The Legacy
Your school has been left a million pounds in the will of an ex-
pupil. What model of investment and spending would you use in order
to ensure the best return on the money?
Age
14 to 16
Challenge level
Problem
Your school has been left £1 000 000 in the will of an ex-pupil ...
The pupil made some conditions on how the money should be invested and used. These were:
- The money should have a lifetime of $50$ years.
- That the school benefits in some way (spends part of the investment) every year.
You are asked to produce models of investment and expenditure based on any balance being invested at a fixed interest rate (it is suggested that you could start with a rate of $4 \%$). Your model could also consider different inflation rates.
What model would you choose to ensure the best return for the school over a period of $50$ years?
If you want some suggested starting points for models you might like to look at these .
Getting Started
What happens when you invest $ £1\; 000 \;000$ at $4 \%$? [You could use this spreadsheet to investigate.]
How long does it take to double the money?
What would happen if we begin to spend money, withdrawing it at the start of each year? You might like to look at this spreadsheet and think about the following:
- Why do you think the interest for the first year isn't £40000?
- Extend the spreadsheet further and, by changing the yearly expenditure, experiment with trying to make the balance each year settle at around £980,000 or giving the £1,000,000 a lifetime of $20$ years.
What happens if you take account of inflation when thinking
about how much you might spend each year? This spreadsheet might be
useful for investigating this idea. You might like to think about
some of the following:
- What is the interest rate in this spreadsheet?
- What is the inflation rate?
- What annual expenditure would leave no (or a very small) balance left at the end of $50$ years? What is the total income/expinditure at the end of $50$ years in this case?
- What interest rates and inflation rates would guarantee an income of £3000000 if you spent the same amount each year (taking into account inflation), with only a small balance, or no balance, at the end?
Student Solutions
No solution in yet with a good explanation
Teachers' Resources
This is an opportunity for pupils to set up their own models using a spreadsheet and investigate what happens when they change variables such as interest, inflation rates and how much is spent each year. Investigating how a small change in interest rates can affect the total income over different time periods can be enlightening. Investigating the impact of spending large amounts of money up-front can also provide valuable insights.
Some models are suggested on the model cards which have varying levels of complexity. Some suggestions about how a series of sessions working on this problem might run are given briefly below.
After spending some time "making sense" of the problem and what is meant by a model, discuss ideas for making good use of the legacy:
- What types of things should the money be spent on?
- Why would the ex-pupil want the money to have a long term impact?
- What is the importance of inflation in any model we try?
- Why would you invest the money?
Students could work in groups to think about the models they
might like to investigate then choose two models to compare. Think
about:
- Total expenditure for each model,
- Benefits and limitations of each model,
- Comparisons between models,
- How a change in inflation might change the recommendations,
- How a change in the time-scale for using the money might affect any recommendation.
A plenary might include a discussion of the elegance or
realism of some of the models and what the use of a spreadsheet had
to offer to the task.