1.8 Portfolio Rebalancing

Question 1

What are the four dynamic (i.e., rule-based) trading strategies to rebalance portfolios
(BH, CM, CPPI, OBPI)

Answer 1

1. buy and hold
2. constant mix
3. constant-proportion portfolio insurance
4. option-based portfolio insurance

Question 2

Buy and hold strategy: Once the initial allocation is made, there is (1). If equities increase in value, the weight of equities in the (2). BH is often used as (3). The floor of the BH strategy is (4).

Answer 2

1. no rebalancing
2. increases
3. a benchmark to compare the relative performance of other strategies
4. the value of the riskless asset

Question 3

Constant mix strategy involves (1). This is done by (2). As such, CM is a (3) startegy

Answer 3

1. rebalancing the portfolio to its target weights on a periodic basis
2. selling the winning assets (whose portfolio weights have increased) and purchasing the losing assets (whose portfolio weights have decreased) so as to revert their relative weights back to the target weights
3. contrarian strategy

Question 4

On a graph, the BH strategy portfolio value is a (1) function of equity values while the CM strategy payoff is a (2) function of equity values

Answer 4

1. linear function
2. concave function

Question 5

On a buy and hold strategy graph (portfolio value against equity value), the slope of the line (e.g. 0.6) indicates the _____. What is the floor / y-intercept?

Answer 5

The % of the total assets that are originally invested in stocks. ie. 60%. As the equity value increases, the protfolio value also increases, at the rate of the proportion of equity in the portfolio.

The floor value represents a portfolio with all funds in the riskless asset / or the value of the riskless asset in the portfolio is the equity portion is worth zero.

Question 6

In a BH strategy graph, the floor value represents (1). In a CM graph, it is (2)

Answer 6

1. The floor value represents a portfolio with all funds in the riskless asset / or the value of the riskless asset in the portfolio is the equity portion is worth zero.
2. it is zero

Question 7

In a CM graph, the slope coefficient is a value of (1), depending on (2)

Answer 7

1. between zero and one
2. the level of risk aversion of the client

Question 8

As stock prices rise, the stock-to-total-assets ratio (1), so stocks (2). As stock prices (3), the stock-to-total-assets ratio decreases, so stocks must be (4).

Answer 8

1. increases
2. must be sold
3. fall
4. purchased

Question 9

How does portfolio volatility differ between the CM and BH strategies?

Answer 9

CM portfolio volatility remains a constant - i.e. if equity is 60% of portfolio and equity vol is 15%, it will remain a constant 9% (0.6 of 15%). For BH strategy, as value of equity increases the proportion of the portfolio in equity increases, so vol increases and approaches the volatility of equity

Question 10

BH strategy is appropriate for an investor who’s risk tolerance….

CM strategy is appropriate for an investor who’s risk tolerance….

Answer 10

BH strategy is appropriate for an investor who’s risk tolerance increases with wealth (as vol will increase with portfolio value). Vol is floored at the starting proportion of equity.

CM strategy is appropriate for an investor who’s risk tolerance is independent of portfolio value. CM strategy has no floor and therefore assumes that the investor has risk tolerance even at low levels of wealth

Question 11

Draw graph for comparing portfolio volatility of BH and CM strategies. (Portfolio vol on y axis and equity value on x axis)

Answer 11

See reading

Question 12

What is the formula for the floor value in the CPPI?

Answer 12

Ft = ATe ^-(T-t)r

i.e. the floor value now is equal to the future required value * e to the power of negative time until future value multiplied by the riskless rate

Question 13

What is the Constant-Proportion Portfolio Insurance (CPPI) Strategy?

Answer 13

The target weight in equities varies directly with the difference between the current portfolio value and some minimum value known as the floor.

The floor is the present value (at riskless rate) of some desired accumulated future portfolio value (AT).

For example, a company has a portfolio managed to service a $2 million liability due in 12 years and wants the portfolio to have at least $2 million available in 12 years (i.e., AT = $2 million, T = 12).

Question 14

With the CPPI strategy, the target weight in equities varies directly with the difference between…

Answer 14

the current portfolio value and some minimum value known as the floor

Question 15

With the CPPI strategy, the difference between the portfolio value and floor value is called the ____. As equities increase in value, (1).

