Reading 1.8 Flashcards
Rebalancing portfolio
Buying & selling its assets to maintain its original asset allocation
Formula for The Time-t value, Vt, of a two asset portfolio
Vt = Ns * St + Mt * Bt
Formula for The Portfolio weights of equity and treasuries (2 assets)
Wt = (Nt * St) / Vt - first asset
1-Wt = (Mt * Bt) / Vt - second asset’s weight
Time t value before rebalancing
Vt-1 = Nt-1 * St + Mt-1 * Bt
Self Financing rebalancing Strategy definition and formula
1) Increased investment into one asset class is achieved by reducing investment into another asset class
2) (Nt -Nt-1) * St + (Mt - Mt-1) * Bt = 0.
Why are portfolios rebalanced?
Portfolio managers are driven to rebalance their portfolios based on their investment returns and their mandates
Buy and hold strategy rebalancing strategy
Portfolio manager does nothing, just establishes the initial asset allocation and hold it
Initial value of portfolio formula and future T value of portfolio formula
1) Vo = N0 * So + Mo * Bo
2) Vt = No * St + Mo * Bt
Constant mix strategy definition and what type of strategy is it and why?
The weight of each asset should stay the same
Contrarian because you sell the ones that have grown the most and buy those who grew the least
Constant mix strategy has a _____ payoff curve, why?
1) concave
2) Because of constant selling and buying
Which strategy, BH or CM outperforms in a volatile and non trending market?
CM - constant mix
For whom is the CM strategy best?
For constant risk averse investors
CPPI definition and type of strategy
1) Constant proportion portfolio insurance strategy is a rebalancing strategy that takes more risk (at least initially) than the BH strategy (and thus generates a reasonable long-term return) and ensures that the portfolio value generally does not fall below a pre-set floor.
2) it is a momentum strategy (opposite of contrarian): it increases the equity position after an increase in equity value, and reduces the position after a decline in equity value.
CPPI formula
St = mCt =m (Vt - Ft) and
Bt = Vt -St
What do the m and Ct stand for in the CPPI formula
1) m = multiplier
2) Cushion - different between floor Ft and portfolio value Vt at time t.
Ct = Vt - Ft
CPPI underperforms which rebalancing strategy in volatile and non trending market?
BH - buy and hold
CPPI outperforms which strategies in what markets?
Trending markets
CPPI is an optimal strategy for which type of investor?
Who’s risk tolerance increases as value of portfolio grows and decreases as portfolio shrinks
Which type is more classical CPPI, behaves like BH, behaves like CM
1) m>1 and F>0
2) m=1 and F>0
3) m<1 and F=0
1) classical CPPI
2) BH
3) CM
2 unique risks of CPPI compared to BH and CM
1) GAP RISK
• This is the risk that the CPPI’s floor is violated when equity value declines by more than 1/m. Gap risk is higher when markets are volatile and when the multiplier is large.
• For instance, a CPPI portfolio with a multiplier of 20 will violate its floor if its risky asset declines by more than 5% (= 1/20) between rebalancing periods.
2) Absorption risk
This risk occurs when the CPPI portfolio value reaches its floor and the portfolio liquidates its equity position. Since the portfolio does not participate in any subsequent recovery in equity values, it does not benefit when equity markets rebound.
3 Variations of the classical CPPI
1) Change the multiplier
2) Reset the floor
3) Stop Loss strategy
How does Change the multiplier variation of the CPPI strategy work? What is the result?
1) M may be expressed as a decreasing function (i.e. when volatility increases you decrease m) of the realized volatility using:
- the past n trading days
- value at risk
- expected shortfall
2) This reduces gap and absorption risk
How does the 2 Reset the Floor variation of the CPPI strategy work? Whats the second one called? Optimal equity position Formula?
1) Set the floor equal to X of the current portfolio value.
FORMULA: St = m* Vt * (1-x)
2) Set the floor equal to X of portfolio’s previous high water mark. TIPP - time invariant portfolio protection
FORMULA: St = m (Vt - xHt)
How does Stop-Loss variation of the CPPI strategy work? Advantage? Other variations?
Formula for equity position?
extreme case of CPPI: the portfolio invests entirely in the risky asset and, if the portfolio value hits the floor, the entire portfolio invests in the riskless asset.
Requires no rebalancing, suitable for when the cost of transactions are high or risky asset is not liquid
Some versions of stop loss allocate again to the risky asset if equity value exceeds Ft by a pre set amount
St =m * Vt { m=1 if Vt > Ft
m=0 if Vt <= Ft
High water mark
A NAV above which the HF manager is compensated.
