Hi Carmen
Happy to help. I have answered your questions below - hopefully this makes things clearer for you.
Purchasing annuities question
Regarding your first question, it's first useful to think about what liquidity is - the ease with which investments can be turned into cash without affecting the market price (ie not having to sell at a lower price to find a buyer). Liquidity risk is therefore the risk of not having cash to meet (benefit or expense) outgo or having to sell illiquid assets at a loss to do this.
Purchasing annuities would be an investment (which would match pensions in payment well or fairly well) for the pension scheme. An annuity cannot (yet or in a CP1 world) be sold, and it can't be surrendered so it's illiquid as it can't be converted to cash. The increase in liquidity risk is because the more annuities that are held, the less the pension scheme assets there are that can be converted to cash; therefore the scheme is more likely to be unable to find the cash to cover any unexpected immediate outgo. For example, a non-pensioner dying could be entitled to a lump sum on death and this could be large relative to the size of the scheme - particularly if the member is a very high earner (death benefit is often a multiple of salary) or the scheme is smaller (and so there are less assets). (Another unexpected payment could be a transfer value needing to be paid.)
The more likely liquidity risk is as I describe above rather than the risk you've outlined, but there may still be liquidity risks involved in the purchase of the annuities. If the pension scheme is buying annuities as part of a plan (to match current pensioner more closely for example), then finding the money to buy the annuities can be done over a longer period with less/no impact from liquidity risk (time could be taken to sell less liquid assets without having to sell cheap). However, if the policy/rule becomes buying annuities on retirement then finding cash to buy annuities for unexpected early retirements could be a liquidity risk. Once an annuity is purchased, any liquidity risk of having to pay regular payments to members in retirement is reduced as the income from annuities can be used to pay pensions (reducing/avoiding the need to liquidate other assets) - the pensions in payment should be best matched by annuities.
Purchasing death in service insurance
Regarding your second question, as described in my answer to the first question above, a death in service lump sum creates a liquidity risk as cash may need to be found at short notice - it's not known when deaths will occur. The risk is that, to the extent cash or liquid assets aren't available to cover these lumps sum benefits, losses could be made by selling less liquid assets, or in the extreme it will not be possible to meet benefit payments.
If insurance is purchased to cover this lump sum benefit, the pension scheme will pay regular known premiums for this insurance rather than larger benefit payments with unknown timing. Liquidity risk from unexpectedly having to have large amounts of cash is therefore reduced.
Regarding your third question, as partly covered at the end of my answer to your first question, the smaller the scheme is, the larger death benefit payments will be relative to the size of the contributions and investment returns received and the assets held. There is then a greater possibility of the scheme not having high enough income or liquid assets to cover these benefits without liquidating less liquid assets. (Pension scheme investments included in assets will ideally be longer term and therefore less liquid as liabilities are mostly longer term.) The greater likelihood of not having enough readily available cash/liquidity for smaller schemes means that liquidity risk is greater making the use of insurance to cover death benefits (removing this risk) greater.
