## Problem

Calculate the college fund required to start a four year college program in 15 years time. The tuition fees are currently 12,000 and are first payable at the start of year 16. Inflation is 4% and the rate of return is 9%. In addition, calculate the lump sum which needs to be invested today to provide the college fund.

## Solution

This problem is solved in three steps.

- Calculate the tuition fees at the start of year 16 allowing for inflation of 4%.
- Calculate the total college fund needed at the start of year 16.
- Calculate the amount which needs to be invested today to create the college fund.

### 1. Calculate the Tuition Fees Allowing for Inflation

The current tuition fee is 12,000 (PV), this needs to be compounded at the inflation rate (g) for a period of 15 years (n), to calculate what the fees will be at the start of year 16. The calculation is carried out using the future value of a lump sum formula as follows:

Tuition fees = FV = PV x (1 + g)^{n}PV = Current tuition fees = 12,000 n = number of years = 15 g = inflation rate = 4% Tuition fees = 12,000 x (1 + 4%)^{15}Tuition fees = 21,611.32

Allowing for inflation of 4%, the current tuition fees of 12,000 will increase to 21,611.32 by the start of year 16.

### 2. Calculate the Total College Fund Needed

The tuition fees will commence at the start of year 16 at the value calculated in step 1 of 21,611.32, they then continue for a further 3 years increasing each year by the inflation rate of 4%.

The college tuition fees are in fact a four year annuity due growing at the rate of inflation (g). The present value of the 4 year annuity at the start of year 16 is give by the present value of a growing annuity due formula as follows:

College fund needed = Pmt x (1 + i) x (1 - (1 + g)^{n}x (1 + i)^{-n}) / (i - g) Pmt = tuition fees = 21,611.32 n = number of years = 4 i = nominal rate = 9% g = growth rate = inflation = 4% College fund needed = 21,611.32 x (1 + 9%) x (1 - (1 + 4%)^{4}x (1 + 9%)^{-4}) / (9% - 4%) College fund needed = 80,677.02

The present value of the college fund needed at the start of year 16 to pay the tuition fees for the next 4 years is 80,677.02

### 3. Calculate the Amount Invested to Create the College Fund

The final step is to calculate the lump amount which needs to be invested today in order to create the college fund of 80,677.02 by the start of year 16. This is simply a matter of discounting the value of the college fund at the start of year 16 back to the present day using the present value of a lump sum formula.

PV = FV / (1 + i)^{n}FV = College fund at start of year 16 = 80,677.02 n = number of years = 15 i = nominal rate = 9% PV = 80,677.02 / (1 + 9%)^{15}PV = 22,148.91

The lump sum amount of 22,148.91 needs to be invested today at a discount rate of 9% in order to provide a college fund of 80,677.02 by the start of year 16

### Alternative Funding Using an Annuity

As an alternative to the lump sum investment of 22,148.91, a 15 year annuity could be taken out to provide the required college fund at the start of year 16. The annual payments for such an annuity are given by the future value of an annuity formula, and are calculated as follows:

Pmt = FV / (( (1 + i)^{n}- 1 ) / i) FV = Value of college fund = 80,677.02 n = number of years = 15 i = nominal rate = 9% Pmt = 80,677.02 / (( (1 + 9%)^{15}- 1 ) / 9%) Pmt = 2,747.77

An annuity of 2,747.77 paid at the end of each year for 15 years will provide the college fund of 80,677.02 at the start of year 16.

## Explanation

The college tuition fees are quoted at current rates and need to be compounded forward at the rate of inflation to provide the value of the fees at the start of year 16. As the fees continue for a further 3 years and increase by inflation each year, the present value of the fees at the start of year 16 can be calculated using the present value of a growing annuity due formula. The annuity due formula is used as the fees are paid at the start of each year.

Having calculated the college fund needed at the start of year 16, the next part of the calculation is to work out what lump sum needs to be invested today to provide for the college fund. As an alternative, the college fund could be provided by investing in an annuity for the next 15 years.

### College Fund Examples

This is one of many time value of money examples, discover another at the links below.