Answer 15

Cushion.

1. The cushion increases, and the weight of equities in the portfolio increases as a result.

Question 16

What is the formula for finding the cushion (CPPI)

Answer 16

cushion = portfolio value – floor value

Question 17

Steps for finding the target equities investment (CPPI)
1. floor
2. cushion
3. target equities

Answer 17

1. find the floor value (present value of future required value found by discounting)
2. find the cushion (current portfolio value minus floor)
3. target equities investment is found by M*cusion (M is the constant proportion of the cushion invested in equities (i.e., the multiplier) - number above 1.0

Question 18

How is the value of the multiplier different between BH, CM and CPPI strategies?

Answer 18

BH it will be one, CM it is less than 1, for CPPI it is more than 1

Question 19

Which portfolio strategy is convex and which is concave?

Answer 19

CM is concave, CPPI is convex

Question 20

In CPPI, what happens to M (constant proportion of the cushion invested in equities) as the portfolio increases in value?

Answer 20

Nothing - M remains unchanged.

Question 21

In CPPI, as M is greater than 1.0, the value invested in a riskless asset is

Answer 21

Lower than the floor value

Question 22

When M is greater than 1 and the portfolio has a floor value of zero, this is equivalent to? Example (2000)

Answer 22

A leveraged bet on a risky asset.

Consider the initial portfolio value of $1,000, floor = 0, and M = 2. In this case, investment in the risky asset = 2 × (1,000 – 0) = 2,000, and investment in cash = −1,000.

Question 23

With CPPI strategy, the decline in equity over XXX will violate the floor.

Answer 23

A decline of more than 1/M. Higher values of M, therefore, are more aggressive and have a higher risk of violating the floor

Question 24

Higher values of M, therefore, are more (1) and have a higher (2)

Answer 24

1. Aggressive
2. risk of violating the floor

Question 25

What are the two unique risks of CPPI strategy?
1. GR
2. AR

Answer 25

1. gap risk - i.e. risk of violating the floor. This occurs when there is not enough time to rebalance the portfolio following a decline in equity price. This risk is higher when M is higher and when markets are more volatile. For example, when M is 10, a 1/10 or 10% decline in market value of the risky asset would violate the floor.
2. Absorption risk - risk of missing out on rising markets when the floor is violated (due to a decline in equities) and subsequent exposure to equities is reduced to zero. The BH and CM strategies on the other hand would never completely remove exposure to equities

Question 26

What are three variations of the CPPI strategy?
1. dm
2. rtf
3. sl

Answer 26

1. dynamic multiplier (increasing the value of M when market volatility is expected to be low and decreasing the value of M when volatility is expected to be high)
2. resetting the floor (allows for the value of the floor to fluctuate, such as the time-invariant portfolio protection (TIPP) strategy where floor is set at a proportion of the previous high-water mark)
3. stop loss (extreme CPPI strategy with 100% allocation (M = 1) to the risky asset until the floor is breached, at which point, the allocation is rotated to 100% cash (i.e., M = 0).

Question 27

What is the stop loss CPPI variation?

Answer 27

It is an extreme CPPI strategy with 100% allocation (M = 1) to the risky asset until the floor is breached, at which point, the allocation is rotated to 100% cash (i.e., M = 0).

Question 28

What is the dynamic multiplier variation CPPI strategy?
What are expectations of future volatility based on? (3)
What risks does this variation seek to reduce?

Answer 28

Results in increasing the value of M when market volatility is expected to be low and decreasing the value of M when volatility is expected to be high.
Future vol can be based on current values of VIX, or realized trailing volatility, or a function of value at risk (VaR)/expected shortfall.
Seeks to reduce both gap risk and absorption risk.

Question 29

What strategy results in increasing the value of M when market volatility is expected to be low and decreasing the value of M when volatility is expected to be high?

Answer 29

Dynamic multiplier variation of CPPI strategy

Question 30

Typically, the floor is set as the present value of some desired minimum future portfolio value. What is another technique? (TIPP)

Answer 30

The time-invariant portfolio protection (TIPP) strategy sets the floor as a proportion of the previous high-water mark

Question 31

What is the TIPP strategy?