Every time the NAV expands, the new NAV is set as the benchmark
Floor definition
Minimum value of the portfolio that the investor is ready to accept
OR
Minimum portfolio value wanted at a future date.
OBPI, how does it work? When does it underperform?
Option Based Portfolio Insurance Strategy
Uses combinations of risky assets, treasuries, and options (traded or synthetic), where options are used to limit the portfolio’s downside risk. As with CPPI, the investor selects the value of the floor
OBPI portfolio has a convex payoff profile and underperforms BH in volatile, non-trending markets.
2 classic forms of OBPI
1) Protective Put = Long put option and risky asset
2) Long call option and risk-less asset
historically implied volatilities have ____ realized volatilities
exceeded
Protective put strategy how does it work? What to consider?
- Investment in a risky asset and a long position in a put option on the asset.
- The put’s strike is the floor, and the strategy works if liquid options on the risky asset exist.
- The portfolio manager needs to consider the put’s up-front cost in the put’s position size and the protection strategy’s design.
Put call parity formula
c + Ke ^ -rT = p + s
c = european call option price
p = european put option price
K = option strike price
r = riskless rate
Using calls and riskless asset portfolio’s initial value is expressed
Vo = Nc + Bo
= Nc + NKe ^ -rT + D
N = Number of call options purchased
Bo = amount invested into the risky asset (amount left after purchasing the calls)
K = minimum portfolio value at maturity to be protected
D = cash left
What does put call parity show?
The value of a put-protected portfolio (on the right side of the equation) is the same as the value of a portfolio of call options and a riskless asset (on the left side of the equation).
Therefore, in addition to using puts and a risky asset, the floor of a portfolio at maturity may be protected using call options and a riskless asset.
Portfolio value at maturity using the long call option and riskless asset
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When is the Put Option replication strategy used?
To which strategy is it similar? How is it different?
When is it preferred to CPPI?
When is it easy to implement?
How is option’s payoff replicated?
1) when trading options is not available
2) similar to CPPI, the multiplier changes in response to time decay and changes in the risky asset’s price and volatility
3) when assets are traded in liquid markets and OPTIONS implied volatilities have a LARGE volatility PREMIUM
4) when long position is held in the asset
5) using option pricing model
The Black-Scholes-Merton (BSM) model shows that a European put on a non-dividend-paying stock may be replicated using _______. Thus, the price of the put p may be expressed as:
1) a long riskless asset position and a short stock position
2) p = Mt - Delta t * St
Mt = amount invested in a safe asset
Delta t = number of shares of the shorted stock
Dynamic trading has a number of advantages and disadvantages compared to buying call or put options.
Advantages
1. Can be implemented when marketable options are not available.
2. Avoids counterparty risk if the investor needs to use over-the-counter options.
3. Avoids paying the risk premium embedded in implied volatilities.
Disadvantages
1. May be costly, since frequent rebalancing increases transaction costs.
2. Exposed to gap risk, which is avoided when options are used.
3. Exposed to estimation risk, since the underlying asset’s volatility needs to be estimated.
There are two advantages of OBPI relative to CPPI
- No absorption risk
• OBPI is not exposed to absorption risk because, in contrast to CPPI, the multiplier in the OBPI strategy is not constant. - Outperforms in volatile, non-trending market
• Since OBPI is not a momentum strategy, it outperforms the CPPI portfolio in a volatile, non-trending market.
• However, CPPI outperforms in a rapidly rising equity market, but has higher volatility.
Both CPPI and OBPI underperform BH in a volatile, non-trending market.
OBPI - who is it perfect for?
for investors with growing but no unbounded risk tolerance as equity grows and low risk tolerance as equity value approaches the floor
Differences and similarities between BH, CPPI and OBPI
• Similar to BH, it has the same upper bound (20%) and lower bound (0%) for volatility.
However, OBPI’s volatility approaches 0% faster when equity value declines.
• Similar to CPPI, it has increasing return volatility. In contrast to CPPI (especially when traded options are used to buy protection), its return volatility has an upper limit (20%).
Payoff curve for BH, CM, CPPI and OBPI
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Portfolio’s uncertain rate of return is expressed as (for illiquid assets)
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Illiquid assets rate of return?
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Portfolio’s target rate of return (illiquid asset)
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Rate of return of portfolio that includes futures contract (for illiquid assets)
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Optimal futures contract position is determined by …
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