Principles of AA Flashcards
Walk through the MVO formula (objective function):
- MVO determines how much allocation each asset gets in order to make the maximum return for a given level of risk.
- Investor’s utility for allocation = Return - (0.5 x λ x σ^2m)
- The risk aversion coefficient (λ) characterizes the investor’s risk–return trade-off; in this context, it is the rate at which an investor will forgo expected return for less variance. A value of λ = 0 corresponds to a risk-neutral investor because it implies indifference to volatility.
- σ^2m is the expected variance. It gets penalised based on the risk aversion coefficient.
- If using % then use 0.005
- 0.06 or 6% = [= 0.10 − 0.5(2)(0.04)]
- Return or Um is the certainty equivalent return - the utility value of the risky return offered by the asset mix, stated in terms of the Rf return the investor would value equally.
- the objective function says that the value of an asset mix for an investor is equal to the expected return of the asset mix minus a penalty that is equal to one-half of the expected variance of the asset mix scaled by the investor’s risk aversion coefficient. Optimization involves selecting the asset mix with the highest such value (certainty equivalent)
Based only on Goddard’s risk-adjusted expected returns for the asset allocations, which asset allocation would she prefer? λ = 2
Bunch of portfolios. Exprected return is 10%. Standard deviation is 20%. So forth with other portfolios.
What if she has 1.2M and needs 60k next year, whats the best portfolio?
- We are using % so do 0.005 x 2 (λ) = 0.01
- 0.01 x (20)^2 = 4
- Return of 10 minus 4 = 6%
- This is the certainty equivalent return for the portfolio.
- Once you calc the middle of the formula you can re-use this for ther portfolio as she will have same (half) aversion.
- This is just a way of finding the best portfolio while taking into account her risk comfort.
- 60k/1.2M means she needs 5%, this is a safety first lens
- (10% - 5%) / 20% is 0.25
- portfolios might have the same utility but this approach shows one portfolio has better prob of meeting or exceeding min return needed.
interpret and evaluate an asset allocation in relation to an investor’s economic balance sheet:
What would HC look like for instance? And how do we integrate it?
- We take your job and house and integrate them as asst classes for the optimisation.
- So your job is 70% UK long-duration inflation-linked bonds, 15% UK corporate bonds, and 15% UK equities.
- Residential real estate was modeled based on a de-smoothed residential property index for London.
- We still work out your aversion co-efficient for the model
- Take all the usual assets like cash at bank etc
- The HC and RE are non-tradeable assets so the optimiser is forced to allocate say 27% of the portfolio to UK RE for instance.
- EF AA Area Graph has big flat rows for fixed assets.
- Expected utility maximised in middle, it accounts for the risk aversion coefficient.
Discuss the use of Monte Carlo simulation and scenario analysis to evaluate the robustness of an asset allocation:
Using MC helps us with problems that cant be solved analytically, the MVO is single period but a lot of problems like risk, rebalancing, taxes, outcomes are all multiperiod. We use MC to see how AA may perform over time. A statistical based process occurs where we use assumptions about distributions etc and repeat hundreds of times, and we could rank the outcomes best to worst.
Uses are:
- Complement limitations of MVO
- Help when we dont know investor’s risk aversion, show future outcomes and likelihood of reaching goals,
- Max dardowns
- See effect of rebalancing or taxes
What are the criticisms of MVO?
The outputs (asset allocations) are highly sensitive to small changes in the inputs.
The asset allocations tend to be highly concentrated in a subset of the available asset classes. (change just 2 asset Re and the whole AA changes, even deletes unchanged assets from mix).
Many investors are concerned about more than the mean and variance of returns, the focus of MVO.
Although the asset allocations may appear diversified across assets, the sources of risk may not be diversified.
Most portfolios exist to pay for a liability or consumption series, and MVO allocations are not directly connected to what influences the value of the liability or the consumption series.
MVO is a single-period framework that does not take account of trading/rebalancing costs and taxes.
What is reverse optimisation? What is it for?
What is an example of it in action?
- MVO weakness is inputs but of these, Returns are the most sensitive. Reverse Opt seeks to address this.
- Reverse Optimisation expains the implied expected returns and it takes a set of AA weights assumed to be optimal, as an input. We take the other inputs and risk aversion, and we find the ‘impied returns’
- Use market cap weighted assets as starting point, or use policy port
- For an example, just use the GMP, all assets are priced by CAPM and have a beta (asset relative to GPM) and Rf rate, and crucially, a global market risk premium.
- So each asset class would have, a Mkt Cap (and weighting) and a return built from CAPM whereby Rf + (beta x Market Risk Premium).
- Using this approach consistently relates the expected returns to their systematic risk.
What is BLM for?
- Thre reverse opt is good starting point but we often have alternative forecasts regarding the expected return for 1 or more assets.
- Black-Litterman starts with the consensus return estimates from reverse optimization. The manager then view adjusts those return estimates up or down and uses their view-adjusted estimates to run MVO and produce the EF and AA. BLM and reverse opt start with weightings therefore not as dependent on initial return estimates, thus they have AA more balanced in final portoflio.
- Allows combining reverse opt returns with their own forecast ie this Asset will do that or outperform this by 100 bips.
- Using BLM is like abackdoor approach, changing the returns this way can subtlely change it in portfolio, doing it through MVO can blow up the allocation.