Answer 31

The time-invariant portfolio protection (TIPP) strategy sets the floor as a proportion of the previous high-water mark

Question 32

Compare the performance of BH, CM and CPPI strategies in up or down trending markets?

Answer 32

1. CPPI - best performance. If the equity market increases in value, the weight of equities and the cushion increases. The increase in cushion under the CPPI results in an additional purchase of equity so that subsequent increases in equity values have a greater positive impact on portfolio value (and v/v)
2. BH - underperforms the CPPI because no purchases or sales of equity are made and there is no amplification of subsequent gain or reduction of subsequent loss as the market continues to trend.
3. CM - worst performance. 0increase in equity values will increase the percentage allocation to equity and require selling equity to restore the initial percentage allocation. This lowers the exposure to subsequent increases in equity in a trending market (and v/v)

Question 33

Compare the performance of BH, CM and CPPI strategies in non trending, mean reverting markets?

Answer 33

1. CM - best performance. An increase in equity values will increase the percentage allocation to equity and require selling equity to restore the initial percentage allocation. This lowers the exposure to subsequent equity decline when the market reverts to its mean. This produces a buy low, sell high result.
2. BH
3. CPPI - worst performance. A rise in equity values will dictate buying more equity, which increases exposure to the subsequent reversal and downturn in equity prices. Produces a buy high sell low outcome.

Question 34

What is the Option-Based Portfolio Insurance (OBPI) Strategy

Answer 34

The OBPI strategy is a convex payoff strategy that uses options to provide portfolio insurance. Options can be market traded or synthetic. One such strategy is the protected put strategy that uses long-put options to provide downside protection on a portfolio with long equity position.

Question 35

What is the name for the convex payoff strategy that uses options to provide portfolio insurance?

Answer 35

Option-Based Portfolio Insurance (OBPI) Strategy

Question 36

What is the trial and error formula to approximate the strike price in the OBPI strategy?

Answer 36

strike price ≈ (floor / portfolio value) × (current stock price + put price)

Question 37

What stock and options combinations give the same outcome? (put protected portfolio and long call)

Answer 37

A long position in a stock and put option (i.e., a put-protected portfolio) can be replicated by a long position in the call option and a risk-free bond with a face value of the strike price

Question 38

Is OBPI strategy convex or concave?

Answer 38

It is convex

Question 39

Assume a portfolio manager wants to implement an OBPI strategy for a portfolio consisting of 100 shares of a stock. The current stock price is $50. 45-calls and 45-puts are available at premiums of $5.78 and $0.70, respectively. The manager wants to protect the portfolio with a floor of $4,500. Construct an OBPI strategy using puts and calls.

Answer 39

Using put options, sell 2 shares (for $100) and purchase 98 puts (for $68.60) leaving net cash of $31.40. If the stock price falls below $45, the manager gets 98 × 45 = 4,410 + cash of 31.40 = $4,441.40, which is just short of the floor.

Using call options, sell all the shares (proceeds = $5,000) and purchase 100 call options (cost = $578). The manager invests the remaining $4,422 at the risk-free rate. If the stock price falls below $45, the manager would still have the risk-free asset (again, just short of the floor).

Question 40

What is one concern with the OBPI strategy?

Answer 40

Options may be too expensive (i.e., the implied volatility that is priced in the option premium is too high). Studies have shown that implied volatility is usually higher than subsequent realized volatility suggesting a risk premium being priced by option writers

Question 41

In OBPI strategy, if not wanting to use an option for the put, what can you do? What is the formula?

Answer 41

If the underlying stock is highly liquid, the portfolio manager can employ dynamic trading instead. A put option can be replicated as a long position in a risk-free bond (X) and short position in delta units of the stock:

Pt = Xt – (Δt × St)

Question 42

What are the advantages and disadvantages (3) of using dynamic trading instead of buying options for OBPI?

Answer 42

Advantages:
1. Can be implemented when marketable options are not available.
2. Avoids counterparty risk in the case of over-the-counter (OTC) options.
3. Avoids the risk premium embedded in overpriced options.

Disadvantages:
1. Requires rebalancing which can be costly.
2. It is exposed to gap risk (which is avoided in the case of options).
3. There is an estimation risk for delta (which relies on an estimate of future volatility).