Cash Flows and Project Appraisal Nov 4, 2014(9:33) Exercises 2.6 Page 35
5.4
Summary.
• Equation of value: NPV(a, ρ) = 0 considered as an equation for i.
• Internal Rate of Return (IRR): disadvantages. Numerical solution: methods for finding a starting value. Linear interpolation.
• Discounted Payback Period (DPP). Payback period.
6 Exercises
(exs2-2.tex)1. (Not examined) Write anR function which takes two arguments:vec.cashflowwhich represents the vector of cash flows, andrwhich represents a vector of interest rates. The function should return a matrix with two columns: the first column calledinterestand the second column calledNPV.
Thus the call
irr( c(-87,25,-40,60,60), seq(0.01,0.3,by=0.001) ) should return a matrix like this:
interest NPV
[1,] 0.010 14.434862816 [2,] 0.011 14.087489548 . . . many lines omitted
[46,] 0.055 0.288410285 [47,] 0.056 0.005780673 [48,] 0.057 -0.275585819 . . . many lines omitted
[290,] 0.299 -43.014001386 [291,] 0.300 -43.120233885 This shows the requiredIRRis 0.056.
2. Compare the following two investment projects:
• Project A: Investment costing £50,000 with returns £8,000 each half-year for the next 4 years.
• Project B: Government bonds which provide a yield of 7% every 6 months.
3. Suppose we have the cash flow a = (−c0,−c1, . . . ,−cm, b0, b1, . . . , bn) at times t0 < t1 <· · · < tm< t′0 < t′1 <
. . . < t′nwhere all cj> 0 and all bj> 0. Prove there is a unique solution for theIRR.
4. An investor is considering two investment projects, A and B. Both involve outlays of £1 million. Project A will provide a single incoming cash payment after 8 years of £1.7 million. Project B will provide incoming cash payments of £1 million after 8 years, £0.321 million after 9 years, £0.229 million after 10 years and £0.245 million after 11 years.
(i) Determine the rate of interest (i′) at which the net present value of the two projects will be equal.
(ii) By general reasoning or by illustrative calculation, show that at a positive rate of interest i∗ where i∗ < i′, project B will have a higher net present value than project A.
(Institute/Faculty of Actuaries Examinations, September 2004)[6]
5. A small technology company set up a new venture on 1 January 2001. The initial investment on that date was £2 million with a further £1.5 million required on 1 August 2001.
It is expected that on 1 January 2002, net income (i.e. income less running costs) will begin at the rate of £0.3 million per annum and that the rate will increase by £0.1million per annum on 1 January of each subsequent year. It is assumed that the net income will be received continuously throughout the project.
The company expects to sell the business on 31 December 2011 for £3 million.
Calculate the net present value of the venture on 1 January 2001 at a rate of interest of 6% per annum, convertible half-yearly. (Institute/Faculty of Actuaries Examinations, April 2001)[8]
6. A computer manufacturer is to develop a new chip to be produced from 1 January 2008 until 31 December 2020.
Development begins on 1 January 2006. The cost of development comprises £9 million payable on 1 January 2006 and £12 million payable continuously during 2007.
From 1 January 2008 the chip will be ready for production and it is assumed that income will be received half yearly in arrear at a rate of £5 million per annum.
(i) Calculate the discounted payback period at an effective rate of interest of 9% per annum.
(ii) Without doing any further calculations, explain whether the discounted payback period would be greater than, less than or equal to that given in part (i) if the effective interest rate were substantially greater than 9% per annum. (Institute/Faculty of Actuaries Examinations, April 2005)[6+2=8]
Page 36 Exercises 2.6 Nov 4, 2014(9:33) ST334 Actuarial Methods c⃝R.J. Reed 7. A company has agreed to build and operate a toll bridge for a regional government. The company will invest £10 million per annum for the first 2 years of the project, the investment being made continuously during this period.
The bridge will then come into operation and the company will start to receive payments at the end of each year, the first payment occurring at the end of year 3 of the project. The amount of payment at the end of year 3 will be £8 million, reducing by £0.5 million in each of the subsequent years until the annual amount is £3 million, after which the annual reduction will be £1 million. When the payments have reduced to zero, the company’s involvement in the project will end.
(i) Calculate the net present value of the project at a rate of interest of 10% per annum effective.
(ii) Explain whether the internal rate of return achieved on this project will be greater or less than 10% per annum effective. (Institute/Faculty of Actuaries Examinations, September 2002)[7+2=9]
8. A motor manufacturer is to develop a new car model to be produced from 1 January 2002 for 6 years until 31 De-cember 2007. The development cost will be £33 million, of which £18 million will be incurred on 1 January 2000,
£10 million on 1 July 2000 and £5 million on 1 January 2001.
The production cost of each car is assumed to be incurred at the beginning of the calendar year of production and will be £9,000 during 2002. The sale price of each car is assumed to be received at the end of the calendar year of production. Both the production costs and the sale prices are assumed to increase by 5% on each 1 January, the first increase occurring on 1 January 2003. It is also assumed that 5,000 cars will be produced each year and that all will be sold.
The sale price of each car produced in 2002 is £12,100.
(i) Calculate the discounted payback period at an effective rate of interest of 9% per annum.
