Portfolio Management

Portfolio ManagementGrenoble Ecole de ManagementMSc Finance2010

Learning Objectives

Mastering the principles of the portfolio management process:

•Balanced portfolio

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Balanced portfolio

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• Equity portfolio + bond portfolio + derivatives + foreign exchange positions + alternative = balanced portfolio

• How to mix different asset classes ?

Equity portfolio

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Equity diversification relies on factor models:

• Sector allocation, geographic allocation

• Small cap versus large cap, low PE versus large PE

• Portfolio construction based on Beta calculations

Equity portfolio

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Weights Beta Beta P

Sector A 6% 0,7 0,042

Sector B 4% 0,9 0,036

Country 1 8% 1,3 0,104

Country 2 4% 1,7 0,068

Country 3 12% 0,8 0,096

Small Cap 22% 1,6 0,352

Value stocks 24% 1,1 0,264

Index 20% 1 0,2

Total 100% 1,16

Equity + Bond portfolio

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• Bond portfolio risk cannot be measured with betas because maturities differ between bonds. They also change during all the life of one bond.

• Some securities may have embedded options, such has calls, puts, sinking fund provisions and prepayments. These features change the security characteristics over time and further limit the use of historical estimates.

• One has to use the concepts of duration, sensitivity and convexity to optimally build a bond portfolio.

Equity + Bond portfolio

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Weights Sensitivity Sensit P

Euro 2Y 50% 1,7 0,85

EURO 10Y 20% 7 1,4

Euro 30Y 5% 16 0,8

Cash 25% 0 0

Total 100% 3,05

Weights Beta Beta P

Sector A 6% 0,7 0,042

Sector B 4% 0,9 0,036

Country 1 8% 1,3 0,104

Country 2 4% 1,7 0,068

Country 3 12% 0,8 0,096

Small Cap 22% 1,6 0,352

Value stocks 24% 1,1 0,264

Index 20% 1 0,2

Total 100% 1,16

+

50% Equity – 50% bonds portfolio

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Weights Beta Beta P

Sector A 3% 0,7 0,021

Sector B 2% 0,9 0,018

Country 1 4% 1,3 0,052

Country 2 2% 1,7 0,034

Country 3 6% 0,8 0,048

Small Cap 11% 1,6 0,176

Value stocks 12% 1,1 0,132

Index 10% 1 0,1

Total 50% 0,58

Weights Sensitivity Sensit P

Euro 2Y 25% 1,7 0,425

EURO 10Y 10% 7 0,7

Euro 30Y 3% 16 0,4

Cash 13% 0 0

Total 50% 1,53

Beta balanced portfolio = 50% beta equity portfolio

Sensitivity balanced portfolio = 50% sensitivity portfolio

Balanced portfolio

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Weights Beta Beta PSector A 29% 0,7 0,203Sector B 8% 0,9 0,072Country 1 1% 1,3 0,013Country 2 0% 1,7 0Country 3 14% 0,8 0,112Small Cap 0% 1,6 0Value stocks 4% 1,1 0,044Index 14% 1 0,14Total 70% 0,58 Weights Sensitivity Sensit PEuro 2Y 6% 1,7 0,102EURO 10Y 9% 7 0,63Euro 30Y 5% 16 0,8Cash 10% 0 0Total 30% 1,53

Weights Beta Beta PSector A 3% 0,7 0,021Sector B 2% 0,9 0,018Country 1 4% 1,3 0,052Country 2 2% 1,7 0,034Country 3 6% 0,8 0,048Small Cap 11% 1,6 0,176Value stocks 12% 1,1 0,132Index 10% 1 0,1Total 50% 0,58 Weights Sensitivity Sensit PEuro 2Y 25% 1,7 0,425EURO 10Y 10% 7 0,7Euro 30Y 3% 16 0,4Cash 13% 0 0Total 50% 1,53

Weights are not useful to measure risk, use only betas and sensitivities.

These two portfolios have different weights but exactly the same exposition to equity market and sensitivity risks.

However risk factors and curve risk are different between the portfolios.

New bond exposure

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Cash Euro 2Y EURO 10Y Euro 30Y0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Pfl 1Pfl 2

Exposure to 2Y has decrease but to 30Y has increased. The fund is positioned for an over performance of the 30Y: flattening of the interest rate curve.

