Equity Performance Evaluation

Return Concepts

Simple returns

\(r_t\) measures the rate of return from \(t-1\) to \(t\). It is the percentage change in price given by:

\[ r_t = \frac{P_t-P_{t-1}}{P_{t-1}} = \frac{P_t}{P_{t-1}}-1 \]

One commonly used equality is

\[ P_t = P_{t-1} (1+r_t) \]

Continuously compounded returns

\(z_t\) measures the log return, also referred to as continuously compounded return. It is the first difference of the natural logarithm of prices.

\[ z_t = \ln P_t - \ln P_{t-1} = \ln \frac{P_t}{P_{t-1}} \]

The conversion between log return and simple return:

\[ \begin{aligned} \color{red}{z_t} &\color{red}{= \ln (1+r_t) }\\ r_t &= e^{z_t} -1 \end{aligned} \]

Q: Why is \(z_t\) continuously compounded return?

A: This is related to the constant \(e\), Euler number.

The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression:

\[ e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n \]

If you put \(\$100\) in a bank with an annual interest rate of \(10\%\) and a yearly compounding period. What you get in a year can be expressed as follows:

\[ D_t = D_0 \left(1+\frac{R}{n}\right)^{nt} = 100\times \left(1+\frac{10\%}{1}\right)^{1\times1} = 110 \] Note that

\(\quad\) \(D_0\) is the initial deposit,
\(\quad\) \(D_t\) is the value of the deposit at time \(t\),
\(\quad\) \(R\) is the annual percentage rate (APR),
\(\quad\) \(n\) is the number of compounding periods in one year,
\(\quad\) \(t\) is the number of years from \(D_0\) to \(D_t\).

(For a bank deposit account, the quoted interest rate often refers to as “simple interest” which ignores compounding. For example, an interest rate of \(5\%\) payable every six months will be quoted as a simple interest of \(10\%\) per annum in the market.)

Q: What if we compound semi-annually?

A: That is when \(e\) equals 2.

\[ D_t = 100\times(1+\frac{10\%}{2})^{2\times1} = 110.25 \]

What if we compound monthly? \(\rightarrow\) \(n=12\)

\[ D_t = 100\times(1+\frac{10\%}{12})^{12\times1} = 110.47 \]

What if we compound daily? \(\rightarrow\) \(n=365\)

\[ D_t = 100\times(1+\frac{10\%}{365})^{365\times1} = 110.52 \]

\(\left(1+\frac{R}{n}\right)^{nt}\) can be rewritten as \(\left(1+\frac{1}{\frac{n}{R}}\right)^{\frac{n}{R} \cdot R \cdot t}\). We have

\[ \left(1+\frac{1}{\frac{n}{R}}\right)^{\frac{n}{R} \cdot R \cdot t} \to e^{Rt} \] as \(\frac{n}{R}\to \infty\).

Under continuous compounding,

\[ \color{red} {D_t = D_0\, e^{Rt}} . \]

R <- 0.1 # annual interest rate
t <- 1   # total time period

n <- 2 # compound frequency in one year
100 * (1+R/n)^{n*t}

## [1] 110.25

n <- 12
100 * (1+R/n)^{n*t}

## [1] 110.4713

n <- 365
100 * (1+R/n)^{n*t}

## [1] 110.5156

100 * exp(R*t)

## [1] 110.5171

Note that

• \(D_t\) under continuous compounding is always larger than those of under fixed compounding frequencies. The higher the frequency, the larger the end value \(D_t\) is. This is due to the earnings from “interest-on-interest”.

• \(\left(1+\frac{R}{n}\right)^{n}-1\) is the effective interest rate. This is the interest rate you get as a proportion to the amount you put in and it depends on the frequency of compounding.

• Continuously compounded interest rate is the effective interest rate when compounded continuously \((n\to\infty).\)

Proof. 1 Show \(\ln (1+x) \approx x\) as \(x\to 0\).

