Which is the Correct Discount Rate? Arithmetic Versus Geometric Mean

2020 ◽  
Vol 53 (3) ◽  
pp. 355-381
Author(s):  
Martin Kipp ◽  
Christian Koziol
Keyword(s):  
Discount Rate ◽  
Historical Data ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Stock Index ◽  
Rigorous Derivation ◽  
Asset Return ◽  
Expected Performance ◽  
German Stock ◽  
The Mean

The paper revisits the two major concepts for average historical returns, i. e., the arithmetic mean and the geometric mean, in order to clarify which approach must be used for which application. Conducting a rigorous derivation with a geometric Brownian motion, we can explain that the appropriate discount rate refers to the mean discrete return and, therefore, to the arithmetic mean rather than the often wrongly applied geometric mean. Likewise, the prominent CAPM relationship between the expected asset return and the expected market return is only valid for the arithmetic mean rather than the geometric mean. Using historical data for the German stock index, we illustrate that an inconsistent application can cause severe deviations from the meaningful ex-ante expected performance of an asset, the true discount rate, the true CAPM risk-adjusted return, and the intended performance scenarios of packaged retail and insurance-based investment products (PRIIPs) within the key information documents (KIDs).

The effects of spatial variability of nitrous oxide emissions from grazed pastures on the sampling distribution of chamber measurements

2015 ◽  
Vol 154 (2) ◽  
pp. 223-241 ◽  
Author(s):  
D. L. GILTRAP ◽  
A. J. R. GODFREY
Keyword(s):  
Nitrous Oxide ◽  
Spatial Variability ◽  
Agricultural Soils ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Nitrous Oxide Emissions ◽  
Sample Mean ◽  
Grazed Pastures ◽  
Urine Patch ◽  
The Mean

SUMMARYChamber sampling is a common method for measuring nitrous oxide (N2O) emissions from agricultural soils. However, for grazed pastures, the patchy nature of urine deposition results in very high levels of spatial variability in N2O emissions. In the present study, the behaviour of the sample mean was examined by simulating a large number (9999) of random N2O chamber samples under different assumptions regarding the underlying N2O distribution. Using sample sizes of up to 100 chambers, the Central Limit Theorem did not apply. The distribution of the sample mean was always right-skewed with a standard deviation varying between 12·5 and 135% of the true mean. However, the arithmetic mean was an unbiased estimator and the mean of the sample mean distribution was close to the true mean of the simulated N2O distribution. The properties of the sample mean distribution (variance, skewness) were affected significantly by the assumed distribution of the emission factor, but not by distribution of the urine patch concentration. The geometric mean was also investigated as a potential alternative estimator. However, although its distribution had lower variance, it was also biased. Two methods for bias correcting the mean were investigated. These methods reduced the bias, but at the cost of increasing the variance. Neither of the bias-corrected estimators were consistently better than the arithmetic mean in terms of skewness and variance. To improve the estimation of N2O emissions from a grazed pasture using chambers, techniques need to be developed to identify urine patch and non-urine patch areas before sampling.

ON DISTINGUISHING BETWEEN RANDOM WALK AND CHANGE IN THE MEAN ALTERNATIVES

2009 ◽  
Vol 25 (2) ◽  
pp. 411-441 ◽  
Author(s):  
Alexander Aue ◽  
Lajos Horváth ◽  
Marie Hušková ◽  
Shiqing Ling
Keyword(s):  
Random Walk ◽  
Structural Breaks ◽  
Stock Index ◽  
Partial Sums ◽  
Test Statistics ◽  
Test Procedures ◽  
Change In The Mean ◽  
German Stock ◽  
The Mean ◽  
Almost All

We study test procedures that detect structural breaks in underlying data sequences. In particular, we wish to discriminate between different reasons for these changes, such as (1) shifting means, (2) random walk behavior, and (3) constant means but innovations switching from stationary to difference stationary behavior. Almost all procedures presently available in the literature are simultaneously sensitive to all three types of alternatives.The test statistics under investigation are based on functionals of the partial sums of observations. These cumulative sum–type (CUSUM-type) statistics have limit distributions if the mean remains constant and the errors satisfy the central limit theorem but tend to infinity in the case when any of the alternatives (1), (2), or (3) holds. On removing the effect of the shifting mean, however, divergence of the test statistics will only occur under the random walk behavior, which in turn enables statisticians not only to detect structural breaks but also to specify their causes.The results are underlined by a simulation study and an application to returns of the German stock index DAX.

scholarly journals The mean conformation tensor in viscoelastic turbulence

2019 ◽  
Vol 865 ◽  
pp. 363-380 ◽  
Author(s):  
Ismail Hameduddin ◽  
Tamer A. Zaki
Keyword(s):  
Geodesic Distance ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Viscoelastic Flows ◽  
Good Representative ◽  
Recent Developments ◽  
Alternative Means ◽  
Principal Stretches ◽  
The Mean ◽  
Conformation Tensor

