the mid-price calculation. This means that the dollar-basedarbitrageur will make riskless gains if the swap points are numerically less than this −36.6 (i.e.if they were 36). So there are zero bounds for this arbitrage, even in the real world (since thesterling investor will make money if the swap points are numerically higher than −36.6, ( i.e.37). This is surprising – normally the transactions costs of the real world mean that arbitragehas a ‘neutral’ area where it is proﬁtable for no-one.
Case 4 – Borrowing one currency to lend the otherThe ﬁnal case we will examine is where an arbitrageur is not a natural lender or borrower.Here, he will have to borrow one currency and lend another, which will mean that he will haveto overcome the interest rate bid/offer spreads to be able to exploit the arbitrage. The basecurrency of the investor is irrelevant (and even could easily be a third currency) – the key isthe boundary forward points beyond which the arbitrage is proﬁtable.The principle of this arbitrage is simple: let’s start with an investor borrowing sterling toinvest in dollars. The maths is identical to Case 2, except that equation (3) in case 2 will have a£LIBOR term in, not a £LIBID. This works through to the following equation (matching (10)in Case 2):(1)$£RHSFWD = $£RHSSPOT × [1 + ($LIBID/12)] / [1 + (£LIBOR/12)]Substituting values:(2)$£RHSFWD = 1.5347 × [1+ (2.0625%/12)] / [1 + (5.0000%/12)]= 1.5347 × (1.001719 / 1.004167)= 1.53096and therefore swap points:(3)1.53096 − 1.5347 = −37.4 pointsSo if the RHS of the swap points rises higher than −37.4 (e.g. 38), then the borrowing sterling/lending dollars arbitrage is proﬁtable.A similar boundary can be established for the LHS of the swap. The maths is identical toCase 3, except that equation (3) in Case 3 will have a $LIBOR term in, not a $LIBID. Thisworks through to the following equation (matching (14) in Case 3):
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(4)$£LHSFWD = $£LHSSPOT × [1 + ($LIBOR/12)] / [1 + (£LIBID/12)]Substituting values:(5)$£LHSFWD = 1.5345 × [1+ (2.1250%/12)] / [1 + (4.9375%/12)]= 1.5345 × (1.001771 / 1.004115)= 1.53092and therefore swap points:(6)1.53092 − 1.5345 = −35.8 pointsSo if the LHS of the swap points falls below −35.8 (e.g. 35), then the borrowing dollars/lendingsterling arbitrage is proﬁtable.These arbitrage possibilities make currency forward rates virtually always a pure mathematicalresult of relative interest rates. The question of whether this always means that interest ratesdetermine forward rates is a sophisticated question, the answer to which involves the relativeweights of money following domestic interest rates or forward rates. Most of the time domesticrates dominate, but sometimes it is clear that forward rate expectations dominate.This has been most visible recently in the attacks against the French franc peg versus DEM(1982 and 1983), sterling peg versus DEM (1992), French franc versus DEM (1993) and HongKong dollar versus US dollar (1998), when domestic interest rates became distorted by viewsof the future exchange rate. At its most extreme, French franc ‘Euro’ (i.e. offshore) overnightrates went to 3000% p.a., equivalent to an 8% devaluation the next day. This was only possiblebecause of exchange control, which kept domestic French rates to about 20% p.a.
APPENDIX 2 – LOGNORMAL RETURNSSetting out the problemIn practical fund management, we are often faced with time series of returns (or asset values),and need to summarise the characteristics of these series with annualised returns and volatility.Let’s take a 1-year period of total monthly returns from some conventional asset (say anequity; Table A2.1) and ask the simplest questions: what are the observed annualised returnand SD?1Table A2.1
1
I use volatility and SD interchangeably.
