102 Session 9

102 Session 9 Time Weighted rate of Return Linked internal rate of return © KSES Exam questions are copyright Faculty & Institute of Actuaries & are used with their permission Source: www.actuaries.org.uk 279

What’s a rate of return? This is a single number that tries to summarise a whole load of numbers. e.g. in a year that had highs and lows and gains and losses, what single measure describes a “good year”, or a “good manager”?

280

What is a money weighted rate of return? This is the average “interest rate” familiar from equations of value.

150 100

Ie it’s the average return on money, where the average is weighted by cashflows (hence the name)

50 0 0

0.5

1

-50

E.g. if 1/(1+i) = v, and our fund is valued at 100 at the start of a year, and 198 at the end of the year, following a cashflow of +49 mid-year,

-100 -150

Then v = 1/(1+MWRR) solves 100 = 49v0.5 + 198v

281

Why is a money weighted rate of return no use for benchmarking? (1) 1200

It’s distorted by cashflows.

800 PRICE

E.g. suppose a fund manager runs a unit trust (which pays no dividends). The unit price halves over 6 months, and then doubles in another six months (back where it started - see the red line)

1000

400 200 0

The average (money weighted return) is 40%, because 100 * 1.40 + 49 * 1.400.5 = 198

CASHFLOWS

Suppose a fund only invests in the unit trust. It has a holding of 100 at the start of the year and adds cash of 49 mid-year. The 100 at the start of the year shrinks to a holding worth 50 mid-year. The new cash makes a holding of 99. The 99 doubles in value to 198 by the end of the year. (See the blue blocks).

600

250 200 150 100 50 0 -50 -100 -150

0

0.5

1

0

0.5

1 282

Why is a money weighted rate of return no use for benchmarking? (2) 1200

Suppose an identical fund took out 49 mid-year, (ie it almost sold up completely so hardly saw any of the “rebound”).

800 PRICE

The 100 at the start of the year shrinks to a holding worth 50 mid-year. The cash withdrawal leaves a holding of just 1. The 1 doubles in value to 2 by the end of the year

1000

400 200 0

The average money weighted annual return would be calculated as -72% because

But both funds were invested in the same asset, managed by the same manager. The (different) MWRRs haven’t measured the return on the asset, and haven’t measured the manager’s performance.

0

0.5

1

0

0.5

1

100 50 CASHFLOWS

100 * (1 – 72%) – 49 * (1 – 72%)0.5 = 2

600

0 -50 -100 -150

283

What is a time-weighted rate of return?

In our example case it is nil.

1200 1000 800 PRICE

This is what the cumulative return on the fund would have been had there been no cashflows. See http://www.investopedia.com/terms/t/tim e-weightedror.asp

600 400 200

The unit price fell and rose.

0 0

0.5

1

Ie the 1 + the total return = =(1 – 50%) * (1 + 100%) = 1 Ie the total TWRR = nil.

284

Eliminating all the cashflows in the TWRR calculation requires a lot of data. The LIRR gets round this by working out a MWRR for various periods, then chaining the returns together.

CASHFLOWS

What is a linked internal rate +100% of return?

If we chose periods of a whole year for our example, we’d just get the rather useless MWRR.

-50%

0

0.5

0

0.5

1

100 50 CASHFLOWS

If we chose periods of half a year, the LIRR would be nil, for both funds.

250 200 150 100 50 0 -50 -100 -150

0 -50

1

-100 -150

285

Specimen Q12

286

Specimen Q12

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Cashflow Value

J

F

M

A

M

J

J

A

S

O

N

D

Rough guess: ignoring 0.2m (cash in), grew from …..to 1.8-0.2 = 1.6 => annual return about …./1.2 ie 33%

Exact calc: the dates that matter are the start & the end. Grew from 1.2 to …… in first period Then grew from (1.4 + …..)= 1.6 to ….. in second period Overall, grew by factor of 1.4 / 1.2 * 1.8 / ….. in a year = growth factor of …….. pa ie …….% pa (close to guess?) 287

Specimen Q12

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Cashflow Value

J

F

M

A

M

J

J

A

S

O

N

D

Rough guess: ignoring 0.2m (cash in), grew from 1.2 to 1.8-0.2 = 1.6 => annual return about 1.6/1.2 ie 33%

Exact calc: the dates that matter are the start & the end. Just though investment management, grew from 1.2 to 1.4 in first period Then grew from (1.4 + 0.2)= 1.6 to 1.8 in second period Overall, grew by factor of 1.4 / 1.2 * 1.8 / 1.6 in a year = growth factor of 1.3125 pa ie 31.25% pa (close to guess) 288

Specimen Q12

Model answer

289

Apr 2000 Q5

290

Apr 2000 Q5

350

Rough guess: ignoring cash in, grew from 180 to 309 – (25+18+16)=…..