This work demonstrates that the popular arithmetic mean conformation tensor frequently used in the analysis of turbulent viscoelastic flows is not a good representative of the ensemble. Alternative means based on recent developments in the literature are proposed, namely, the geometric and log-Euclidean means. These means are mathematically consistent with the Riemannian structure of the manifold of positive-definite tensors, on which the conformation tensor lives, and have useful properties that make them attractive alternatives to the arithmetic mean. Using a turbulent FENE-P channel flow dataset, it is shown that these two alternatives are physically representative of the ensemble. By definition, these means minimize the geodesic distance to realizations and exactly preserve the scalar geometric mean of the volume and of the principal stretches. The proposed geometric and log-Euclidean means have clear physical interpretations and are attractive quantities for turbulence modelling.

scholarly journals Theory of Means

2018 ◽  
Author(s):  
Daniel S Goldman
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
General Nature ◽  
The Mean

There are many notions of means, the arithmetic mean and geometric mean are quite familiar to many. There are also a few generalizations of these concepts, including the mean developed by V. M. Tikhomirov. However, these means fail to capture the general nature of means. A new definition is presented and it is shown that the Tikhomirov mean is a form of this mean.

scholarly journals Impact of engineering properties of grass seeds in developing post-harvest operations and machineries

2021 ◽  
Vol 22 (3) ◽  
pp. 395-399
Author(s):  
Sanjay Kumar Singh ◽  
Sheshrao Kautkar ◽  
Amit Patil
Keyword(s):  
Geometric Mean ◽  
Seed Mass ◽  
Seed Storage ◽  
Arithmetic Mean ◽  
Engineering Properties ◽  
Mean Values ◽  
Post Harvest ◽  
Mean Diameter ◽  
Length Width ◽  
The Mean

Engineering properties of grass seeds are most important for the development of post-harvest mechanization and operations. Therefore engineering properties of fluffy as well as true seeds were determined in view of its important in development of post-harvest mechanization. The mean values of length, width, thickness, arithmetic mean diameter, geometric mean diameter, sphericity, surface area, volume, thousands seed mass and bulk density of fluffy Deenanath grass seed were observed in the range of 5.23-7.17 mm, 2.10-3.44 mm, 1.17-2.49 mm, 3.07- 4.13 mm, 2.53- 3.69 mm,  41.01-60.13 %, 19.12-43.70 mm2, 3.70-18.24 mm3, 0.789-0.849 g and 7.41-7.89 kg/m3 respectively. However, for true seeds of Deenanath grass, the range of these values varied from 2.23-2.65 mm, 0.69-0.95 mm, 0.47-0.69 mm, 1.16-1.40 mm, 0.93-1.17 mm,  38.69-47.33 %, 2.67-4.31 mm2, 3.60-9.64 mm3, 0.468-0.488 g and 602.97-624.29 kg/m3 respectively moisture level of 9 % db. Determined properties of fluffy as well as true seeds of Deenanath would be utilized to develop threshing, cleaning, grading, seed storage and packaging operations and machineries.

DMSA-scintigraphy in paediatrics: Is the evaluation of the geometric mean necessary for the calculation of the differential renal function?

2001 ◽  
Vol 40 (04) ◽  
pp. 107-110 ◽  
Author(s):  
B. Roßmüller ◽  
S. Alalp ◽  
S. Fischer ◽  
S. Dresel ◽  
K. Hahn ◽  
...  
Keyword(s):  
Renal Function ◽  
Geometric Mean ◽  
Region Of Interest ◽  
Posterior View ◽  
Renal Abnormality ◽  
Anterior View ◽  
Differential Renal Function ◽  
The Mean ◽  
The Right ◽  
To Receive

SummaryFor assessment of differential renal function (PF) by means of static renal scintigraphy with Tc-99m-dimer-captosuccinic acid (DMSA) the calculation of the geometric mean of counts from the anterior and posterior view is recommended. Aim of this retrospective study was to find out, if the anterior view is necessary to receive an accurate differential renal function by calculating the geometric mean compared to calculating PF using the counts of the posterior view only. Methods: 164 DMSA-scans of 151 children (86 f, 65 m) aged 16 d to 16 a (4.7 ± 3.9 a) were reviewed. The scans were performed using a dual head gamma camera (Picker Prism 2000 XP, low energy ultra high resolution collimator, matrix 256 x 256,300 kcts/view, Zoom: 1.6-2.0). Background corrected values from both kidneys anterior and posterior were obtained. Using region of interest technique PF was calculated using the counts of the dorsal view and compared with the calculated geometric mean [SQR(Ctsdors x Ctsventr]. Results: The differential function of the right kidney was significantly less when compared to the calculation of the geometric mean (p<0.01). The mean difference between the PFgeom and the PFdors was 1.5 ± 1.4%. A difference > 5% (5.0-9.5%) was obtained in only 6/164 scans (3.7%). Three of 6 patients presented with an underestimated PFdors due to dystopic kidneys on the left side in 2 patients and on the right side in one patient. The other 3 patients with a difference >5% did not show any renal abnormality. Conclusion: The calculation of the PF from the posterior view only will give an underestimated value of the right kidney compared to the calculation of the geometric mean. This effect is not relevant for the calculation of the differntial renal function in orthotopic kidneys, so that in these cases the anterior view is not necesssary. However, geometric mean calculation to obtain reliable values for differential renal function should be applied in cases with an obvious anatomical abnormality.

