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Geometric Returns of negative interest rates

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 is one of the risk factors.

EURO overnight LIBOR rates are as given below –

May 12, 2017 - 0.42571%

May 11, 2017 - 0.42214%

May 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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Matthew Graves

Active Member
Subscriber
Not sure about the arithmetic average of the log return, seems like a strange thing to do to me. Are you instead trying to find the geometric average?
In your first example this would be:



which is what you were expecting.

You can follow the same procedure for the interest rates.
 
Not sure about the arithmetic average of the log return, seems like a strange thing to do to me. Are you instead trying to find the geometric average?
In your first example this would be:



which is what you were expecting.

You can follow the same procedure for the interest rates.

Thanks for your comments.

I understand the formula mentioned by you gives holding period Arithemetic Returns with the holding period = 2.

I have mentioned above that it is just a crude method of computing average returns. However, my main concern is computations of log returns of mixture of rates e.g. if rates vary in the range
( - 1% to 1%). If the successive rates differ in sign, how do I compute the log returns (since log returns way of computing returns, being strongly advocated rightly by Mr Kevin Dowd).

Regards

Ashok
 
Geometric return = (ending value/beginning value)^(1/N) - 1

This formula gives something like CAGR. What if there are the intermittent returns and you need to compute the holding period returns?

Anyways, I repeat my concern is computing log returns for two days e.g. suppose today's rate is 0.5% while yesterdays rate was say -0.11%. How do I compute LN( 0.5/ (-0.11) ) if I have decided to use the log returns and not the Arithmetic returns?

Regards

Ashok
 

Matthew Graves

Active Member
Subscriber
Anyways, I repeat my concern is computing log returns for two days e.g. suppose today's rate is 0.5% while yesterdays rate was say -0.11%. How do I compute LN( 0.5/ (-0.11) ) if I have decided to use the log returns and not the Arithmetic returns?

Log returns only make sense when the underlying asset price can be positive only, as Dowd himself says. This not the case for interest rates.
 
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Log returns only make sense when the underlying asset price can be positive only, as Dowd himself says. This not the case for interest rates.

Dear Mr Graves

Thanks a lot for your comments.

Totally Agree with you. But the problem is in my portfolio, I may be having equity as well as debt securities. If I need to construct the correlation matrix, I need to have risk factor returns. I can't have equity returns generated using log returns while debt security risk factor returns using say Arithmetic returns. Isn't it?

What Mr Dowd mentioned is -

upload_2017-5-17_18-8-55.png

Hence, if I decide to use log returns, I may have to deal with negative rates too. Unfortunately, we have to face such operational problems in real life world.

Anyways, thanks a lot for your valuable input. Guess we wait for Mr David to respond.

Thanks again:)

Regards

Ashok
 

emilioalzamora1

Well-Known Member
Listen, a logarithm of the returns (in your case 0.5% and -0.11%) makes NO sense at all. A log return is calculated from prices having LN(t:0/t:-1) or written as LN(today's price of the asset/yesterday's price of the asset).

There are several contributions to this topic, let me add few important lines about pros/cons of log returns:

  • The assumption of a log-normal distribution of returns, especially over a longer term than daily (say weekly or monthly) is unsatisfactory, because the skew of log-normal distribution is positive, whereas actual market returns for, say, S&P is negatively skewed (because we see bigger jumps down in times of panic).
  • It’s difficult to logically combine log returns with fat-tailed distributional assumptions, even for daily returns, although it’s very tempting to do so because assuming “fat tails” sometimes gives you more reasonable estimates of risk because of the added kurtosis.
  • The compounded return (having a sequence of returns as a product, (1+r1)*(1+r2)*(1+r3)....is also not satisfactory as the product of normally distributed variables is NOT normal.
    Instead, the sum of normally-distributed variables is normal (BUT ONLY when all variables are uncorrelated), which is useful when we recall the following logarithmic identity:
log(1+r) = ln(p at t:0/p at t:-1) = log (p at t:0) - log (p at t:-1)

Example:
Price t:0 = 1430
Price t:-1 = 1409

ln(1430/1409) = 0.0148
log(1430) - log(1409) = 0.0064
log(1+0.0148) = 0.00638

See the very close difference between 0.0064 and 0.00638!