Let \(f(x)=\ln (1+x)\), its first-order Taylor expansion at \(x\) close to \(0\) is:

\[ \begin{aligned} f(x) \approx f(0) + f^\prime(0)(x-0) \end{aligned} \]

The first derivative of \(f(x)\) is

\[ f^\prime(x) = \frac{1}{1+x}. \]

Hence

\[ \begin{aligned} f(x) &\approx f(0) + f^\prime(0)(x-0) \\ &= \ln(1) + 1\cdot x \\ &= x. \end{aligned} \]

\(\square\)

From prices to returns.

f_name <- "data/Titlon_equity_price_2014-2023_daily.csv"
titlon_data <- read_csv(f_name)
## group by ISIN, calculate returns
titlon_group <- titlon_data %>% group_by(ISIN)
groups <- titlon_group %>% group_split()
group_key <- titlon_group %>% 
    group_keys() %>% 
    mutate(id = row_number())
group_key

## # A tibble: 501 × 2
##    ISIN            id
##    <chr>        <int>
##  1 AU000000CSS3     1
##  2 AU0000057408     2
##  3 BMG0451H1170     3
##  4 BMG0670A1099     4
##  5 BMG067231032     5
##  6 BMG0702P1086     6
##  7 BMG1466R1732     7
##  8 BMG1466R2078     8
##  9 BMG173841013     9
## 10 BMG1738J1247    10
## # ℹ 491 more rows

# subset companies with more than 3 years' data
isin_vec <- titlon_group %>% tally() %>% 
    filter(n>(252*3)) %>% 
    pull(ISIN)
id_vec <- sapply(isin_vec, function(isin) which(group_key$ISIN==isin))
# data for one equity
i <- 4
groups[[id_vec[i]]]

## # A tibble: 951 × 46
##    Date       `Internal code` SecurityId CompanyId Symbol ISIN         Name     
##    <date>               <dbl>      <dbl>     <dbl> <chr>  <chr>        <chr>    
##  1 2020-02-19         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  2 2020-02-20         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  3 2020-02-21         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  4 2020-02-24         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  5 2020-02-25         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  6 2020-02-26         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  7 2020-02-27         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  8 2020-02-28         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  9 2020-03-02         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
## 10 2020-03-03         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
## # ℹ 941 more rows
## # ℹ 39 more variables: BestBidPrice <dbl>, BestAskPrice <dbl>, Open <dbl>,
## #   High <dbl>, Low <dbl>, Close <dbl>, OfficialNumberOfTrades <dbl>,
## #   OfficialVolume <dbl>, Price <dbl>, AdjustedPrice <dbl>, Dividends <dbl>,
## #   LDividends <dbl>, CorpAdj <dbl>, DividendAdj <dbl>, Currency <chr>,
## #   NumberOfShares <dbl>, Exchange <chr>, NOKPerForex <dbl>, mktcap <dbl>,
## #   OSEBXmktshare_prevmnth <dbl>, OSEBXAlpha_prevmnth <dbl>, …

groups[[id_vec[i]]] %>% tail()

## # A tibble: 6 × 46
##   Date       `Internal code` SecurityId CompanyId Symbol ISIN         Name      
##   <date>               <dbl>      <dbl>     <dbl> <chr>  <chr>        <chr>     
## 1 2023-11-17         2015535    1305295     12748 BWE    BMG0702P1086 BW ENERGY…
## 2 2023-11-20         2015535    1305295     12748 BWE    BMG0702P1086 BW ENERGY…
## 3 2023-11-21         2015535    1305295     12748 BWE    BMG0702P1086 BW ENERGY…
## 4 2023-11-22         2015535    1305295     12748 BWE    BMG0702P1086 BW ENERGY…
## 5 2023-11-23         2015535    1305295     12748 BWE    BMG0702P1086 BW ENERGY…
## 6 2023-11-24         2015535    1305295     12748 BWE    BMG0702P1086 BW ENERGY…
## # ℹ 39 more variables: BestBidPrice <dbl>, BestAskPrice <dbl>, Open <dbl>,
## #   High <dbl>, Low <dbl>, Close <dbl>, OfficialNumberOfTrades <dbl>,
## #   OfficialVolume <dbl>, Price <dbl>, AdjustedPrice <dbl>, Dividends <dbl>,
## #   LDividends <dbl>, CorpAdj <dbl>, DividendAdj <dbl>, Currency <chr>,
## #   NumberOfShares <dbl>, Exchange <chr>, NOKPerForex <dbl>, mktcap <dbl>,
## #   OSEBXmktshare_prevmnth <dbl>, OSEBXAlpha_prevmnth <dbl>,
## #   OSEBXBeta_prevmnth <dbl>, SMB <dbl>, HML <dbl>, LIQ <dbl>, MOM <dbl>, …

