QFIP-140-19: Modern Investment Management: An Equilibrium Approach Ch 10

Question 1

State the liability return framework in MIM Ch 10.

Answer 1

• The noise term:
  
  • Assumed to be uncorrelated with the bond index
  • Will often assume the noise term has zero mean (when the current cash flow projections reflect all available information, the expected change in benefit obligation due to changes in projected payouts is zero)
• Beta parameter:
  
  • It reflects uncertainty in the value of the liabilities due to changes in interest rates
  • Used to duration-match the liability and bond indexes
  • A publicly traded bond index is used as a proxy for the liability return, and the index is levered to match the duration of the liability stream

Question 2

Similarities b/w Surplus and Shareholder Equity

Answer 2

• Can be seen as the value that would be left to shareholders if the company used all of its assets to pay off its liabilities
• One important difference between surplus and shareholder equity is that the value of equity is floored at zero due to limited liability for shareholders
• Surplus, however, can be negative
• The surplus of a pension plan can be negative, which can be mitigated by either a contribution from the sponsor to the plan or from asset returns that exceeds the returns on the liabilities

Question 3

How can the Sharpe Ratio of a stock be calculated using the risk-free rate and expected value/standard deviation of a stock? State the two disadvantages of the Sharpe ratio.

Answer 3

Question 4

What is the risk-adjusted change in surplus? What does the value of St represent?

Answer 4

Question 5

Surplus Risk in Static Analysis in the Presence of Liabilities

Answer 5

• Can measure surplus risk as a fraction of the asset value: sigma_t[S_t+1]\A_t for various equity allocations and funding ratios
• Lower funding ratios translate into greater surplus risk
• In general a greater equity allocation causes more surplus risk because equities are not very correlated with the liability; however, a small allocation to equity is often beneficial for diversification, particularly for overfunded plans

Question 6

What is the equity allocation that minimizes surplus risk? How does the optimal equity allocation vary as the funding ratio increases?

Answer 6

Formula sheet (10.A.14)

• Note, we are simply minimizing surplus risk here (and not optimizing RACS yet).
• A fund with a deficit is better off investing in bonds, because they offer a better hedge against changes in liability value, leading to lower surplus volatility
• A fund with a surplus, however, many want to invest in bonds up to a point so as to duration-match the liabilities, which offers the best possible hedge against changes in liability value. Beyond that point, the fund may be better off by investing an incremental dollar in equities rather than bonds due to the diversification effect between equities and bonds

Question 7

State the equity allocation needed to prevent expected surplus from shrinking. How does the equity allocation vary as the funding ratio increases?

how does the expected change in surplus change with the equity allocation?

Answer 7

Formula Sheet (10.A.16)

• The expected change in surplus is linearly increasing in the equity allocation (for a given funding ratio, assuming that the expected return on equity is larger than bonds)
• This formula gives us the critical point (equity allocation) where the exp change in surplus is zero

Question 8

Summarize the results of RACS optimization

Answer 8

• The RACS is strictly increasing with respect to the equity allocation for underfunded and exactly funded plans
• RACS “peaks” for overfunded plans
  
  • Based on their numerical results, RACS is maximized for an overfunded plan at an equity allocation of about 30%
  • A plan with sufficient funds (i.e. an overfunded plan) can invest Beta*Lt in the bond index (which effectively matches the liabilities) and then maximize the Sharpe ratio for the remaining assets
• The optimal equity allocation increases as noise is introduced
• Equity is most attractive for underfunded plans with very uncertain liabilities (a lot of noise), when the objective is RACS maximization
• Key Point: Suppose a pension fund desires to optimize RACS. The more underfunded a plan is, and the more uncertain future liabilities are, the more attractive equity appears relative to fixed income

Question 9

Optimizing RACS in a Dynamic Model

Answer 9

• So far, we have thought about optimal allocations at one specific point in time, for a pension plan with a static funding ratio
• However, we can extend this framework to understand dynamic changes in funding ratios
• When the funding ratio decreases (e.g. due to poor asset returns), a fund attempting to maximize RACS should invest more in equities
• If the noise in liabilities increases (e.g. due to legislative uncertainty), a fund attempting to maximize RACS should invest more in equities

Question 10

State how the asset and liability values at time t + 1 are computed in the dynamic analysis model. What does p represent?

Answer 10

Question 11

State how to compute the excess return required to maintain the funding ratio under the dynamic analysis model.

Answer 11

• The required return will be larger for larger payout rates and low funding ratios
• The large excess return requirements for underfunded plans highlights the need for large equity allocations for underfunded plans

Question 12

State how the expected future funding ratio is computed. How can this be used to determine the required excess return to reach fully funded status?

Answer 12

Question 13

Funding Probabilities

Answer 13

• The reading does not define the term “funding probability”, but you can think of it loosely as the probability the pension plan is at least fully funded over some given time horizon
• Underfunded plans reduce the probability of staying underfunded by allocating more to equity (since it has a higher expected return)
• Overfunded plans actually increase the probability of becoming underfunded by allocating more to equity
  
  • This is due to the fact that equity has more volatility and is less correlated with the liability
  • Overfunded plans could instead maintain a strong funding ratio by purchasing bonds that match the liabilities
  • Large equity allocations often outweight the benefits for overfunded plans

Question 14

Three key drivers of long-term performance in the presense of Liabilities

Answer 14

1. Bond/equity split
   
   • Underfunded plans (assets < liabilities) often have an optimal allocation to equity that is higher, in order to improve their funding status
   • Overfunded plans typically have lower optimal equity allocations
   • Here, we are looking at optimality from an RACS-maximiziation perspective
2. Duration of the bond portfolio
   
   • Matching the duration of the bond portfolio to that of liabilities is important for all plans
3. Level of diversification (i.e. international/global equity diversification)
   
   • There is a tradeoff – the higher Sharpe ratio is favorable, but the lower correlation with the liability index is unfavorable
   • Overfunded plans typically benefit more from global equity diversification – they can benefit from the higher Sharpe ratio of global equity
   • Underfunded plans are better off investing domestically in order to benefit from the higher correlation of liabilities with domestic assets
   • Additionally, note that fixed income diversification was not attractive for any of the plans studied. The effect of the increase in Sharpe ratio of assets from moving to global fixed income is more than offset by the lower correlation of liabilities with nondomestic assets