Having said that, the below identity leads us to a pleasant algorithmic benefit; formula for calculating compounded returns:

log(1+rintial) + log (1+r2)........+log(1+rfinal) = log (p initial) - log (p final)


wich is the compounded return over n periods is merely the difference in log between initial and final periods.

where
1. rinitial = initial return of the sample
2. rfinal = final return of the sample
3. p initial = initial price (of the asset) of the sample
4. p final = last price (of the asset) of the sample



Pro log-returns: The approximate raw-log equality says that when returns are very small (common for trades with short holding durations), the following approximation ensures they are close in value to raw returns: log(1+r) ~ r, r < 1.
 
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Listen, a logarithm of the returns (in your case 0.5% and -0.11%) makes NO sense at all. A log return is calculated from prices having LN(t:0/t:-1) or written as LN(today's price of the asset/yesterday's price of the asset).

There are several contributions to this topic, let me add few important lines about pros/cons of log returns:

  • The assumption of a log-normal distribution of returns, especially over a longer term than daily (say weekly or monthly) is unsatisfactory, because the skew of log-normal distribution is positive, whereas actual market returns for, say, S&P is negatively skewed (because we see bigger jumps down in times of panic).
  • It’s difficult to logically combine log returns with fat-tailed distributional assumptions, even for daily returns, although it’s very tempting to do so because assuming “fat tails” sometimes gives you more reasonable estimates of risk because of the added kurtosis.
  • The compounded return (having a sequence of returns as a product, (1+r1)*(1+r2)*(1+r3)....is also not satisfactory as the product of normally distributed variables is NOT normal.
    Instead, the sum of normally-distributed variables is normal (BUT ONLY when all variables are uncorrelated), which is useful when we recall the following logarithmic identity:
log(1+r) = ln(p at t:0/p at t:-1) = log (p at t:0) - log (p at t:-1)

Example:
Price t:0 = 1430
Price t:-1 = 1409

ln(1430/1409) = 0.0148
log(1430) - log(1409) = 0.0064
log(1+0.0148) = 0.00638

See the very close difference between 0.0064 and 0.00638!

Having said that, the below identity leads us to a pleasant algorithmic benefit; formula for calculating compounded returns:

log(1+rintial) + log (1+r2)........+log(1+rfinal) = log (p initial) - log (p final)


wich is the compounded return over n periods is merely the difference in log between initial and final periods.

where
1. rinitial = initial return of the sample
2. rfinal = final return of the sample
3. p initial = initial price (of the asset) of the sample
4. p final = last price (of the asset) of the sample



Pro log-returns: The approximate raw-log equality says that when returns are very small (common for trades with short holding durations), the following approximation ensures they are close in value to raw returns: log(1+r) ~ r, r < 1.

Dear emilioalzamora1

If you allow, I wish to bring to your kind notice few points -

(1) I have initiated this thread by quoting the overnight EURO Libors which are negative. Though pure intention of raising this thread was totally academic, however, this is actual problem we are facing. No doubt the EURO Libors are having negative values. However, one can't deny the fact that going forward depending on economic conditions, they may become positive rates. You may note that I have used the word "Hypothetical" too in the thread.

I am assuming such an hypothetical situation (though very rare, but can't rule out the possibility) where the rates became positive from negative. I am considering this instance and I am assuming very small values in the neighborhood of zero. Hence I have mentioned the rates 0.5% (say at t:eek:) and -0.11% (say at t-1).

(2) You have mentioned that "Listen, a logarithm of the returns (in your case 0.5% and -0.11%) makes NO sense at all".

Response :

Please note that I have used the word "rates". These are not returns. I am not computing the log of returns.

(3) The assumption of a log-normal distribution of returns, especially over a longer term than daily (say weekly or monthly) is unsatisfactory, because the skew of log-normal distribution is positive, whereas actual market returns for, say, S&P is negatively skewed (because we see bigger jumps down in times of panic).

Response :

In my thread nowhere I have mentioned that I intend to use log normal distribution. If I use Var covar or for that matter Monte carlo, I may assume risk factors returns are normally distributed (i.e. prices or rates follow log normal).

If I use Historical simulation, I still compute the returns and for 1 day horizon, I assume that tomorrow one of these past scenarios will occur and these scenarios are nothing but the returns observed over last one year.

Is it that in case of Historical simulation, I can't compute returns using log of ratio of prices (or rates)? At least I am not aware. Many spreadsheets which are available on internet for Historical simulation do use log returns.

(4) the sum of normally-distributed variables is normal (BUT ONLY when all variables are uncorrelated)

Response :

I am exactly not sure about this. I feel any linear combination of two Normal variables is Normal. If these two variables are not independent i.e. "Rho" is non zero,

May be (X + Y) ~ N(MEAN = mean X + mean Y, VARIANCE = V(X) + V(Y) + 2*Cov(X,Y)) where Cov(X, Y) = Rho * Sqrt(V(X) * V(Y)).