the_group <- groups[[id_vec[i]]] 
the_group

## # A tibble: 951 × 46
##    Date       `Internal code` SecurityId CompanyId Symbol ISIN         Name     
##    <date>               <dbl>      <dbl>     <dbl> <chr>  <chr>        <chr>    
##  1 2020-02-19         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  2 2020-02-20         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  3 2020-02-21         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  4 2020-02-24         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  5 2020-02-25         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  6 2020-02-26         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  7 2020-02-27         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  8 2020-02-28         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
##  9 2020-03-02         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
## 10 2020-03-03         2015535    1305295     12748 BWE    BMG0702P1086 BW Energ…
## # ℹ 941 more rows
## # ℹ 39 more variables: BestBidPrice <dbl>, BestAskPrice <dbl>, Open <dbl>,
## #   High <dbl>, Low <dbl>, Close <dbl>, OfficialNumberOfTrades <dbl>,
## #   OfficialVolume <dbl>, Price <dbl>, AdjustedPrice <dbl>, Dividends <dbl>,
## #   LDividends <dbl>, CorpAdj <dbl>, DividendAdj <dbl>, Currency <chr>,
## #   NumberOfShares <dbl>, Exchange <chr>, NOKPerForex <dbl>, mktcap <dbl>,
## #   OSEBXmktshare_prevmnth <dbl>, OSEBXAlpha_prevmnth <dbl>, …

ticker <- the_group$Symbol[1]
ticker

## [1] "BWE"

the_group$Name[1]

## [1] "BW Energy Limited"

# convert to xts
library(quantmod)
the_group_xts <- xts(the_group[,c("AdjustedPrice")], order.by=the_group$Date)
# the_group_xts <- xts(the_group[,c("Price")], order.by=the_group$Date)
the_group_xts %>% str()

## An xts object on 2020-02-19 / 2023-11-24 containing: 
##   Data:    double [951, 1]
##   Columns: AdjustedPrice
##   Index:   Date [951] (TZ: "UTC")

the_group_xts

##            AdjustedPrice
## 2020-02-19        24.150
## 2020-02-20        24.525
## 2020-02-21        24.370
## 2020-02-24        21.800
## 2020-02-25        20.355
## 2020-02-26        19.742
## 2020-02-27        17.350
## 2020-02-28        17.028
## 2020-03-02        17.294
## 2020-03-03        19.550
##        ...              
## 2023-11-13        29.300
## 2023-11-14        28.850
## 2023-11-15        28.750
## 2023-11-16        26.100
## 2023-11-17        25.850
## 2023-11-20        25.550
## 2023-11-21        24.500
## 2023-11-22        24.000
## 2023-11-23        24.100
## 2023-11-24        24.250

# from daily to monthly
prices_monthly <- the_group_xts %>% to.monthly(indexAt = "last", OHLC=FALSE)
prices_monthly %>% head(20)

##            AdjustedPrice
## 2020-02-28        17.028
## 2020-03-31         9.070
## 2020-04-30        12.740
## 2020-05-29        14.554
## 2020-06-30        17.168
## 2020-07-31        17.170
## 2020-08-31        20.500
## 2020-09-30        16.750
## 2020-10-30        14.800
## 2020-11-30        20.920
## 2020-12-30        27.600
## 2021-01-29        24.180
## 2021-02-26        25.160
## 2021-03-31        26.880
## 2021-04-30        27.750
## 2021-05-31        23.950
## 2021-06-30        26.200
## 2021-07-30        26.750
## 2021-08-31        27.950
## 2021-09-30        27.900

prices_monthly %>% tail(20)