300 250 200 150 100 50 0 1/97

7/97

1/98

7/98

1/99

7/99

1/2000

=> 3 year return about ……/180 ie 39%

Exact calc: Just though investment management, grew from 180 to ….. in first six months. Over next year, shrank from ……+25 = 237 to 230. Grew from …….+18 = 248 to 295 in next year. Shrank from ……..+16= 311 to 309 in last six months. Growth factor over period = 212/…… * 230/237 * 295/248 * 309/311 = 1.3509 (close to guess) => annual growth factor = 1.3509(……) = 1.105 => TWRR = ……..%

291

Apr 2000 Q5

350

Rough guess: ignoring cash in, grew from 180 to 309 – (25+18+16)=250

300 250 200 150 100 50 0 1/97

7/97

1/98

7/98

1/99

7/99

1/2000

=> 3 year return about 250/180 ie 39%

Exact calc: Just though investment management, grew from 180 to 212 in first six months. Over next year, shrank from 212+25 = 237 to 230. Grew from 230+18 = 248 to 295 in next year. Shrank from 295+16= 311 to 309 in last six months. Growth factor over period = 212/180 * 230/237 * 295/248 * 309/311 = 1.3509 (close to guess) => annual growth factor = 1.3509(1/3) = 1.105 => TWRR = 10.5%

292

Apr 2000 Q5

Model answer

293

Apr 2001 Q2

294

Apr 2001 Q2

800

Rough guess: ignoring cash in, 11% pa would increase initial fund from 400 to 400 * 1.11^3 = …..

700 600 500 400 300 200 100 0 1/98

7/98

1/99

7/99

1/2000

7/2000

Exact calc: Just though investment management, Total growth factor is …… / 400 * ……/ (460 + 50) * …… / (500 + 40) * And total growth factor = 1.11^…. (11% pa TWRR) => X = £………m (close to guess?)

1/2001

Add cashflows back in (without any return), guess X= …..+ 50+ 40+ 60 = 700

X / (650 + 60) 295

Apr 2001 Q2

800

Rough guess: ignoring cash in, 11% pa would increase initial fund from 400 to 400 * 111^3 = 550.

700 600 500 400 300 200 100 0 1/98

7/98

1/99

7/99

1/2000

Exact calc: Just though investment management, Total growth factor is 460 / 400 * 500 / (460 + 50) * 650 / (500 + 40) * And total growth factor = 1.11^3 (11% pa TWRR)

7/2000

1/2001

Add cashflows back in (without any return), guess X= 550+ 50+ 40+ 60 = 700

X / (650 + 60)

=> X = £715.5m (a bit more than guess – OK allowing for additional return on cash in)

296

Apr 2001 Q2

Model answer

297

Apr 2003 Q8

298

Apr 2003 Q8(i)a

60

Rough guess: ignoring cash in, grew from 40 to 53 – (4+2)= 47

50 40 30 20 10 0 1/00

7/00

1/01

7/01

1/02

=> 2 year return is approx. 47/40 ie 18%

Exact calc: Just though investment management, Growth factor over whole 2 years is 43 / 40 * 49 / (43 + 4) * 53 / ( 49 + 2 ) = 1.1647 (a bit below guess – OK as cashflow in got some return too) => Average annual growth factor (just due to management) = TWRR = 1.1647(1/2) – 1 = 7.92%

299

Apr 2003 Q8(i)a

Model answer

300

Apr 2003 Q8(i)b

60

Rough guess: will be close to TWRR. Will be more or less than TWRR?

50 40 30 20 10 0 1/00

7/00

1/01

7/01

1/02

Will be slightly less. 2nd half of 2001 was worse (on TWRR measure) than 1st half of 2001. MWRR for 2001 is average of frist & second half of year. Average will be more heavily weighted to (poor) second half by the extra money that came in mid-2001.

Exact calc: MWRR over first year is 43/40 = 1.075 MWRR over first year is i: if v = 1 / (1 + i) Then (43 + 4)(1 + i) + 2(1 + i) (1/2) = 53 ie (1 + i) (1/2) (using quadratic formula) = 1.04085 and so 1 + i = 1.08337. Hence 1 + LIRR pa = ( 1.075 * 1.08337 ) (1/2) = 1.07918 Ie LIRR is just less than TWRR (as guessed)

301

Apr 2003 Q8(i)b

Abbreviated Model answer

302

Apr 2003 Q8(ii)

60

TWRR will be identical to LIRR when calculations are identical.

50 40 30 20 10 0 1/00

7/00

1/01

7/01

Intervals free of mid-interval-cashflow are: 1.1.00 to 1.1.01 1.1.01 to 1.7.01 1.7.01 to 31.12.01

1/02

Ie MWRR calcs (mid-period) must be identical to TWRR calcs ie simple end_value / start_value ratios. So, the intervals chosen for the LIRR can’t include any mid-interval cashflows. (This is a bit unlikely as the point of the LIRR sum is to get round the heavy data requirements of the TWRR calculation). 303

Apr 2003 Q8(ii)

Model answer

304

Sep 2003 Q2

305

Sep 2003 Q2

Model answer

306

Key question Get 100% on April 2003 Q8 It doesn’t matter how many times you see the answers. Your goal is to get someone else to understand the solution.

307

Next session: par yields

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