What is a risk budget and what is the goal of it?
What do risk budget tools do and what does knowing marginal contribution allow?
- A risk budget is the total amount of risk to be allocated to the portfolio’s constituent parts
- Risk needs to be used in relation to seeking return, max the return for risk whether we are lloking at market risk and AA or at active risk in implementation
- The risk budget tools assist in finding the optimal use of risk in the pursuit of return. A risk budget specifies the total amount of risk and how much of that risk should be budgeted for each allocation.
- Finding for any portfolio positiion or holding the marginal contribution to risk (ie total risk or active risk etc) lets us see the effect on the portfolios total risk and whether it is a good move, and how the risk budget is being spent etc.
In the world of risk budgeting we have a big chart with every asset line by line with their respective allocation. We then see how ideally they all contribute the SAME excess return relative to their contribution to overall risk (the portfolio SD). We actually know the assets Return, Risk, the Rf and the Beta for each asset.
So each column has some info, define these:
MCTR:
ACTR:
Ratio of excess return to MCTR:
Bonus: If A is two times the weighting of B, how could B have a bigger ACTR?
- MCTR is: Asset’s BETA x Portfolio SD (doesnt change)
- ACTR is the above times the weighting of the asset class. When you add every asset ACTR up the total will equal the portfolio’s SD.
- Ratio of excess return to MCTR is the Asset’s expected return, minus the risk free rate, divided by the MCTR for the asset. Every single asset should have same answer here for there to be an optimal AA from a risk budget perspective. Matches the sharpe ratio of tangency portfolio, too.
- If we know the MCTR is beta x port SD and this flows to the ACTR by multiplying by the size of the weighting, the only thing we can change is the beta. The beta would have to be over 2 times bigger.
Diversification by asset class may not provide diversification by risk source, what is the solution?
- Use factor based allocation. Risk factors can be market risk premium (CAPM), market cap, value v growth, momentum v reversion, duration, credit risk, volatility. The oppurtunity set changes to factors - based on observed market premiums and anomalies.
- We can make our own EF with the risk factor returns, risk and correlation.
- It isnt necessarily better than asset class aproach. This way has lower correlations. They have similar EFs.
How to go about the liability relative AA? What is basis risk?
- Look at the nature of the liabilities - Fixed v contingent cashflows, legal v quasi, DUration, Size, economic factors like inflation, timing, regulations.
The characteristics of the liability drive the composition of the portfolio. The DR must be chosen which eastablishes the PV. High quality corp bond DR or treasuries? Will there be a surplus?
- The basis risk is degree of mismatch between hedging and liability portfolios.
- Calc MV of assets. Project future liability cash flows. Discount them back. Find the funding ratio and surplus value.
- Lower DR means bigger liability and therefore lower funding ratio. DR usually set by regs
What is the surplus optimisation approach?
- MVO by itself doesnt consider liabiltiies. So we use LR AA.
- Surplus opt means doing an MVO just on the surplus. The EF has total surplus amount in Y axis and surplus RISK (SD) in the X axis.
- It assumes that relationship between A and L is approximated though correlation coefficient.
- It exploits natural hesges that may exist between A & L as a result of their systematic risk characteristics.
- Usually the PM wants to maintain low surplus vol, and not risk being underfunded, to keep the surplus vol to 0.25 Billion or 10% for instance. The risk is presented in terms of surplus.
- You do the same objective function with risk aversion
- You do all the usual steps like you would an MVO, like CME etc. For the liabilities we can assume they have same returns and vol as corporate bonds (or use factors). We work out the correlations.
- Liability returns measure the time value of money for the liabilities plus any expected changes in the discount rate over the planning horizon.
- We compare it to AO allocation. For conservative, the portfolios are different, for aggressive, they are both dominated by PE. Check current portfolio, where it falls on EF.
What is the main type of Hedging/Return-Seeking portfolio and the other types?
- The basic type is the 2 portfolio approach - when you have a surplus. You have a majority in the hedge portfolio, usually put in long bonds. Cashflows are hedged by cashflow matching, duration matching or immunization. The return seeking portfolio is the surplus and managed independently. This is used usually for conservative investors ie insurance or overfunded pension plans.
- Or a variant, when there is no surplus (not enough assets or DR is too low). Partial hedge where capital for the hedging portfolio is reduced to make higher expected returns. Or increase the hedging allocation as the funding ratio increases (glide path). These are less conservative.
- You coud underfund the mimicking portfolio and allocate more funds to return seeking but monitor the surplus and shft more to the mimicking portfolio if the surplus deteriorates.
How do you form the hedging portfolio?
- The assets have to be driven by same factors as liabilities to keep them growing the same.
- So if L driven by inflation, include inflation linked T bonds.
- This is complicated by the fact that the DR might be driven by long govt bond index but inflation (or eco growth) drives the L behaviour.
- The barriers are if it is underfunded (may need to do glide path or contribute more, but can’t do 2 portfolio method here) and also sometimes it is hard to hedge things like weather. Surplus opt does not req overfunded status.
Integrated Asset-Liability approach
- You can decide on your liabilities (banks, hedge funds, insurers, reinsurers). Choose what you insure.
- A & L are integrated in enterpirse risk mgmt. You end up looking at probabilities.
- When you consider new projects etc you consider the mirror asset for it, and you have multiperiod models with a set of prjections.