##            AdjustedPrice
## 2022-04-29         27.50
## 2022-05-31         28.84
## 2022-06-30         25.56
## 2022-07-29         27.42
## 2022-08-31         24.82
## 2022-09-30         21.62
## 2022-10-31         26.30
## 2022-11-30         27.06
## 2022-12-30         25.14
## 2023-01-31         27.90
## 2023-02-28         29.24
## 2023-03-31         26.92
## 2023-04-28         28.60
## 2023-05-31         27.25
## 2023-06-30         25.80
## 2023-07-31         30.00
## 2023-08-31         25.65
## 2023-09-29         27.35
## 2023-10-31         27.90
## 2023-11-24         24.25

# price plot
plot(prices_monthly, main = sprintf("Monthly Price: %s", ticker))

# calculate monthly return by hand 
simple_ret <- (diff(prices_monthly)/lag(prices_monthly)) %>% setNames("simple_return") # simple return
log_ret <- diff(log(prices_monthly)) %>% setNames("log_return")   # log return, aka, continuously compounded return
merge(prices_monthly, simple_ret, log_ret)

##            AdjustedPrice simple_return   log_return
## 2020-02-28        17.028            NA           NA
## 2020-03-31         9.070 -0.4673478976 -0.629886784
## 2020-04-30        12.740  0.4046306505  0.339774386
## 2020-05-29        14.554  0.1423861852  0.133119220
## 2020-06-30        17.168  0.1796069809  0.165181316
## 2020-07-31        17.170  0.0001164958  0.000116489
## 2020-08-31        20.500  0.1939429237  0.177261211
## 2020-09-30        16.750 -0.1829268293 -0.202026628
## 2020-10-30        14.800 -0.1164179104 -0.123771078
## 2020-11-30        20.920  0.4135135135  0.346078458
## 2020-12-30        27.600  0.3193116635  0.277110134
## 2021-01-29        24.180 -0.1239130435 -0.132289928
## 2021-02-26        25.160  0.0405293631  0.039729587
## 2021-03-31        26.880  0.0683624801  0.066127084
## 2021-04-30        27.750  0.0323660714  0.031853325
## 2021-05-31        23.950 -0.1369369369 -0.147267516
## 2021-06-30        26.200  0.0939457203  0.089791087
## 2021-07-30        26.750  0.0209923664  0.020775063
## 2021-08-31        27.950  0.0448598131  0.043882726
## 2021-09-30        27.900 -0.0017889088 -0.001790511
## 2021-10-29        27.550 -0.0125448029 -0.012624153
## 2021-11-30        21.100 -0.2341197822 -0.266729495
## 2021-12-30        20.100 -0.0473933649 -0.048553225
## 2022-01-31        22.900  0.1393034826  0.130417095
## 2022-02-28        23.300  0.0174672489  0.017316450
## 2022-03-31        27.000  0.1587982833  0.147383505
## 2022-04-29        27.500  0.0185185185  0.018349139
## 2022-05-31        28.840  0.0487272727  0.047577308
## 2022-06-30        25.560 -0.1137309293 -0.120734683
## 2022-07-29        27.420  0.0727699531  0.070244045
## 2022-08-31        24.820 -0.0948212983 -0.099622894
## 2022-09-30        21.620 -0.1289282836 -0.138030968
## 2022-10-31        26.300  0.2164662350  0.195950127
## 2022-11-30        27.060  0.0288973384  0.028487684
## 2022-12-30        25.140 -0.0709534368 -0.073596420
## 2023-01-31        27.900  0.1097852029  0.104166486
## 2023-02-28        29.240  0.0480286738  0.046910946
## 2023-03-31        26.920 -0.0793433653 -0.082668130
## 2023-04-28        28.600  0.0624071322  0.060537213
## 2023-05-31        27.250 -0.0472027972 -0.048353197
## 2023-06-30        25.800 -0.0532110092 -0.054679029
## 2023-07-31        30.000  0.1627906977  0.150822890
## 2023-08-31        25.650 -0.1450000000 -0.156653810
## 2023-09-29        27.350  0.0662768031  0.064172957
## 2023-10-31        27.900  0.0201096892  0.019910160
## 2023-11-24        24.250 -0.1308243728 -0.140210071

