1. Measuring Investment Performance
MTH5124 Actuarial Mathematics I
6 October 2017

2. Take two investment managers……
Fund Manager A
Fund Manager B
Who is the best manager manager?

3. Syllabus objective
(ix) Show how discounted cashflow techniques can be used in investment project appraisal…….
5. Calculate the money-weighted rate of return, the time- weighted rate of return and the linked internal rate of return on an investment or a fund.

4. Example Who is the best fund manager? Fund Manager A Fund Manager B
Fund Value
1 Jan 2016
£1000
Net Cashflow £0
31 Dec 2016
£1080
£1020
Who is the best fund manager?

5. Example Who is the best fund manager? Fund Manager A Fund Manager B
Fund Value
1 Jan 2016
£1000
Cashflow In
£200
Cashflow Out
31 Dec 2016
£1080
£1020
Who is the best fund manager?

6. Problem
Who is the best fund manager?

7. Background
The managers both invest principally in the same underlying asset portfolios (eg UK equities).
The managers each have the same performance benchmark (eg the FTSE All-Share Total Return Index).
Over the 12 month period the performance benchmark returned 8% (this is an example not a real portfolio).

8. Manager A – Money Weighted Rate of Return
The money weighted rate of return is equivalent to the internal rate of return (IRR) on the fund.
Let i be the monthly rate of return on the fund (and assume months of equal length).
The equation of value is:
F12 = 1000(1+i) (1+i) (1+i)3 = 1080
Find i by estimation

9. Manager A – Money Weighted Rate of Return
F12 = 1000(1+i) (1+i) (1+i)3 = 1080 Initial estimate = 1% per month F12 = nd Estimate = 0.50% per month F12 = rd Estimate (by interpolation) = 0.648% per month F12 = 1074, etc Annual Money Weighted Return = 8.9% pa

10. Manager A – Time Weighted Rate of Return
Calculate the return earned by the manager on the fund in the periods between the cashflows: Eg From Start Jan to End April: Start Value = 1000 Cashflows = none End Value = 1180 Increase in Fund = 𝐴= =1.180 ….an 18.0% return in 4 months
Terminology
Ft- represents fund value at time t immediately before any cash flow into and out of the fund.
Ft+ represents fund value at time t immediately after any cash flow into and out of the fund.

11. Manager A – Time Weighted Rate of Return
From To
Fund + Cashflow at Start
Fund at End of Period
Increase
Factor
% Increase
1 Jan
30 April
1000
1180
1.1800
+18.00%
1 May
30 Sept
980
753
0.7684
-23.86%
1 Oct
31 Dec
953
1082
1.1354
+13.54%
Time weighted rate of return = x x = 0.029
So, over one year, the annual time weighted rate of return is 2.9% (in comparison to a benchmark return of 8%).

12. Generalise – Money Weighted Rate of Return
Fund value at time O = F 0 Net cashflows into fund = Ctk at times t1 , t2 ….tn Fund value at time T ( T > tn ) is F T Call i the effective annual rate of interest earned in the interval [0,T] Then: F T = F 0 (1 +i)T + Ct1 (1 +i)T-t1 + Ct2 (1 +i)T-t2 + …………………………. + Ctn (1 +i)T-tn Then i is the money weighted rate of return (MWRR). The MWRR is the IRR on the fund over the period 0 to T. Remember The cashflows measured are those to and from the fund manager. Income from assets would be retained by the fund manager.

13. Generalise – Time Weighted Rate of Return
Fund value at time O = F 0 Net cashflows into fund = Ctk at times 0, t1 , t2 ….tn Fund value at time T ( T > tn ) is F T Fund value at time tk , including new cashflow, is Ftk+ At tk , immediately before the cashflow, the fund value is defined to be Ftk- The time weighted rate of return (TWRR) is i per year, where: The fund would be valued every time there is a cashflow; with modern systems this will usually be daily. _

14. TWRR or MWRR?
(Graph reflects capital return only).

15. TWRR or MWRR? Time Weighted Money Weighted
Not influenced by timing of cashflows
Reflects the return earned by the investor
Best way to judge an investment manager or to compare investment managers
Used to compare performance against a market index
Can’t be calculated directly
- use iterative techniques to get best estimate
Can be very data intensive (eg if daily cashflows)
Less data needed – but there may be no solution!
…important measure but not perfect…

16. Cashflows can help managers!
A great time for the manager to pay out cash.
A great time for the manager to receive cash.
(Graph reflects capital return only).

17. Linked Internal Rate of Return
A simplified method sometimes used to approximate to TWRR
Rate of return calculated over different intervals (0,t1), (t1, t2)….. (tn-1, tn)
Annual rate of return in interval (tr-1, tr) is ir
Linked internal rate of return is i, where:
it may be calculated by an approximate method (ie not an accurate periodic IRR)
The linked IRR should be close to the true TWRR if:
the time intervals over which the returns ir are calculated are short; and
cashflows are fairly small relative to the size of the fund.

18. Exercise
Complete the following table: Who is the best investment manager?
2016
Fund Manager A
Fund Manager B
Benchmark Return
8.0%
Time Weighted Rate of Return
2.9% ??
Money Weighted Rate of Return
8.9%

19. “Investing should be more like watching paint dry or watching grass grow. If you want excitement, take $800 and go to Las Vegas.” 
Paul Samuelson
American Economist & Winner of Nobel Prize for Economic Sciences

20. Key Points
An introduction to the theory of measuring investment performance
The size and timing of cashflows does effect calculated returns if an IRR (or MWRR) is used
Industry standard methodologies for measuring fund performance are based on time weighted measures.
Measuring performance over a year is not long enough to make a sound judgement (in most cases)

21. Measuring Investment Performance
MTH5124 Actuarial Mathematics I
6 October 2017