Period
Returns
123456789101112
2.4142%4.9287%1.3992%2.5029%−1.4179%6.6249%5.5299%4.4657%2.9206%0.1610%−3.5356%−3.6404%
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ReturnThe compound return is the product (less 1) of (1 + period return) for the 12 periods. Thereader might like to ascertain himself that this is 24.02%. We do not need to do any explicitannualising because the 12 periods are one year’s worth. There are two other obvious arithmeticor hybrid methods, which I mention to dismiss:Arithmetic average = Average of monthly returns × 12 = 22.35%Hybrid = {[(Average of monthly returns) + 1] ˆ12} − 1 = 24.79%2I dismiss these measures because they fail our basic deﬁnition; the asset value after one yearof the asset above is $124.02 for a starting value of $100.Note that in this example the arithmetic average return is lower than the compound; this iscommonly (but not universally) the case for positive return series when annualising monthlydata over only one year. However, when we annualise longer series of data, the effect is reversed,and compound returns are typically lower than arithmetic. This effect is emphasised by highvolatility series, a phenomenon often referred to as ‘volatility drag’.3VolatilityThis is a much less straightforward problem. Firstly, we should note that the answers inthe returns section above were all given without mentioning logs.4 In volatility calculations,however, we will ﬁnd that logs are essential. Let’s go back to ﬁrst principles.The series in Table A2.1 was generated by a lognormal random generator model. For eachmonth, the model selected a random point on a normal distribution5 with mean R and volatility V . In this model, R = 0.007943 and V = 0.040346. We will come back to why thesevalues were chosen later. The randomly generated values for the 12 months were as givenin Table A2.2. Note that while I have labelled these values ‘log values’ there is nothingTable A2.2Period
Log values
123456789101112Sum of logs
0.0238550.0481100.0138950.024721−0.0142810.0641470.0538240.0436880.0287880.001609−0.035997−0.0370830.215276
√I use ‘ˆ’ to mean ‘raise to the power of’. This means x and x ˆ 0.5 are identical.McCutcheon and Scott (1986) show the mathematics of volatility drag on p. 293.4Throughout this appendix ‘log x’ or ‘ln x’ means log base e of x, and ‘exponential of x’ or ‘exp(x)’ means ex . e = 2.7182818 . . . .5In Microsoft Excel this can be calculated by using =NORMINV(RAND(),R,V), where R = mean return and V = volatility.23
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particularly ‘log’ about them at this stage; they are normally distributed, for example. However,to create a lognormal distribution of returns, we take these numbers and treat them as if theywere logs. This means that to ‘apply’ them to asset values over time we take the exponentialof each of these values, and multiply the end-month asset values by these exponential values.Table A2.3 shows this mathematical process clearly in each of its stages. We can recognisecolumn (e) as our series of returns.6 We also note that returns = exp(logs) − 1. At this juncturewe can also succinctly show that compounding returns using geometric linking exactly matchesthe lognormal model. The annualised return using compounding is 24.02%. The sum of thelogs in column (b) is 0.21527. Exp(0.21527) − 1 = 24.02%. For longer periods than a year (sayfrom Table A2.4 later) the reader might like to take the average monthly log return and multiplyby 12. The exp()−1 result will be identical to the annualised compound return produced bygeometric linking [={(asset value at end/asset value at start) f /n }−1, where f = data frequencyper year and n = no of periods].Table A2.3(a)Period
123456789101112Sum of logs
(b)Random log value
(c)Exponential value[=exp(b)]
(d)Asset value ($)[=c(t)d(t−1)]
0.0238550.0481100.0138950.024721−0.0142810.0641470.0538240.0436880.0287880.001609−0.035997−0.0370830.215276
1.0241421.0492871.0139921.0250290.9858211.0662491.0552991.0446571.0292061.0016100.9646440.963596
