Cash Flows and Project Appraisal Nov 4, 2014(9:33) Exercises 2.8 Page 39 Example 7.3a. An investment fund is valued once a year at the beginning of April. You are given the following information about its performance:
Date Value of fund Deposits
April 1, 1996 £50,000
October 1, 1996 £20,000
April 1, 1997 £71,000 April 1, 1998 £97,000
Find the linked internal rate of return for the period of April 1, 1996 to April 1, 1998 using subintervals of a year.
(University of Warwick ST334 Examinations, 2012) Solution. Use units of £1,000. Let i1 denote the annual effective rate of interest earned by the fund in the interval from April 1, 1996 to April 1, 1997. Then 50(1 + i1) + 20(1 + i1)1/2= 71. This is the quadratic 50x2+ 20x− 71 = 0 which has solution x = 1.00830. Hence 1 + i1= x2= 1.01666889. Also 1 + i2= 97/71.
Hence if iLdenotes the LIRR, we have (1 + iL)2= 1.01666889× 97/71 and hence 1 + iL= 1.17854578768. Hence the LIRR is 17.86%.
7.4
Summary.
• Money Weighted Rate of Return (MWRR): same as internal rate of return.
f
0(1 + i)
t+ c
t1(1 + i)
t−t1+ · · · + c
tn(1 + i)
t−tn= f
t• Time Weighted Rate of Return (TWRR): requires value of fund at times of all cash flows.
(1 + i)
t= f
t1−0f
0f
t2−0f
t1· · · f
tn−0f
tn−1f
tf
tn(7.4a)
= (1 + i
1)
t1(1 + i
2)
t2−t1· · · (1 + i
n)
t−tn−1• Linked Internal Rate of Return (LIRR): used if value of fund not known at times of all cash flows.
8 Exercises
(exs2-3.tex)1. The following table gives information concerning an investment fund.
Calendar Year 1997 1998 1999 2000
£ millions £ millions £ millions £ millions
Value of fund at 30 June — 460 500 650
Net cash flow received on 1 July — 50 40 60
Value of fund at 31 December 400 550 600 X
If the time weighted rate of return on the fund during the period from 31 December 1997 to 31 December 2000 is 11% per annum effective, calculate X, the value of the fund on 31 December 2000.
(Institute/Faculty of Actuaries Examinations, April 2001)[3]
2. The following table gives information concerning an investment fund.
Calendar Year 2000 2001 2002
£ million £ million £ million
Value of fund on 1 January before cash flow 100 80 200
Net cash flow received on 1 January 20 30 10
Value of fund on 31 December 80 200 200
Calculate the effective time weighted rate of return over the three year period.
(Institute/Faculty of Actuaries Examinations, September 2003)[3]
3. You are given the following information in respect of a pension fund:
Calendar Value of fund at Value of fund at Net cash flow received on
Year 1 January 30 June 1 July
1997 £180,000 £212,000 £25,000
1998 £261,000 £230,000 £18,000
1999 £273,000 £295,000 £16,000
2000 £309,000
Calculate the annual effective time weighted rate of return earned on the fund over the period from 1 January 1997 to 1 January 2000. (Institute/Faculty of Actuaries Examinations, April 2000)[4]
Page 40 Exercises 2.8 Nov 4, 2014(9:33) ST334 Actuarial Methods c⃝R.J. Reed 4. A fund has a value of £120,000 on 1 January 2001. A net cash flow of £20,000 was received on 1 November 2001 and a further net cash flow of £48,000 was received on 1 May 2002. Immediately before receipt of the first cash flow, the fund had a value of £137,000 and immediately before the second cash flow the fund had a value of £173,000.
The value of the fund on 31 December 2002 was £205,000.
(i) Calculate the annual effective time weighted rate of return earned on the fund for the period 1 January 2001 to 31 December 2002.
(ii) Discuss the relative strengths and weaknesses of using the time weighted rate of return as opposed to the money weighted rate of return when comparing the performance of two investment managers over the same period.
(Institute/Faculty of Actuaries Examinations, April 2004)[3+3=6]
5. An investment fund had a market value of £2.2 million on 31 December 2001 and £4.2 million on 31 December 2004.
It had received a net cash flow of £1.44 million on 31 December 2003.
The money weighted rate of return and the time weighted rate of return for the period from 31 December 2001 to 31 December 2004 are equal (to two decimal places).
Calculate the market value of the fund immediately before the net cash flow on 31 December 2003.
(Institute/Faculty of Actuaries Examinations, April 2005 )[7]
6. A fund had a value of £21,000 on 1 July 2003. A net cash flow of £5,000 was received on 1 July 2004 and a further cash flow of £8,000 was received on 1 July 2005. Immediately before receipt of the first net cash flow, the fund had a value of £24,000, and immediately before receipt of the second net cash flow the fund had a value of £32,000. The value of the fund on 1 July 2006 was £38,000.
(i) Calculate the annual effective money weighted rate of return earned on the fund over the period 1 July 2003 to 1 July 2006.
(ii) Calculate the annual effective time weighted rate of return earned on the fund over the period 1 July 2003 to 1 July 2006.
(iii) Explain why the values in (i) and (ii) differ.
(Institute/Faculty of Actuaries Examinations, April 2007)[3+3+2=8]
7. A life insurance fund had assets totalling £600m on 1 January 2003. It received net income of £40m on 1 Jan-uary 2004 and £100m on 1 July 2004. The value of the fund was:
£450m on 31 December 2003; £500m on 30 June 2004; and £800m on 31 December 2004.
(i) Calculate, for the period 1 January 2003 to 31 December 2004, to three decimal places:
(a) the time weighted rate of return.
(b) the linked internal rate of return, using subintervals of a calendar year.