On Statistics of Log-Ratio of Arithmetic Mean to Geometric Mean for Nakagami-m Fading Power

2012 ◽  
Vol E95-B (2) ◽  
pp. 647-650
Author(s):  
Ning WANG ◽  
Julian CHENG ◽  
Chintha TELLAMBURA
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Log Ratio

scholarly journals Operating Rooms efficiency: leveraging data to improve management of elective surgery

2020 ◽  
Vol 30 (Supplement_5) ◽  
Author(s):  
C Quercioli ◽  
G A Carta ◽  
G Cevenini ◽  
G Messina ◽  
N Nante ◽  
...  
Keyword(s):  
Historical Data ◽  
Elective Surgery ◽  
District General Hospital ◽  
Operating Rooms ◽  
Accurate Estimation ◽  
Surgical Time ◽  
Accurate Analysis ◽  
Management Efficiency ◽  
Scheduling Model ◽  
The Mean

Abstract Background Careful scheduling of elective surgery Operating Rooms (ORs) is crucial for their efficient use, to avoid low/over utilization and staff overtime. Accurate estimation of procedures duration is essential to improve ORs scheduling. Therefore analysis of historical data about surgical times is fundamental to ORs management. We analyzed the effect, in a real setting, of an ORs scheduling model based on estimated optimum surgical time in improving ORs efficiency and decreasing the risk of overtime. Methods We studied all the 2014-2019 elective surgery sessions (3,758 sessions, 12,449 interventions) of a district general hospital in Siena's Province, Italy. The hospital had3 ORs open 5 days/week 08:00-14:00. Surgery specialties were general surgery, orthopedics, gynecology and urology. Based on a pilot study conducted in 2016, which estimated a 5 times greater risk of having an OR overtime for sessions with a surgical time (incision-suture)&gt;200 minutes, from 2017 all the ORs were scheduled using a maximum surgical time of 200 minutes calculated summing the mean surgical times for intervention and surgeon (obtained from 2014-2016 data). We carried out multivariate logistic regression to calculate the probability of ORs overtime (of 15 and 30 minutes) for the periods 2014-2016 and 2017-2019adjusting for raw ORs utilization. Results The 2017-2019 risk of an OR overtime of 15 minutes decreased by 25% compared to the 2014-2016 period (OR = 0.75, 95%CI=0.618-0.902, p = 0.003); the risk of a OR overtime of 30 minutes decreased by 33% (OR = 0.67, 95%CI= 0.543-0.831, p &lt; 0.001). Mean raw OR utilization increase from 62% to 66% (p &lt; 0.001). Mean number of interventions per surgery sessions increased from 3.1 to 3.5 (p &lt; 0.001). Conclusions This study has shown that an analysis of historical data and an estimate of the optimal surgical time per surgical session could be helpful to avoid both a low and excessive use of the ORs and therefore to increase the efficiency of the ORs. Key messages An accurate analysis of surgical procedures duration is crucial to optimize operating room utilization. A data-based approach can improve OR management efficiency without extra resources.

scholarly journals Faecal contamination of drinking water during collection and household storage: the need to extend protection to the point of use

2003 ◽  
Vol 1 (3) ◽  
pp. 109-115 ◽  
Author(s):  
Thomas F. Clasen ◽  
Andrew Bastable
Keyword(s):  
Drinking Water ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Faecal Contamination ◽  
Household Level ◽  
Safe Water ◽  
Female Head ◽  
Point Of Use ◽  
Thermotolerant Coliforms ◽  
And Storage

Paired water samples were collected and analysed for thermotolerant coliforms (TTC) from 20 sources (17 developed or rehabilitated by Oxfam and 3 others) and from the stored household water supplies of 100 households (5 from each source) in 13 towns and villages in the Kailahun District of Sierra Leone. In addition, the female head of the 85 households drawing water from Oxfam improved sources was interviewed and information recorded on demographics, hygiene instruction and practices, sanitation facilities and water collection and storage practices. At the non-improved sources, the arithmetic mean TTC load was 407/100 ml at the point of distribution, rising to a mean count of 882/100 ml at the household level. Water from the improved sources met WHO guidelines, with no faecal contamination. At the household level, however, even this safe water was subject to frequent and extensive faecal contamination; 92.9% of stored household samples contained some level of TTC, 76.5% contained more than the 10 TTC per 100 ml threshold set by the Sphere Project for emergency conditions. The arithmetic mean TTC count for all samples from the sampled households was 244 TTC per 100 ml (geometric mean was 77). These results are consistent with other studies that demonstrate substantial levels of faecal contamination of even safe water during collection, storage and access in the home. They point to the need to extend drinking water quality beyond the point of distribution to the point of consumption. The options for such extended protection, including improved collection and storage methods and household-based water treatment, are discussed.

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