MIM Ch 10: Strategic Asset Allocation in the Presence of Uncertain Liabilities

Question 1

State the liability return framework in MIM Ch 10.

Answer 1

RL,t Rf ,t pRB,t Rf ,t q 􀀀 t

RL,t is the total return on the liability index at time t

Rf ,t is the risk-free rate of return

RB,t is the total return on the bond index

t is the noise term, with mean return and volatility

The parameter reflects the duration of the liability relative to the specified bond
index
Ÿ
Pension Plan Duration
Bond Index Duration

Question 2

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 2

SRi
i Rf
i

Two disadvantages of the Sharpe ratio:
1. Does not consider the liabilities
2. Does not work well in a multi-period time setting

Question 3

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

Answer 3

St At Lt is the surplus

The RACS is computed as:
RACSt
Et rSt􀀀1 St p1 􀀀 Rf qs
t rSt􀀀1s

This RACS is the return relative to the liabilities, whereas the traditional Sharpe
ratio evaluates the asset return relative to cash

Question 4

Summarize the results of RACS optimization.

Answer 4

The RACS is strictly increasing with respect to the equity allocation for
underfunded and exactly funded plans

RACS “peaks” for overfunded plans
Ÿ Based on their numerical results, RACS is maximized for an overfunded plan at an
equity allocation of about 30%
Ÿ A plan with sufficient funds (i.e. an overfunded plan) can invest Lt in the bond index
(which effectively matches the liabilities) and then maximize the Sharpe ratio for the
remaining assets

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 5

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

Answer 5

The optimal (in terms of minimizing surplus risk) portion of the fund to invest in
equities can be calculated with the following formula:
$’%
1
Lt
At
,/-
p2
B BEq
2
E 􀀀 2
B 2BE

The equity allocation increases with the funding ratio (A/L) when the objective is to
minimize surplus risk
Note: The formula above is provided on the formula sheet. Therefore, it is ok if you
want to skip memorizing this formula.

Question 6

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

Answer 6

The minimum equity allocation needed to prevent surplus from shrinking is:
B
$’%

Lt
At
1
,/-
􀀀
Lt
At
Rf p1 q
E B

This equity allocation decreases as the funding ratio (A/L) increases
Ÿ Underfunded plans (assets liabilities) have higher minimum equity allocations
Ÿ Overfunded plans have lower minimum equity allocations
Note: The formula above is provided on the formula sheet. Therefore, it is ok if you
want to skip memorizing this formula.

Question 7

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

Answer 7

p is the fixed fraction of the liability value paid at the end of each period (i.e.
payout factor)

At􀀀1 At p1 􀀀 RA,t􀀀1q pLt p1 􀀀 RL,t􀀀1q

Lt􀀀1 Lt p1 􀀀 RL,t􀀀1qp1 pq

Question 8

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

Answer 8

Et rRx,t􀀀1s
Ft p1 pq 􀀀 p
Ft
1

Question 9

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 9

E0pFt q F0

1􀀀x
1p
t
􀀀 p
1p 1􀀀x
1p q
t
x􀀀p

Set the above equation equal to 1 and solve for x to get the required excess
return to reach fully funded status
Note: The formula above is provided on the formula sheet. Therefore, it is ok if you
want to skip memorizing this formula.