100102.41107.46108.97111.69110.11117.40123.90129.43133.21133.42128.71124.02
(e)Returns[={d(t)/d(t−1)}−1] or[c−1]%2.41%4.93%1.40%2.50%−1.42%6.62%5.53%4.47%2.92%0.16%−3.54%−3.64%
Turning again to volatility, we know that the SD of the population from which these returns were drawn was 0.040346. However, we also know that they were transformed by theexponential function to reach percentage returns. So to get to the volatility of this return series,we could go through the same transformation process. Firstly, we take the SD of column (b).7The reader might like to ascertain for himself that it is 0.033646. But this is the volatility of the1-month log values, whereas we want the annualised volatility. The Central Limit Theorem6We could have done it a different way: we could have taken the log of the initial asset value of $100 [ln($100) = ln$4.60517]and added the numbers in column (b) in sequential order to give us a ln(asset value) each month. Log asset values are not particularlyintuitive, but once exponentiated they come out ﬁne. Using this method, asset value at the end of the year is: exp(4.60517+0.215276)= exp(4.820446) = 124.02.n
(li −l )2SD = i=1n−1where li = ith period log value, l = average value of l in the sample, and n = number of observations in thesample.√There is some debate about the denominator being n or n − 1. Using n − 1 gives an unbiased estimate of variance (but notSD = variance). However, I will use it as most investment management users do. See however the AIMR section.7
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tells us that the SD of the mean√of a sample size of n independent observations from a normalpopulation with SD= V is V / n. However, we are not averaging here, we are summing√– wewant the sum of√the 12 observations to make up the year’s return. We therefore get nV / n, orrearranging, V n. So the annualised SD of this series is 0.033646 × (12 ˆ 0.5) = 0.116553.Note that we can only sum returns if they are log returns; if we try to √sum percentage returnsto annualise, we are being inconsistent with geometric linking. So nV/ n is only valid for logreturns, not percentage returns. The users of the conventional percentage SD calculation haveprobably not realised this.But this value is in logs. To convert to percentages as deﬁned, we have to transform using theexponential function: SD = exp(0.116553) −1 = 12.36%. So you would think that, followingthis logic, the volatility of this series was 12.36%. I should just say that the population SDof 0.040346 is derived√ from wanting to generate a 15% annualised volatility series. If V =0.040346, then exp( 12V)−1 = 15%.
Industry PracticeFaced with the return series in Table A2.1, I suspect when asked to calculate the sample√SD,most industry participants would take a straight SD of the percentages, and multiply by 12.This comes to 11.79%.8 There are a number of points to make here.The ﬁrst is that if returns are lognormal, then an SD cannot be calculated as if the returns areadditive (i.e. as the SD of the % period returns). Just imagine a very high volatility investmentof SD = 50%. What happens if the returns are negative 3 SDs? Does the value of the investmentgo to less than zero? Figure A2.1 shows the range of percentage changes for a series with amean annual return of 10% and SD of 15% for (a) lognormally calculated percentage changesfor −4 to +4 SDs (the SD applied to the log of the % returns and converted back to %) and (b)additive percentages for the same range (the SD applied directly to the % returns). It showsquite clearly that additive percentages underestimate the risk of high returns, and overestimatethe risk of low returns. In short, additive percentages show no convexity.9We should note that the 1 SD positive value for the log series is 26.5% = exp[ln(1.1) +ln(1.15)] − 1, whereas for the additive series it is 25% (10% + 15%). This is my guess as toindustry practice: what evidence do we have for this?
