#### Ashok_Kothavle

##### Member

Dear Mr David

Kevin Dowd in his book Measuring Market Risk (2nd Edition) has mentioned the advantages of using the Geometric returns over Arithmetic returns (3rd chapter). In fact I always quote following example –

Suppose an asset was trading at the prices as given below –

May 15, 2017 - $50

May 14, 2017 - $100

May 13, 2017 - $50

If I consider the Arithmetic returns, my return on May 14th over May 13th was (100-50)/50*100 = 100%. The return on May 15th over May 14th is (50-100)/100*100 = -50%. Hence, my average return is (100%+(-50%))/2 = 25% (Do understand it’s a crude method of arriving at average return). However this is misleading as asset was trading at 50 and now its trading at the same level i.e. 50. Hence, my actual return is zero.

On the other hand, if I use Geometric return, my return on May 14th over May 13th = LN(100/50) = 69.31%. Similarly, my return on May 15th over May 14th is = LN(50/100) = - 69.31% and hence my average return = (69.31%-69.31%)/2 = 0.

As Mr Dowd had mentioned, for longer horizon, we must use Geometric returns than Arithmetic returns. However, problem is suppose

EURO overnight LIBOR rates are as given below –

May 12, 2017

May 11, 2017

May 10, 2017

......... and so on

In this case,

From academic point of view, can you comment on this or suggest some resource for the same i.e. how to obtain returns if the risk factor is a series of mix rates i.e. positive and negative or for that matter even ‘0’ value (may be hypothetical).

Regards

Ashok

Kevin Dowd in his book Measuring Market Risk (2nd Edition) has mentioned the advantages of using the Geometric returns over Arithmetic returns (3rd chapter). In fact I always quote following example –

Suppose an asset was trading at the prices as given below –

May 15, 2017 - $50

May 14, 2017 - $100

May 13, 2017 - $50

If I consider the Arithmetic returns, my return on May 14th over May 13th was (100-50)/50*100 = 100%. The return on May 15th over May 14th is (50-100)/100*100 = -50%. Hence, my average return is (100%+(-50%))/2 = 25% (Do understand it’s a crude method of arriving at average return). However this is misleading as asset was trading at 50 and now its trading at the same level i.e. 50. Hence, my actual return is zero.

On the other hand, if I use Geometric return, my return on May 14th over May 13th = LN(100/50) = 69.31%. Similarly, my return on May 15th over May 14th is = LN(50/100) = - 69.31% and hence my average return = (69.31%-69.31%)/2 = 0.

As Mr Dowd had mentioned, for longer horizon, we must use Geometric returns than Arithmetic returns. However, problem is suppose

**EURO overnight LIBOR**is one of the risk factors.EURO overnight LIBOR rates are as given below –

May 12, 2017

**- 0.42571**%May 11, 2017

**%****-**0.42214May 10, 2017

**- 0.42571**%......... and so on

In this case,

**the rates are negative**, however, still computing the Geometric returns won’t be a problem as the ratio is positive and we can take log of positive value only. However, if I have mix rates like some small positive values, some small negative values etc, then sometimes the ratio will be negative and it won’t be possible to obtain log value of negative ratio.From academic point of view, can you comment on this or suggest some resource for the same i.e. how to obtain returns if the risk factor is a series of mix rates i.e. positive and negative or for that matter even ‘0’ value (may be hypothetical).

Regards

Ashok

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