 12%, R = 18%

σ = 0.20, σ = 0.28,ρ =0.6

Er =( 0.4 () 12 ) +( 1– 0.4 () 18 )

= 15.6%

σ =( 0.4 ) ( 0.2 ) +( 0.6 )( 0.28 )

+2( 0.4 () 0.6 () 0.2 () 0.28 () 0.6 )

= 0.051

Standard Deviation of a Portfolio

2 = (0.4)2 (0.2)2 + (0.6)2 (0.28)2+ 2 (0.4) (0.6) (0.2) (0.28) (0.6) = 0.051

= [0.051]1/2 = 22.58%

Note that the standard deviation of the two stock portfolio is less than the weighted average of the two standard deviations [(0.4 × 0.2) + (0.6 × 0.28)] = 24.8%

= (0 .2 ) (0 .2 8 ) (0 .6 )

= 0 .0 3 3 6

1 2211 2c o v

1 21 2

1 2

c o vρ =

σ σ

Expected Return and Standard Deviation of a Portfolio

• The expected return of the portfolio is the weighted average return of the individual assets

• The standard deviation of the portfolio is less than the weighted average standard deviation as long as the correlation coefficient is less than +1

The Development of the Efficient Frontier

• Markowitz realized that we can create portfolios with the same level of risk and higher rates of return, or the same level of return and lower levels of risk, by diversifying our investments across many stocks

• The lower the correlation coefficient (the closer it is to negative 1), the greater the benefits of diversification

Efficient Frontier

• A portfolio is efficient if no other portfolio offers a higher level of expected return with the same or lower risk. Alternatively, no other portfolio offers lower risk for the same or higher return.

• The efficient frontier and the concept of efficient portfolios is the key concept in portfolio theory and capital market theory

EREfficient Frontier

Optimal Portfolio on a Efficient Frontier

• Steep indifference curves imply a highly risk averse investor while flatter indifference curves imply less risk aversion

• The investor’s optimal portfolio lies where the efficient frontier is tangent to the highest indifference curve