Reconciling Portfolio Value Growth with Cumulative Returns
Article Quant Q&A · Author: Daniel Ryback
Summary
The document asks why an active strategy’s portfolio value can appear to exceed a passive strategy’s while its calculated returns and cumulative returns look weaker. It provides monthly portfolio value series and shows discrete returns computed as the percentage change between consecutive observations, followed by compounding those returns. The central issue is that portfolio value levels and return series only make a valid comparison when they describe the same investment, dates, starting capital, and treatment of cash flows.
The supplied active series contains large drops and apparent additions, while the passive series begins at a different value and rises more smoothly. Such flows or inconsistent initial values can make raw portfolio levels incomparable and can distort returns if treated as investment performance. The document poses the discrepancy but includes no answer or validated diagnosis. It also does not provide enough information about contributions, withdrawals, scaling, or strategy execution to determine the cause; returns should be calculated from comparable, flow-adjusted portfolio values.
Key ideas
- Portfolio value levels and percentage returns can tell different stories when the series use inconsistent starting capital or scaling.
- Discrete returns are calculated from changes between consecutive portfolio values and then compounded for cumulative performance.
- Cash contributions and withdrawals can distort apparent returns if they are treated as investment gains or losses.
- The active series contains abrupt value changes that may reflect flows or other accounting effects.
- The document does not establish a definitive explanation without details about cash flows and portfolio construction.
Tags
Full text
# trading strategy outperforms passive strategy in absolute terms, but in returns vice versa. how could this be?
# trading strategy outperforms passive strategy in absolute terms, but in returns vice versa. how could this be?
Background: I want to compare two trading strategies in term of profitability. The red one is an active trading strategy, which involves many entry and exit to a specific market. The blue one is a passive strategy.
Both portfolios start at some unit 3. The units on the x-axis are months. As you can see, the active one clearly outperforms the passive strategy in absolute terms.
But when I compare both portfolios in terms of returns (edit: discrete returns ) and cumulative returns (something like cumprod(1+Return)):
It seems that the opposite is the case: the passive strategy works better.
How could this be?
Here is part of the code in using R:
```
library(quantmod)
library(tseries)
# calculating returns
portfolio.active <- c(3e+07, 6680419, 6980132, 7268161, 7519643, 7739132, 7941601,
8119644, 8286440, 8435480, 8595353, 8772177, 9000774, 27468724.0476,
9162873.0476, 9503273.0476, 9791440.0476, 10097455.0476, 10415108.0476,
10682368.0476, 10932171.0476, 11190461.0476, 11462988.0476, 11746256.0476,
12009560.0476, 34328954.4748696, 34768365.0921479, 35236579.0753889,
35677036.3138312, 36108728.4532286, 36549254.940358, 36940331.9682198,
37325742.7650882, 37687802.4699096, 38047092.8534561, 38444050.8555604,
38856683.6680768, 18480549.6680768, 18699486.6680768, 18927669.6680768,
19157485.6680768, 19415315.6680768, 19665072.6680768, 19874970.6680768,
20094206.6680768, 20321975.6680768, 20534633.6680768, 41857739.8826469,
42284688.8294499, 42704716.7384891, 43167351.1698227, 43670970.266804,
44206667.5020768, 25078572.5020768, 25262687.5020768, 25460188.5020768,