Risk factors

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Yield curve changes include:• a parallel shift in the yield curve• change in the slope of the yield curve.• a twist of the yield curve

Derivatives: Futures

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When a transaction is initiated , the portfolio manager puts up a certain amount of money to meet the initial margin requirement . Both buyer and seller of futures contract must deposit margin.

Weights Beta Beta PSector A 3% 0,7 0,021Sector B 2% 0,9 0,018Country 1 4% 1,3 0,052Country 2 2% 1,7 0,034Country 3 6% 0,8 0,048Small Cap 11% 1,6 0,176Value stocks 12% 1,1 0,132Index ETF 10% 1 0,1Future Index -15% 1 -0,15Total 50% 0,43 Weights Sensitivity Sensit PEuro 2Y 25% 1,7 0,425EURO 10Y 10% 7 0,7Euro 30Y 2,5% 16 0,4Cash 11,5% 0 0Margin 1% 0 0Total 50% 1,53

Hedge ratio = Beta Portfolio. Immunizing the portfolio to equity risk requires a short position on futures of 58%

Derivatives: Options

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Derivatives: Options

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Weights Beta Beta P

Sector A 3% 0,7 0,021

Sector B 2% 0,9 0,018

Country 1 4% 1,3 0,052

Country 2 2% 1,7 0,034

Country 3 6% 0,8 0,048

Small Cap 11% 1,6 0,176

Value stocks 12% 1,1 0,132

Index ETF 10% 1 0,1

Future Index -15% 1 -0,15

Option Index -2% 1 -0,02Total 50% 0,41 Weights Sensitivity Sensit P

Euro 2Y 25% 1,7 0,425

EURO 10Y 10% 7 0,7

Euro 30Y 2,5% 16 0,4

Cash 11,5% 0 0

Margin 1% 0 0

Total 50% 1,53

We buy put options to cover the market risk:

• Delta is 0.25• minimum quantity 10 • the value of the index we cover is 5000

or 0.005% of the portfolio.

Therefore the share of the portfolio we cover with one option is h = 0.005%*10*0.25 = 0.013%.

Then to cover 2% of the portfolio we need to buy 160 put options.

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Derivatives: Options

Market performances deform the delta of the option. In this example the delta has a value of 0.6 for an index value of 4500 or 0.0045%.

Hedging reaches 4.3% of the portfolio. (160*0.0045%*10*0,6)

Delta neutral policy would call for selling put options as to maintain the level of hedging unchanged.With 160 options I cover 4.3% of the portfolio. To cover 2% I need 74 options therefore I can sell 86 options.

Weights Beta Beta PSector A 3% 0,7 0,021Sector B 2% 0,9 0,018Country 1 4% 1,3 0,052Country 2 2% 1,7 0,034Country 3 6% 0,8 0,048Small Cap 11% 1,6 0,176Value stocks 12% 1,1 0,132Index ETF 10% 1 0,1Future Index -15% 1 -0,15Option Index -4,3% 1 -0,043Total 50% 0,39 Weights Sensitivity Sensit PEuro 2Y 25% 1,7 0,425EURO 10Y 10% 7 0,7Euro 30Y 2,5% 16 0,4Cash 11,5% 0 0Margin 1% 0 0

Total 50% 1,53

Strong assumption: all other assets prices are unchanged

Foreign exchange position

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Weights Beta Beta PSector A 3% 0,7 0,021Sector B 2% 0,9 0,018Country 1 4% 1,3 0,052Country 2 2% 1,7 0,034Country 3 6% 0,8 0,048Small Cap 11% 1,6 0,176Value stocks 12% 1,1 0,132Index ETF 10% 1 0,1Future Index -15% 1 -0,15Option Index -4,3% 1 -0,043Total 50% 0,39 Weights Sensitivity Sensit PEuro 2Y 25% 1,7 0,425EURO 10Y 10% 7 0,7Euro 30Y 2,5% 16 0,4Cash 11,5% 0 0Margin 1% 0 03-month Fw 1,4

EUR IR -3% 0 0US IR 3% 0 0

Total 50% 1,53

Foreign exchange position based on forward contract. Equivalent to borrowing in one currency and lending in the other one. No interest rate risk, risk might be measured by the beta.