MSCIMSCI, the index provider, has published very comprehensive index methodology10 and, insummary, it uses additive returns to calculate the annualised return of the SD. Their formulais: n(TR j − TR)2√ j=1× 12(A2.1)Annualised volatility =n−1In Microsoft Excel terms =STDEV(Series)∗ (12 ˆ 0.5).Convexity in returns (and asset values) is characteristic of lognormal processes, and for the mathematically-minded the seconddifferential of the curve in Figure A2.1 is positive for the convex process, zero for the additive process.10www.msci.com/risk/index.html89
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271
100.0%80.0%
Logs
Returns %
60.0%
Additive percentages
40.0%20.0%Mean0.0%
−20.0%−40.0%−60.0%4 SDs
3 SDs
2 SDs
1 SD
Mean
1 SD
2 SDs
3 SDs
4 SDs
Normalised range of returnsFigure A2.1 Lognormal return distribution for a series with mean 10% and SD 15%
where TR j symbolises the monthly total return, TR is the average total return and n is thenumber of data points used in the calculation.MSCI’s TR value is a percentage monthly return; they spell out in great detail their methodology for annualising TR, and they do so correctly using geometric linking and annualisingby raising the product of (1 +TR) to the power of 12/m, where m is the number of months tobe annualised. All of which is ﬁne, and ﬁts exactly with log methodology as described above.It is all the more worrying then that they abandon their log methodology when it comes tothe calculation formula above for the SD of the series, and calculate the SD of the % TR j ,rather than via ln(1 + TR j ). Their formula therefore treats returns as if they were additive, andincorporates no convexity.Interestingly, MSCI go on in their methodology document to show a Value at Risk (VAR)calculation as follows:VAR = (1 − e[−1.645σ ] ) × 100
(A2.2)
where 1.645 implies a 95% conﬁdence level, σ is the monthly standard deviation (volatility) of the monthly total returns and e is a constant term having an approximate value of2.718.Assume Poland has a VAR of $15. This implies that for every $100 invested in Poland, thereis a 5% chance of losing $15 or more in a given month. This is particularly odd, because theintroduction of the e-power function (or exp()) would be absolutely correct if the SD was anSD of log returns, but is absolutely mathematically wrong for an SD of total return %.Rewriting the formula (A2.2) using the exp function:VAR = [1 − exp(−1.645SD)] × 100
(A2.3)
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We could expand this formula to:VAR = [1 − exp(−1.645{SD(exp(ln(x)) − 1)})] × 100
(A2.4)
where x is normally distributed, because the TR are already exp() of the log returns, so wehave a ‘double exponentiation’. We also have the problem of pro-rata-ing up from monthly toannual. This will be addressed later.If MSCI were to stick with the logic of their additive SD calculation, then the VAR wouldbe:VAR = [1.645SD] × 100
(A2.5)
which would be 19.39 from our series above. The irony is that with the same data that MSCIhave, we can calculate the correct result.We calculate an SD of the log returns, which are ln(1+T R j ). I will call this SD(ln). VAR isthen:VAR = [1 − exp(−1.645{SD(ln)})] × 100
(A2.6)
Using the series in Table A2.3, the correct calculation of the VAR is 17.45, whereas MSCI’scalculation would be 17.63. Do not be fooled by this relatively small error; it is not stable andit could be much larger depending on the data set and the annualisation process. More of thislater.AIMRWhat about the Association for Investment Management and Research (AIMR)? AIMR isthe US performance reporting standards setter, and is the leading partner in the new globalperformance standard – GIPS. AIMR has published its preferred methodology for the calculation of SD in Appendix B of the 1993 Performance Presentation Standards. It describes SDformulae in two places (external and internal risk). The basic formulae are the same, and are asfollows.‘Standard deviation of portfolio performance over time (S p ) is a measure of volatility. . . calculated as follows: n (CASSETi − M E AN (CASSET ))2 i=1(A2.7)Sp =nwhere CASSETi is the asset weighted composite return in the ith time period, and n is the numberof periods in the study.’AIMR do not mention the lognormality issue at all, although like MSCI they imply that returnsare lognormal by their insistence on geometric linking for returns over time. Like MSCI, theytoo see the SD of percentage returns as the appropriate measure. By way of an aside, AIMRtake some time to justify the use of n rather than n − 1 as the denominator. It is a statisticaldebate at a tangent to the present issue – so I will ignore it.