25670408.5020768, 25900891.5020768, 26185010.5020768, 26468094.5020768,
26791336.5020768, 27124813.5020768, 27445686.5020768, 27797816.5020768,
28169749.5020768, 28530619.5020768, 28851193.5020768, 29171192.5020768,
29524403.5020768, 29857673.5020768, 30123254.5020768, 52942557.9825139,
53849640.4759476, 54747134.4838801, 55628563.3490706, 56548288.9297752,
57481335.6971165, 58443190.0477816, 59434776.1722589, 60455073.163216,
61472733.5614635, 62515720.9408897, 63555565.7658731, 64621180.7518809,
65678814.0768534, 44286709.0768534, 44509648.0768534, 44736037.0768534,
44940967.0768534, 45122736.0768534, 45299951.0768534, 45497061.0768534,
68541626.0566368, 69672562.8865713, 70826805.0117255, 72040304.2709264,
73279397.5043863, 74510491.38246, 75774686.0529157, 77045174.9557362,
78316420.3425059, 79621694.014881, 80943414.135528, 82281678.5825687,
83625612.666084, 84910659.5807195, 70986088.5807195, 71134875.5807195,
71259144.5807195, 71382746.5807195, 86070512.2301142, 87344355.8111198,
72406044.8111198, 72574841.8111198, 72734093.8111198, 72876348.8111198,
73013152.8111198, 73157845.8111198, 73342328.8111198, 73526765.8111198,
73704118.8111198, 73898514.8111198, 74081870.8111198, 74300600.8111198,
74525126.8111198, 74756092.8111198, 74992670.8111198, 75240955.8111198,
75485445.8111198, 75736352.8111198, 75978887.8111198, 76218754.8111198,
76455194.8111198, 94467009.4053774, 96003672.7583715, 97418126.8703449,
98674820.7069723, 99865496.8768364, 101027265.490504, 102209284.496743,
103354028.483106, 104263543.933757, 105014241.45008, 105668830.221786,
106264097.965369, 106848550.504178, 107464710.478752, 108105916.584609,
108725723.839694, 91768810.839694, 109049526.184452, 109660203.531085,
110252368.630153, 94067536.630153, 94240657.630153, 94440021.630153,
94687041.630153, 94938224.630153, 95175768.630153, 95433874.630153,
95743316.630153, 96077069.630153, 96389386.630153, 96724956.630153,
97052499.630153, 97410241.630153, 98093042.630153, 98769541.630153,
99552139.630153, 100431061.630153, 101345311.630153, 102241667.630153,
103005198.630153, 103578174.630153, 129996171.902754, 130572488.264856,
131216645.87363, 131938337.425934, 132712375.672167, 133628091.064305,
94297376.0643046, 95030984.0643046, 95750240.0643046, 96575296.0643046,
97249725.0643046, 97755334.0643046, 141077560.446541, 142591792.928667,
144226845.487583, 145871031.526141, 147674969.949348, 149584899.560693,
151524517.091663, 153666063.599892, 155935199.13905, 158279424.966107,
160706376.148921, 163234823.133664, 165841139.143032, 168577517.938892,
127864050.938892, 128484705.938892, 129103980.938892, 129759917.938892,
130434392.938892, 131118780.938892, 131759054.938892, 132479828.938892,
133227604.938892, 134109332.938892, 134952558.938892, 136083238.938892,
137206719.938892, 138478067.938892, 139996228.938892, 141580951.938892,
143068867.938892, 144160194.938892, 145172631.938892, 146430708.938892,
147709646.938892, 149254901.938892, 150654198.938892, 151985116.938892,
240918508.942675, 241825968.659692, 242366046.656365, 242519545.152581,
242543797.107096, 242648899.419176, 242891548.318595, 243061572.402418,
243191205.241033, 243337119.964177, 243483122.236156, 208208237.236156,
208482926.236156, 208784318.236156, 209112574.236156, 209601255.236156,
210075285.236156, 210553892.236156, 210973504.236156, 211499146.236156,
211986447.236156, 212568324.236156, 213018802.236156, 213301380.236156,
251743071.638948, 251868943.174768, 251978086.383477, 252095676.157123,
252213320.805996, 252339427.466399, 252448774.551634, 252532924.143151,
252583430.72798, 252617108.518744, 252650790.79988, 252684477.571986,
