Charity programme funding after an investment shortfall
A charity receives a legacy of £500,000 on 1 January 2025. It plans to pay £55,000 on 1 January each year for ten years, with the first payment made immediately. The trustees assume the fund will earn 4.5% effective annually throughout. After four payments have been made, on 1 January 2029, the fund is worth £285,000, not the £350,000 that the original 4.5% assumption would have produced. The charity can now earn only 3.5% effective annually. The next payment is due immediately. The trustees want to know whether the original plan was sound, what level payment can now be sustained for the remaining six payments, and how to explain the position to donors who compare the £285,000 fund with the £330,000 of payments still to be made.
1. Using an effective rate of 4.5%, calculate the present value at 1 January 2025 of the ten planned payments and determine whether the original £500,000 was sufficient. Show the annuity factor you use.
Read the worked answer
Use an annuity-due because the first payment is immediate. At 4.5%, v=1/1.045=0.956938 and v^10=0.643927. The annuity-due factor is a_10=(1-v^10)/(1-v)=(1-0.643927)/0.043062=8.269. PV=55,000 x 8.269=£454,795. The £500,000 legacy exceeds this by about £45,205, so the original plan was affordable on the stated assumptions. The calculation assumes the 4.5% rate applies throughout and that all ten payments are made; it does not allow for expenses or investment risk.
2. At 1 January 2029, immediately before the fifth payment, £285,000 is available and the fund will earn 3.5% thereafter. Calculate the maximum level payment, made immediately and annually thereafter for six payments in total, that can be supported. Compare it with the planned £55,000.
Read the worked answer
At 1 January 2029, before the fifth payment, six payments remain, including the immediate one. Use a six-payment annuity-due at 3.5%: v=1/1.035=0.966184, v^6=0.81350, so a_6=(1-0.81350)/0.033816=5.515. Set P x 5.515=285,000, giving P=£51,676, about £51,700. This is £3,324, or 6.0%, below the planned £55,000. The trustees must either reduce the annual payment to about £51,700, inject about £18,300 of present value, or shorten the programme. The present value of the planned £55,000 payments is 55,000 x 5.515=£303,325, so the present-value shortfall is 303,325-285,000=£18,325.
3. Explain why comparing the £285,000 fund with the £330,000 of remaining undiscounted payments is misleading, and state two assumptions in the calculation that the trustees should monitor.
Read the worked answer
Comparing £285,000 with £330,000 ignores the time value of money: the remaining payments are spread over six years, and the fund can earn 3.5% on balances. The relevant comparison is the present value of the promised payments, about £303,300 at 3.5%, against £285,000, a shortfall of about £18,300, not £45,000. The undiscounted comparison therefore exaggerates the deficit. Two assumptions to monitor are the 3.5% reinvestment rate, which may fall further, and the timing and amount of payments, including whether the charity can delay or reduce them. A third is that the £285,000 is net of any expenses. If the rate falls, the sustainable payment falls because the annuity factor rises.