# calculate monthly return using `PerformanceAnalytics::Return.calculate`
library(PerformanceAnalytics)
merge(Return.calculate(prices_monthly, method = "discrete"),
      Return.calculate(prices_monthly, method = "log") ) %>%
    fortify() %>% 
    setNames(c("Date", "Simple Ret", "Log Ret")) %>% 
    knitr::kable(digits = 5, escape=F, caption="Using `PerformanceAnalytics::Return.calculate`") %>%
    kable_styling(bootstrap_options = c("striped", "hover"), full_width = F, latex_options="scale_down") %>% 
    scroll_box(height = "500px")

Table 1: Using PerformanceAnalytics::Return.calculate

Date	Simple Ret	Log Ret
2020-02-28
2020-03-31	-0.46735	-0.62989
2020-04-30	0.40463	0.33977
2020-05-29	0.14239	0.13312
2020-06-30	0.17961	0.16518
2020-07-31	0.00012	0.00012
2020-08-31	0.19394	0.17726
2020-09-30	-0.18293	-0.20203
2020-10-30	-0.11642	-0.12377
2020-11-30	0.41351	0.34608
2020-12-30	0.31931	0.27711
2021-01-29	-0.12391	-0.13229
2021-02-26	0.04053	0.03973
2021-03-31	0.06836	0.06613
2021-04-30	0.03237	0.03185
2021-05-31	-0.13694	-0.14727
2021-06-30	0.09395	0.08979
2021-07-30	0.02099	0.02078
2021-08-31	0.04486	0.04388
2021-09-30	-0.00179	-0.00179
2021-10-29	-0.01254	-0.01262
2021-11-30	-0.23412	-0.26673
2021-12-30	-0.04739	-0.04855
2022-01-31	0.13930	0.13042
2022-02-28	0.01747	0.01732
2022-03-31	0.15880	0.14738
2022-04-29	0.01852	0.01835
2022-05-31	0.04873	0.04758
2022-06-30	-0.11373	-0.12073
2022-07-29	0.07277	0.07024
2022-08-31	-0.09482	-0.09962
2022-09-30	-0.12893	-0.13803
2022-10-31	0.21647	0.19595
2022-11-30	0.02890	0.02849
2022-12-30	-0.07095	-0.07360
2023-01-31	0.10979	0.10417
2023-02-28	0.04803	0.04691
2023-03-31	-0.07934	-0.08267
2023-04-28	0.06241	0.06054
2023-05-31	-0.04720	-0.04835
2023-06-30	-0.05321	-0.05468
2023-07-31	0.16279	0.15082
2023-08-31	-0.14500	-0.15665
2023-09-29	0.06628	0.06417
2023-10-31	0.02011	0.01991
2023-11-24	-0.13082	-0.14021

Cumulative returns

For simple return, the cumulative return from time \(0\) to time \(T\) is given by:

\[ r_{0:T} = \Pi_{t=1}^T (1+r_t) -1. \]

For log return

\[ z_{0:T} = \sum_{t=1}^T z_t \]

Note that we have to back out the simple return from the log return

\[ r_{0:T} = \exp(z_{0:T}) -1 \]

return_monthly_simple <- Return.calculate(prices_monthly, method = "discrete")
return_monthly_simple[1,] <- 0
cumulative_returns_simple <- cumprod(1 + return_monthly_simple) - 1
# plot price, monthly ret, and cumu ret
par(mfrow=c(3,1))
plot(prices_monthly, main = sprintf("Monthly adjPrice: %s", ticker))
plot(return_monthly_simple, 
     main = sprintf("Monthly Return: %s", ticker))
plot(cumulative_returns_simple, 
     main = sprintf("Monthly Cumulative Return: %s", ticker))