252701323.203824, 252709746.581265, 252726593.897703, 252735018.1175,
252743442.618104, 252768716.962366, 252844547.577454, 252911972.790142,
252979415.982886, 253055309.80768, 253131226.400623, 253215603.47609,
253300008.677248)
portfolio.passive <- c(6680419, 6980132, 7268161, 7519643, 7739132, 7941601, 8119644,
8286440, 8435480, 8595353, 8772177, 9000774, 9223828, 9510477,
9850877, 10139044, 10445059, 10762712, 11029972, 11279775, 11538065,
11810592, 12093860, 12357164, 12611636, 12826134, 13020162, 13205887,
13411967, 13586583, 13738107, 13890114, 14033013, 14176096, 14379071,
14632186, 14858552, 15077489, 15305672, 15535488, 15793318, 16043075,
16252973, 16472209, 16699978, 16912636, 17117129, 17309938, 17499182,
17669704, 17838570, 18030597, 18248752, 18432867, 18630368, 18840588,
19071071, 19355190, 19638274, 19961516, 20294993, 20615866, 20967996,
21339929, 21700799, 22021373, 22341372, 22694583, 23027853, 23293434,
23552161, 23807208, 24037208, 24239999, 24430692, 24636197, 24838827,
25024759, 25167083, 25292732, 25396830, 25531909, 25750800, 25982456,
26197851, 26420790, 26647179, 26852109, 27033878, 27211093, 27408203,
27587764, 27761184, 27940101, 28100963, 28275165, 28449068, 28580697,
28738500, 28872981, 29016478, 29153880, 29281553, 29397565, 29511990,
29657419, 29806206, 29930475, 30054077, 30189731, 30345050, 30481739,
30650536, 30809788, 30952043, 31088847, 31233540, 31418023, 31602460,
31779813, 31974209, 32157565, 32376295, 32600821, 32831787, 33068365,
33316650, 33561140, 33812047, 34054582, 34294449, 34530889, 34784441,
35030472, 35288003, 35537392, 35777282, 35993896, 36182795, 36309410,
36428366, 36551784, 36678261, 36799218, 36930272, 37056381, 37192817,
37315361, 37443448, 37559161, 37678255, 37796705, 37943123, 38116244,
38315608, 38562628, 38813811, 39051355, 39309461, 39618903, 39952656,
40264973, 40600543, 40928086, 41285828, 41968629, 42645128, 43427726,
44306648, 45220898, 46117254, 46880785, 47453761, 47874224, 48446901,
49082644, 49735039, 50465013, 51353940, 52223225, 52956833, 53676089,
54501145, 55175574, 55681183, 56126394, 56426107, 56695598, 57013205,
57401606, 57732898, 58059337, 58331657, 58599952, 58940697, 59221343,
59514294, 59891103, 60293258, 60784791, 61405446, 62024721, 62680658,
63355133, 64039521, 64679795, 65400569, 66148345, 67030073, 67873299,
69003979, 70127460, 71398808, 72916969, 74501692, 75989608, 77080935,
78093372, 79351449, 80630387, 82175642, 83574939, 84905857, 85860403,
86533153, 86693118, 86848759, 86933744, 86987679, 87166596, 87400230,
87530962, 87793484, 88192304, 88594919, 88869608, 89171000, 89499256,
89987937, 90461967, 90940574, 91360186, 91885828, 92373129, 92955006,
93405484, 93688062, 94046149, 94442209, 94722786, 94994784, 95253442,
95412993, 95558123, 95756774, 95872809, 96042503, 96225790, 96353440,
96481780, 96623966, 96805068, 96962756, 97096340, 97134658, 97129276,
97141765, 97210259, 97308354, 97369925, 97419651, 97417259, 97374617
)
ts.plot(ts(portfolio.active, start=1990, frequency=12)/1000000,ts(portfolio.passive, start=1990, frequency=12)/1000000, col=c("red","blue"), ylab="in million dollars")
abline(h=0)
return.ma.trading <- ROC(portfolio.active, type ="discrete")
return.buy.and.hold <- ROC(portfolio.passive, type ="discrete")
return.ma.trading[is.na(return.ma.trading)] <- 0
return.buy.and.hold[is.na(return.buy.and.hold)] <- 0
cumret.ma.trading <- cumprod(1+return.ma.trading)
cumret.buy.and.hold <- cumprod(1+return.buy.and.hold)
par(mfrow=c(2,2))
plot.ts(return.ma.trading)
plot.ts(return.buy.and.hold)
plot.ts(cumret.ma.trading)
plot.ts(cumret.buy.and.hold)
```Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.