(ii) Explain why the linked internal rate of return is higher than the time weighted rate of return.
(Institute/Faculty of Actuaries Examinations, September 2006)[8+2=10]
8. A pension fund had assets totalling £40 million on 1 January 2000. It received net income of £4 million on 1 Jan-uary 2001 and £2 million on 1 July 2001. The value of the fund totalled:
£43 million on 31 December 2000; £49 million on 30 June 2001 and £53 million on 31 December 2001.
(i) Calculate for the period 1 January 2000 to 31 December 2001, to 3 decimal places:
(a) the time weighted rate of return per annum
(b) the linked internal rate of return, using sub-intervals of a calendar year.
(ii) State both in general, and in this particular case, when the linked internal rate of return will be identical to the time weighted rate of return. (Institute/Faculty of Actuaries Examinations, April 2003)[3+5+2=10]
9. A pension fund makes the following investments(£m):
1 January 2004 1 July 2004 1 January 2005 1 January 2006
12.5 6.6 7.0 8.0
The rates of return on money invested in the fund were as follows:
1 January 2004 to 1 July 2004 to 1 January 2005 to 1 January 2006 to 30 June 2004 31 December 2004 31 December 2005 31 December 2006
5% 6% 6.5% 3%
You may assume that 1 January to 30 June and 1 July to 31 December are precise half-year periods.
(i) Calculate the linked internal rate of return per annum over the 3 years from 1 January 2004 to 31 December 2006, using semi-annual sub-intervals.
(ii) Calculate the time weighted rate of return per annum over the 3 years from 1 January 2004 to 31 December 2006.
(iii) Calculate the money weighted rate of return per annum over the 3 years from 1 January 2004 to 31 Decem-ber 2006.
(iv) Explain the relationship between your answers to (i), (ii) and (iii) above.
(Institute/Faculty of Actuaries Examinations, September 2007)[3+3+4+2=12]
Cash Flows and Project Appraisal Nov 4, 2014(9:33) Exercises 2.8 Page 41 10. The value of the assets held by an investment fund on 1 January 2012 was £1.3 million.
On 30 September 2012, the value of the assets was £1.9 million.
On 1 October 2012, there was a net cash outflow from the fund of £0.9 million.
On 31 December 2012, the value of the assets was £0.8 million.
(i) Calculate the annual effective time-weighted rate of return (TWRR) for 2012.
(ii) Calculate the annual effective money-weighted rate of return (MWRR) for 2012 to the nearest 1%.
(iii) Explain why the MWRR is significantly higher than the TWRR.
(Institute/Faculty of Actuaries Examinations, April 2013)[2+3+2=7]
11. The value of the assets held by an investment fund on 1 January 2011 was £2.3 million. On 30 April 2011, the value of the assets had risen to £2.9 million and, on 1 May 2011, there was a net cash inflow to the fund of £1.5 million.
On 31 December 2011, the value of the assets was £4.2 million.
(i) Calculate the annual effective time-weighted rate of return (TWRR) for 2011.
(ii) Calculate, to the nearest 0.1%, the annual effective money-weighted rate of return (MWRR) for 2011.
(iii) Explain why the TWRR is significantly higher than the MWRR for 2011.
(Institute/Faculty of Actuaries Examinations, April 2012)[2+4+2=8]
12. In an accumulation fund, income is retained and used to increase the value of the units. Suppose the price in pence of the units in two accumulation funds, fund A and fund B, at various dates was as follows:
Date 1/4/2001 1/4/2002 1/4/2003 1/4/2004 1/4/2005 1/4/2006
Fund A: unit price 80 90 98 180 190 160
Fund B: unit price 70 67 66 100 110 130
Suppose the rate of inflation3is given by the index in the following table:
Date 1/4/2001 1/4/2002 1/4/2003 1/4/2004 1/4/2005 1/4/2006
Inflation index 100 102 105 117 120 122
An individual invests £1000 in fund A on 1/4/2001 and a further £1000 in fund A on 1/4/2002. On 1/4/2003 he withdraws £1000 from fund A and invests this £1000 in fund B.
(a) Calculate the real time weighted rate of return of the total investment for the period 1/4/2001 to 1/4/2006.
(b) Derive an equation for the real money weighted rate of return of the total investment for the period 1/4/2001 to 1/4/2006. Use this equation to determine whether the real money weighted rate of return is greater or less than the real time weighted rate of return. (University of Warwick ST334 Examinations, 2014) 13. An investment fund is valued at £120 million on 1 January 2010 and at £140 million on 1 January 2011. Immediately after the valuation on 1 January 2011, £200 million is paid into the fund. On 1 July 2012, the value of the fund is
£600 million.
(i) Calculate the annual effective time-weighted rate of return over the two-and-a-half year period.
(ii) Explain why the money weighted rate of return would be higher than the time weighted rate of return.
(Institute/Faculty of Actuaries Examinations, September 2012)[3+2=5]
14. (i) State the strengths and weaknesses of using the money weighted rate of return as opposed to the time weighted rate of return as a measure of an investment manager’s skill.
(ii) An investor had savings of £41,000 in an account on 1 January 2006. He invested a further £12,000 in this account on 1 August 2006. The total value of the account was £45,000 on 31 July 2006 and was £72,000 on 31 December 2007.
Assuming that the investor made no further deposits or withdrawals in relation to this account, calculate the annual effective time weighted rate of return for the period 1 January 2006 to 31 December 2007.
(Institute/Faculty of Actuaries Examinations, September 2008)[3+2=5]
3 Real and money rates of interest are considered in section 4 of chapter 5 on page 89.
Page 42 Exercises 2.8 Nov 4, 2014(9:33) ST334 Actuarial Methods c