Case 043Investment and portfolio riskHard

An underfunded pension fund must close its gap in ten years without risking a large loss in any one year. Set the return target and find the equity share that meets it inside the loss limit.

MSCIRemote · 2013

The situation

Nirvelan Employees' Pension Fund has assets equal to 90% of the present value of its pension promises, and those liabilities grow 7.5% a year as benefits accrue and discount unwinds. The trustees want the fund fully funded in ten years, without new contributions from the sponsor.

The investment committee assumes equities return 11% a year with 18% volatility and bonds 7% with 5% volatility, with a correlation of 0.2. The trustees' risk limit: in a one-in-twenty bad year, the fund should not lose more than 10% of its assets. Treat returns as normal, and all figures as the committee's assumptions, not forecasts.

Your task

What return does the fund need, what range of equity shares meets both the target and the loss limit, and what mix would you put to the trustees?

Quick check

What yearly return closes a 90% funding level in ten years when liabilities grow 7.5%?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The fund needs about 8.64% a year, which takes at least 41% in equities; the loss limit caps equities at about 63%. Allowing for volatility drag, the compound target needs nearer 55% equity, so the workable band is about 55% to 63%. A 60% equity mix meets both tests with a small margin on each, and the trustees should hear how thin that margin is.

Step 1What return does closing the gap require?

Someone who owes a lender an amount that grows 7.5% a year and holds savings worth 90% of it must grow the savings faster than 7.5% just to catch up. Assets must rise from 90 to match liabilities rising from 100 at 7.5% a year: 1.075 times the tenth root of 100 over 90, less one, is about 8.64%. That is only about 1.1 points above liability growth, but it has to be earned every year for ten years.

At 8.64% a year the gap closes in ten years; at 7% it widens1001502000246810YearsLiabilities at 100 todayliabilities 206assets at 8.64%assets at 7%: 177, 86% funded
With liabilities growing 7.5% a year from 100, assets starting at 90 must grow 8.64% a year to catch them in year 10; at 7%, an all-bond return, the fund ends about 86% funded, worse than today.
Step 2What equity share do the two constraints allow?

The return target sets a floor. Expected return is 7% plus 4 points for each unit of equity, so reaching 8.64% needs at least 41% in equities. The loss limit sets a ceiling. A one-in-twenty bad year is expected return less 1.645 standard deviations; the loss must stay under 10%. Volatility rises with equity, so solve for the share where 1.645 times volatility less expected return equals 10%: about 63%.

The relationship
σp=w2σE2+(1−w)2σB2+2w(1−w)ρσEσB1.645 σp−μp≤10%\sigma_p = \sqrt{w^2\sigma_E^2 + (1-w)^2\sigma_B^2 + 2w(1-w)\rho\sigma_E\sigma_B} \qquad 1.645\,\sigma_p - \mu_p \le 10\%
wshare of assets in equities
\sigma_E, \sigma_Bvolatility of equities, 18%, and bonds, 5%
\rhocorrelation between them, 0.2
\mu_pexpected return of the mix, 7% + 4% x w
What it says in wordsThe mix's volatility comes from both assets and their correlation, and the loss limit says a one-in-twenty bad year must not cost more than 10% of assets.
The return target sets the floor, the loss limit sets the ceiling: 41% to 63% equity7%8%9%10%11%4%8%12%16%20%Volatility of the fund's assets, one yearExpected returntarget 8.64%loss limit: 1.645 x vol - return = 10%all bondsall equity41% equity63% equity
As the equity share rises from 0 to 100%, expected return climbs from 7% to 11% and volatility from 5% to 18%; the 8.64% target needs at least 41% equity and the 10% loss limit allows at most 63%, so the feasible mixes lie between the two.
Step 3What does the simple answer leave out?

Expected yearly return is not what compounds. A fund that gains 20% and then loses 20% is down 4%. The compound return is roughly the expected return less half the variance, so hitting 8.64% compounded needs about 55% in equities, not 41%. The workable band shrinks to about 55% to 63%. At 60% equity the mix expects 9.4%, compounds at about 8.75%, and its one-in-twenty loss is about 9.3%: inside both tests, but only just.

Two further cautions for the trustees. Returns are not normal; equity crashes are more frequent than a normal curve implies, so the 10% limit will be breached more often than one year in twenty. And the right risk for a pension fund is surplus risk, assets against liabilities: long bonds that move with the liabilities are less risky than their asset volatility suggests. A liability-relative study might allow the same return with less measured risk, which is the next step to propose.

Where candidates lose it

The first trap is to set the target equal to liability growth, 7.5%. That only keeps the fund 90% funded forever; closing the gap needs the extra 1.1 points.

The second is to meet the return target with the cheapest possible equity share, 41%, using expected rather than compound returns. Volatility drag means the fund would most likely fall short, and the answer that says so is the one interviewers remember.

What the interviewer asks next

  • The sponsor offers to contribute 1% of liabilities a year. How does the equity range change?
  • How would you measure risk against liabilities rather than on assets alone?
  • The trustees want no losing years at all. What would you tell them?

Asked at MSCI, Risk Management, Remote, 2013 (Wall Street Oasis): What risk-return targets would you set for an institutional investor?

Company names and figures are illustrative.