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273
Textbooks and academic literatureThere is a huge volume of textbook and academic literature from which to draw samples ofcommon usage. I have picked two textbooks at random: Haugen (1993) and Brigham andHouston (1998). The latter is particularly relevant because it is a recommended text for theCFA – the US (and now worldwide) industry standard investment management qualiﬁcation.Both of these books unequivocally calculate SDs of investment returns using the (incorrect)methodology shown earlier, or generically: n (ri − r )2 i=1SD =(A2.8)n−1where ri = ith period return in %, r = average value of r , and n = number of observations inthe sample.However, a number of commentators remark (in passing) that log returns are the ‘correct’measure, but then go on to say that ‘percentages are good enough’. An example of this comesfrom Gartland and Letica (1997) in the bond market literature: ‘The intuitive approach tocalculating a percentage volatility is to ﬁnd the standard deviation of daily returns, assuminga normal distribution. This approach is equivalent to the lognormal assumption as long as thedistribution can be characterized as being equally normal and lognormal and the changes inprices are taken on a small interval, such as daily.’These comments seem to abound – that percentages are ﬁne as long as the volatilities and/ortime period are small. Of course this is wrong. A daily SD of 1% (and most commentators wouldsay that 1% is ‘reasonably small’) annualises to 17.48% if correctly calculated via logs, and to16.19% if incorrectly calculated direct. A 2 SD movement upwards is 38.01% (if calculatedcorrectly via logs), and only 32.37% if calculated incorrectly with direct percentages. Theseare not second-order errors.What about the founders of modern portfolio theory? Markowitz (1952), a model of clarityof thought struggling against computational odds, does discuss logs in relation to compoundreturns and their geometric mean. However, he is principally concerned with his emergingtheory of efﬁciency, and the mathematics of combining stocks to form efﬁcient portfolios.Because period returns are easy to observe and intuitively appealing, he chooses to stick toreturns as his core ‘random variable’. While recognising the inadequacy of adding returns overtime rather than compounding, he seeks to provide a quadratic formula for approximating thecompound return from return data rather than effecting a fundamental transformation of hisreturn data into logs. It is quite possible that this approximation (adopted, I suspect, largelybecause of the computational difﬁculties of wholesale transformations into and out of logs)has carried on down the years just because no one has thought hard about an alternative. So inthe end, Markowitz calculates the SD of returns directly, not via logs.Investment banks and option tradersOne ﬁnal industry sector has not yet been considered. This is one where the exact calculation ofvolatilities for the pricing of options and other exotic derivatives is critical, since large volumesof trades are transacted on the basis of these calculations every day. It consists of the optionsdesks of the major investment banks.
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From a brief survey of custom, practice and use in this sector,11 it is clear that logs areuniversally used for SD calculations, but that, bizarrely, the logs are not converted back topercentages, but left as logs with a ‘%’ sign on the end. This is extremely confusing, but aslong as this is consistently and universally applied, no errors will arise. In particular, annualisingwill be intuitive, and the errors described in the section below will not arise. However, if thisconvention is universally applied to returns and SDs, then a return of 10% under this conventionwould turn $100 into $110.52 at the end of one year. This is because the 10% here is a log, andthe asset value will be multiplied by exp(0.1) to get the next value. Exp(0.1) is 1.1052. Thisis equivalent to continuously compounding 10% over inﬁnitesimal time. While mathematicalmodels ﬁnd this easy to do, investors ﬁnd continuously compounding hard to handle and evenharder to re-create with data!This treatment is conﬁrmed in the textbooks written for the investment banking community.Fitzgerald (1987), writing about option pricing, describes a fully consistent log calculation ofSD, but right at the end he says:√‘Annual volatility = 0.003625 × 365= 0.069 or 6.9%’His use of % is confusing, as it implies that it refers to returns, when in fact it refers to ln(1 + r )where r = returns. It is also interesting that he uses 365 as the annualising factor.So the investment banking conventions violate our investment management convention of10% return meaning you have $110 at the year-end from your $100. I see nothing wrong in theinvestment banking community continuing to use logs as the basis of everything they do, butI do not believe their usage would translate well to the