But honestly I am not sure about this. Sort of gut filling and I will definitely come back over this.

(5) I am bit confused with your example

ln(p at t:0/p at t:-1) = log (p at t:0) - log (p at t:-1)

Example:
Price t:0 = 1430
Price t:-1 = 1409

ln(1430/1409) = 0.0148
log(1430) - log(1409) = 0.0064
log(1+0.0148) = 0.00638


Response :

ln(p at t:0/p at t:-1) = log (p at t:0) - log (p at t:-1)

Here are considering base = 10 on RHS of equation?

I am getting LHS = ln(p at t:0/p at t:-1) = 0.014794 which is not equal to RHS if we consider the base = 10.

This is my earnest request to you can you elaborate if possible why we are using Log with base 10. Am aware of the relation

LN(X) = Log(X, base = 10) / Log(2.718282, base 10) or LOG(X, Base = 10) = LN(X) / LN(10). Does this relationship to play any role here. Honestly, I am ignorant about this esp

log(1+0.0148) = 0.00638 = log(1 + LN(1430/1409)). I wasn't aware of this actually. Qyestion is why do we need to do it.

Its a well known fact that

LN(1 + Arithmetic Return) = Geometric Return

Is this what you are trying to lead to?


And finally

Log(p1 / p10) = Log(p1/p2*p2/P3*.....*P9/P10)

= Log(p1/p2) + Log(p2/p3) .........+Log(p9/p10)

= Log(p1) - Log(p2) + log(p2) - log(p3) + log(p3) - log(p4) +..........- log(p9) + log(9) - log(p10)

= Log(p1) - Log(p10)


Thanks for your comments.

With warm regards

Ashok
 
Hi everyone,

Thanks a lot for your valuable feedback. This forum does indeed help persons like me to clarify so many doubts. May be I need to get my concepts clear. May I suggest that we wait for Mr David to respond on this.

Thanks again

Regards

Ashok
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @Ashok_Kothavle With respect to your original post, I don't recall ever seeing interest rates as direct inputs into a log (ie, continuously compounded returns). As mentioned above, my experience with the log return is ln[P(1)/P(0)] or ln([P(1)+D]/P(0)], which is to say, using prices not interest rates. My first instinct here is, if you really want to do this, can't you just transform (translate) the interest rates into prices, even if for artificial purposes? After all, this is really what a Treasury bill price does for a rate, or a Eurodollar futures price does for a rate. Any given interest rate can be transformed into many different non-zero prices; e.g., 2% --> (100 -2) = 98.00 like a money market "discount" instrument; or r --> exp(-r*T) as a zero-coupond bond. In short, any f(.) could transform the rate. As ln(P1/P0) = r such that exp(r) = P1/P0, could you just use P1 = P(0)*exp(r); i.e., assume P(0) is 100 or index, and transform 100 into 100*exp(r*a) where constant a is a sensitivity. (?)

Otherwise, I'd be tempted to push back on trying too hard to take log returns of rates; e.g., why aren't simple differences better?

FWIW, when I consulted and we generated performance measures, we did not try to represent returns (i) any time that values crossed the zero axis (e.g., positive to negative or vice verse) or (ii) even when (too) small bases were involved--e.g., P1 growing from a P0 that is too small--because our examples showed the output tended to distort the findings. I hope that's helpful!
 
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Dear Mr David,

OMG I get so much to learn from this forum. I am sure, if I continue with BT for another few months, I shall be an enlightened person. Thanks for your valuable inputs. Those will go long in refining my whole approach towards Market Risk and overall financial risk management. I have so much to learn esp when it comes to Market risk.

One of my resource has suggested me to use change in basis instead of computing returns when it comes to interest rates.

Thanks again.

Regards

Ashok
 
Through some resource, I got following feedback –

“The return on interest rates should be calculated as a simple difference between two rates. Interest rates themselves are treated as some kind of Return instead of Asset price. Hence, use of Arithmetic as well as Geometric returns in case of interest rates is not proper.”

E.g. in the first message, the Overnight EURO rates have been quoted as.

May 12, 2017 ~ (– 0.43571%)

May 11, 2017 ~ (– 0.42214%)

Thus the return is simply (– 0.43571% – (– 0.42214%)) = - 0.01357%

emilioalzamora has rightly mentioned “logarithm of the returns (in your case 0.5% and -0.11%) makes NO sense at all”.

It was my ignorance. Thanks again Mr David and emilioalzamora.


Regards

Ashok
 
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