No Alex there is not a "private pension debt", there is however a deficit between current funding levels and actuarial calculations of future liabilities.
Pension funds invest in a wide range of assets including but not limited to government bonds (gilts), Corporate bonds, shares & warrants, property, various derivatives including interest rate swaps, mortality swaps etc. The investment in property has little to do with capital values or the movement thereof but a lot to do with yield (rental income).
A good example of the sort of range of instruments and providers used by pension funds can be found here
http://www.pensionprotectionfund.org.uk/About-Us/Pages/Investment.aspx