For a currency quoted directly (EUR/USD = 1.40):

F=S(1+r*)/(1+r)

With F and s respectively the direct quotation of the forward and spot of the domestic currency and r* and r the short term interest rate of the foreign and domestic currencies.

Profit = S – F ; Max[0,(S-F)] if optional ; realized discount or premium will increase the cash account

Alternative: Long – short position

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Weights Beta Beta PSector A 3% 0,7 0,021Sector B 2% 0,9 0,018Country 1 4% 1,3 0,052Country 2 2% 1,7 0,034Country 3 6% 0,8 0,048Small Cap 11% 1,6 0,176Value stocks 12% 1,1 0,132Index ETF 10% 1 0,1Future Index -15% 1 -0,15Option Index -4,3% 1 -0,043

Total 50% 0,39 Weights Sensitivity Sensit PEuro 2Y 25% 1,7 0,425EURO 10Y 10% 7 0,7Euro 30Y 2,5% 16 0,4Cash 10,5% 0 0Margin 2% 0 03-month Fw 1,4

EUR IR -3% 0 0US IR 3% 0 0

Total 50% 1,53

Overlay/alternative L/S Small Cap 20% 1,6 0,32Large Cap -20% 0,9 -0,18Portable alpha Fund HugeReturns 10% 1,3 0,13Future Index -10% 1 -0,1Etc…

Total Equity 50% 0,56

• Introduction of an arbitrage position or long-short position. Either risk neutral (zero-beta) or cash neutral strategies. Cash neutral arbitrage of small cap versus large cap in the example.

• Introduction of a portable alpha strategy: buy a fund and sell the market as to capture only the over performance of the fund (alpha) but not the market risk. This increase the position in future therefore the margin requirement increase.

This might be cash consumer if the size of the fund cannot increase.

Learning Objectives

Mastering the principles of the portfolio management process:

•Portfolio insurance

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Portfolio insurance aims at hedging portfolio from market downside risk.

Stop loss

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• Stop loss: i) a maximum acceptable loss (which defines the floor of the value of the portfolio) is set at the beginning of the investment period ii) once this floor is reached, the fund is fully invested in the risk free rate to maturity.

• main limitation: if the drawdown happens at the beginning of the investment period you may gain only the risk-free rate missing the subsequent returns of the market.

• modified stop loss enables one to transfer only a

portion of the NAV of the fund. This portion is related to the distance to maturity. The closer to maturity, the larger the portion invested in the risk free rate.

Constant Proportion Portfolio Insurance

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CPPI is a dynamic trading strategy aiming at protecting a portfolio while taking profit of market returns.

• A predefined share of the value of the initial investment is guaranteed. This guarantee (the floor at maturity) is based on the investment in a zero-coupon bond.

With L the floor at maturity (in t periods) and r the discount factor, the initial value of the floor is F:

Constant Proportion Portfolio Insurance

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With Wt the value of the fund at time t, we define the cushion C as the difference between the value of the fund and the floor at t0

Then the exposure to risky markets, e, is defined using the cushion and a coefficient m. Initially the exposure is:

At any point in time we define the exposure as:

Constant Proportion Portfolio Insurance: data manipulation

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Years Index price Portfolio Floor Cushion Exposure Reserve

0 2000 10000 8000 2000 6000 4000

1 2200 10760 8320 2440 7320 3440

2 2000 10232 8653 1579 4738 5494

3 1800 9978 8999 979 2938 7041

4 1600 9933 9359 574 1723 8210

5 1400 10046 9733 313 939 9107

6 1200 10276 10123 154 462 9815

7 1000 10592 10527 65 194 10398

8 800 10969 10949 21 62 10907

Initial portfolio value 10000, final floor 10949, coefficient 3, tolerance 10%, risk free rate 4%

The floor might be also revaluated as to protect superior interim value of the fund (at a rate of 4% in this example). This called ratchet management.

Summary

• Building a portfolio is a dynamic process based on risk analysis (Betas, duration, default…).

• Combining and rebalancing these assets in an optimal way is key (optimization).

• Looking at weights in a portfolio is useless, only risk and relative risks matter.

• There exist methodologies to insure portfolios against market risks.

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