(ii) Without doing any further calculations, explain whether the discounted payback period would be greater than, equal to, or less than the period calculated in (i) if the effective rate of interest were substantially less than 9%
per annum. (Institute/Faculty of Actuaries Examinations, April 2000)[9+3=12]
9. An investment project gives rise to the following cash flows. At the beginning of each of the first three years,
£180,000 will be invested in the project. From the beginning of the first year until the end of the the twenty-fifth year, net revenue will be received continuously. The initial rate of payment of net revenue will begin at £25,000 per annum. The rate of payment is assumed to grow continuously at a rate of 6% per annum effective.
(i) Calculate the net present value of the project at an effective rate of interest of 7% per annum.
(ii) Calculate the discounted payback period of the project at an effective rate of interest of 7% per annum.
(iii) Calculate the annual effective rate of growth of net revenue which would be required if the project is to have a zero net present value at an effective rate of interest of 7% per annum.
(Institute/Faculty of Actuaries Examinations, September 2000)[6+5+6=17]
10. (i) An investor is deciding to invest in a project. Explain why the discounted payback period is a poorer decision criterion than net present value assuming the investor is not short of capital.
An investor is considering two projects A and B. Project A involves the investment of £1 million at the outset.
The only income to be received will be a payment of £3.5 million after ten years. Project B also involves the investment of £1 million at the outset. Income will be received from this project continuously. In the first year the rate of payment will be £0.08 million, in the second year £0.09 million, in the third year £0.10 million, with the rate increasing by £0.01 million each year thereafter until the tenth year, after the end of which no further income will be received.
(ii) Calculate the net present value of both investment projects at a rate of interest of 4% per annum effective.
(iii) Show that the discounted payback period of project A is after that of project B (no further calculation is neces-sary).
(iv) In the light of your answer to (i) above, explain which project is the more desirable to an investor with unlimited capital, and why. (Institute/Faculty of Actuaries Examinations, April 2002)[2+10+3+2=17]
11. A car manufacturer is to develop a new model to be produced from 1 January 2016 for six years until 31 Decem-ber 2021. The development costs will be £19 million on 1 January 2014, £9 million on 1 July 2014 and £5 million on 1 January 2015.
It is assumed that 6,000 cars will be produced each year from 2016 onwards and that all will be sold.
The production cost per car will be £9,500 during 2016 and will increase by 4% each year with the first increase occurring in 2017. All production costs are assumed to be incurred at the beginning of each calendar year.
The sale price of each car will be £12,600 during 2016 and will also increase by 4% each year with the first increase occurring in 2017. All revenue from sales is assumed to be received at the end of each calendar year.
(i) Calculate the discounted payback period at an effective rate of interest of 9% per annum.
(ii) Without doing any further calculations, explain whether the discounted payback period would be greater than, equal to, or less than the period calculated in part (i) if the effective rate of interest were substantially less than 9% per annum. (Institute/Faculty of Actuaries Examinations, April 2013)[9+2=11]
Cash Flows and Project Appraisal Nov 4, 2014(9:33) Section 2.7 Page 37 12. (i) In respect of an investment project, define
(a) the discounted payback period (b) the payback period
(ii) Discuss why both the discounted payback period and the payback period are inferior measures compared with the net present value for determining whether to proceed with an investment project.
(iii) A consortium of investors is considering bidding to host a major athletics event. The project will be regarded as viable if it provides a positive net present value at a rate of interest of 10% per annum effective. The consortium estimates the following cash flows will be generated by the event (all figures in £100 million):
Costs
Initial costs of building To be incurred continuously at a rate of 1 per annum for five years starting on 1 January 2006
Running costs of the event To be incurred continuously at a rate of 1 per annum for 3 months starting on 1 January 2011.
Cost of making bid 0.2 to be incurred on 1 January 2004.
Revenue
Sale of television rights To be received continuously at a rate of 0.3 per annum for 3 months starting on 1 January 2011.
Other revenue from sale of merchan-dise, marketing rights, tickets, etc
Assumed to be received in the middle of each year from 2004 to 2015 inclusive. The revenue from this source is expected to start at 0.1 per annum and increase each year by 0.1 up to and including 2011. The same revenue is expected in 2012 as in 2011. After 2012 revenue is expected to decline by 0.2 per annum until 2015, after which year no further revenue will be received from this source.
Revenue from sale of the stadium and other infrastructure
This will be received on 1 January 2015
Determine the sale price of the stadium and other infrastructure that would have to be achieved for the project to be considered viable. (Institute/Faculty of Actuaries Examinations, September 2003)[3+3+11=17]
13. A company is considering investing in the following project. The company has to make an initial investment of 3 payments, each of £105,000. The first is due at the start of the project, the second 6 months later, and the third payment is due one year after the start of the project.
After 15 years, it is assumed that a major refurbishment of the infrastructure will be required, costing £200,000. The project is expected to provide no income in the first year, an income received continuously of £20,000 in the second year, £23,000 in the third year, £26,000 in the fourth year and £29,000 in the fifth year. Thereafter, the income is expected to increase by 3% per annum (compound) at the start of each year.
The income is expected to cease at the end of the 30th year from the start of the project. The cash flow within each year is assumed to be received at a constant rate.
(i) Calculate the net present value of the project at a rate of interest of 8% p.a. effective.
(ii) Show that there is no discounted payback period within the first 15 years, assuming an effective rate of interest of 8% p.a.
(iii) Calculate the discounted payback period for the project, assuming an effective rate of interest of 8% p.a.
(Institute/Faculty of Actuaries Examinations, May 1999. Specimen Examination.)[8+8+6=22]