investment management community.Finally, as an aside, I am not sure that many participants in the investment banking community really understand that they are dealing with logs, particularly since everything they do isexpressed as x%. I see logs as a particular language (call it Latin), and percentages as anotherlanguage (call it English). There is an exact translation process to convert one to the other. Butfor option traders to describe an implied volatility as ‘12%’, when this is in fact a log value of0.12 (which would translate to a return volatility of exp(0.12)−1 = 12.75%) in my view runsthe risk of using Latin words that look like English words but have a different meaning. I willalways use the convention of x% when referring to percentages, and y.yyy when referring tologs.It is possible that the reader is already convinced by the theoretical arguments presentedabove. However, there may be a strong element of ‘so what?’ from readers who see theerrors I have demonstrated to be rather esoteric, and the SD of % returns to be a reasonableapproximation of the more precise log-based SD. I hope that the following example willconvince waverers. Let us suppose we are dealing with a very volatile stock – say an Internetstock – whose volatility we wish to measure. The monthly returns for 2 years are given inTable A2.4. This stock has observed annualised volatilities as follows:1. Correct (ln) calculation2. SD on return percentages3. SD of log
SD= 109.03%SD = 78.60%SD = 0.7373
11See, for example, JP Morgan (1996). There is a very good analysis of the relative merits of percentage and log returns in section4.1. They choose log returns for all their calculations, but they do note that this is not as convenient as percentages for portfoliocross-sectional analysis. They do not ‘re-translate’ log returns back into percentages, but nevertheless they use ‘x.x x%’ notation.
Appendices
275
I suspect that the differences between values 1 and 2 is large enough to worry the most hardened‘approximator’, and it of course brings into question the whole concept of percentage changesthat can in theory be larger than 100%. Raw percentages cannot cope with this level of volatility;by contrast, log returns can cope with any level of volatility. The ﬁrst thing that any modellerwill notice when faced with annualised volatility of 109% (or indeed 78%) is that there seems tobe a high chance each year that the value of the stock will go negative. Since this is impossiblein a limited liability company’s equity, he is forced to override at zero value, or put in someother ‘fudge factor’ to prevent this possibility.Table A2.4Period
Returns
Period
Returns
123456789101112
29.12%−23.96%27.42%−10.09%62.09%−19.81%−25.54%−16.53%−22.20%−6.82%2.04%3.60%
131415161718192021222324
−18.05%19.67%14.34%−15.19%6.21%2.71%4.10%22.76%39.29%9.34%−20.34%18.67%
The third value quoted is the un-exponentiated SD of the logs. This is ‘Latin’, and whatoption traders would quote. The uninitiated might note that the second and third values arequite close, and so perhaps all this is a fuss about nothing (mirroring the close VAR values inthe MSCI methodology). But let’s subtly change the speciﬁcation of the series above, and weshall see how wrong this complacency is.Let us suppose instead that the series in Table A2.4 is annual data rather than monthly. Thisis clearly a less volatile series, and obligingly we have 24 years’ worth of data. What are thethree volatilities now?1. Correct (ln) calculation2. SD on return percentages3. SD of log
SD=23.72%SD =22.69%SD =0.2128
Here the relative relationships have changed, with the third value now the most distant fromeither of the other two. This changed relationship is caused by the annualising process; it is asymptom of convexity, and is considered in the next section in more detail.AnnualisingWhen we annualise independent returns for periods less than a year, the industry universallyuses the rules of the Central√Limit Theorem, which,√ as mentioned previously, boils down tomultiplying monthly SD by 12 and daily SD by 262 to get annual SD.1212In some markets the convention is to use 250, approximately the number of working days (weekdays excluding Bank Holidays)in a year, rather than 262, approximately the number of weekdays in a year.
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If we take our lognormal model, which is additive, then we can use the Central LimitTheorem and achieve accurate results. However, in the process of converting the log returnsinto percentages that we can use and which we understand, we have to exp() those log returns.In fact, ‘lognormal returns’ is a slight misnomer – so called because if you transform the returnseries using logs you get a normal distribution. But if you start the other way, with the normaldistribution, returns are the exponential of this normal series. You could call this ‘exponentiatednormal returns’. This ﬁts the skew of returns, which is upwards.Returning to the annualising question, we√have to exp() the log series