Mathematics of Finance - net present value

Question

A small company set up a new investment project on 1 January 2017. The initial investment on that date was £2.5 million with a further £2 million required after nine months, on 1 October 2017.

It is expected that from 1 January 2019 to 1 January 2032 inclusive, the company will receive net income from the project once a year. The net income on 1 January 2019 is expected to be £0.3 million, increasing by £0.1 million per annum on 1 January of each subsequent year.

The company expects to sell the business on 31 December 2032 for £3 million.

Calculate the net present value of the project on 1 January 2017 at a rate of interest of 12% per annum effective.

(Total 6 marks)

Answer 1

The total investment during 2017=£4,500,000.

The present value of the expected returns at 12% per annum starting at 2019 is 

2019 £300000/1.12²=£239158.16

2020 £400000/1.12³=£284712.10

2021 £500000/1.12⁴=£317759.04

2022 £600000/1.12⁵=£340456.11

2023 £700000/1.12⁶=£354641.78

2024 £800000/1.12⁷=£361879.37

2025 £900000/1.12⁸=£363494.91

2026 £1000000/1.12⁹=£360610.03

2027 £1100000/1.12¹⁰=£354170.56

2028 £1200000/1.12¹¹=£344971.32

2029 £1300000/1.12¹²=£333677.62

2030 £1400000/1.12¹³=£320843.87

2031 £1500000/1.12¹⁴=£306929.72

2032 £1600000/1.12¹⁵=£292314.02

TOTAL Benefits=£4575618.61. Net Present Value=£4575618.61-£45000000=£75,618.61.

However, since the second investment was made 9 months after 1 January, 2017, it could be argued that the present value of the second investment in January was 2000000/1.09 (¾ of 12%) =£1,834,862.39 making the total investment on 1 January 2017, £4,334,862.39 instead of £4,500,000. This would change the NPV to £240,756.22.