Fig. 1: Simple returns

return_monthly_log <- Return.calculate(prices_monthly, method = "log")
return_monthly_log[1,] <- 0
cumulative_returns_log <- exp(cumsum(return_monthly_log)) - 1 # back out the simple return
par(mfrow=c(3,1))
plot(prices_monthly, main = sprintf("Monthly adjPrice: %s", ticker))
plot(return_monthly_log, 
     main = sprintf("Monthly Return: %s", ticker))
plot(cumulative_returns_log, 
     main = sprintf("Monthly Cumulative Return: %s", ticker))

Fig. 2: Log returns

The two types act very differently when it comes to aggregation. Each has an advantage over the other:

• simple returns aggregate across assets
  The simple return of a portfolio is the weighted sum of the simple returns of the constituents of the portfolio.

• log returns aggregate across time
  The log return for a time period is the sum of the log returns of partitions of the time period. For example, the log return for a year is the sum of the log returns of the days within the year.

Multiperiod returns

Multiperiod returns are the generalized case of cumulative return by allowing any holding period \(k\le T\).

The \(k\)-period simple return from time \(t-k\) to \(t\) is given by

\[ r_{t}(k) = \frac{P_t-P_{t-k}}{P_{t-k}}. \]

It can be expressed as in terms of one-period returns as follows:

\[ \begin{aligned} r_t(k) &= \prod_{j=0}^{k-1}(1+r_{t-j}) -1 \\ &= (1+r_t)(1+r_{t-1})\cdots (1+r_{t-k+1})-1 . \end{aligned} \]

We always like to talk in terms of annual performance as people like to know how much they can expect to make a year in percentage terms. That is why in most of the fund reports, you will find a standard metric called annualized returns. It is also known as the Compound Annual Growth Rate (CAGR) or the Geometric Annual Return.

Annualized return under simple returns \(r_t^A\) is given by:

\[ \begin{split} (1+r_t^A)^k &= 1+r_t(k) \\ r_t^A &= \big(1+r_t(k) \big)^{\frac{1}{k}}-1 = \left[\prod_{j=1}^{k-1}(1+r_{t-j})\right]^{\frac{1}{k}}-1 . \end{split} \]

The \(k\)-period log return is given by

\[ z_t(k) = \ln \frac{P_t}{P_{t-k}}. \]

It is the sum of the \(k\) one-period log returns:

\[ z_t(k) = \sum_{j=0}^{k-1} z_{t-j} = z_t + z_{t-1} + \cdots + z_{t-k+1} \]

Annualized return under continuously compounding \(z_t^A\) is given by:

\[ \begin{aligned} k\,z_t^A &= z_t(k) \\ z_t^A &= \frac{1}{k}z_t(k) = \frac{1}{k} \sum_{j=0}^{k-1} z_{t-j}, \end{aligned} \] which is the average one-period log returns.

To summarize in one table

	Simple returns	Log returns	Back out simple returns
Single period	\(r_t=\frac{P_t}{P_{t-1}}-1\)	\(z_t=\ln \frac{P_t}{P_{t-1}}\)	\(r_t = \exp(z_t)-1\)
Multiperiod	\(r_t(k) = \prod_{j=0}^{k-1}(1+r_{t-j}) -1\)	\(z_t(k) = \sum_{j=0}^{k-1} z_{t-j}\)	\(r_t(k) = \exp\big(z_t(k)\big)-1\)
Annualized	\(r_t^A = \left[\prod_{j=0}^{k-1}(1+r_{t-j})\right]^{\frac{1}{k}}-1\)	\(z_t^A = \frac{1}{k} \sum_{j=0}^{k-1} z_{t-j}\)	\(r_t^A = \exp(z_t^A)-1\)

Note that the annualized return here assumes the single period returns are annual returns.

In case of higher frequency data than annual, i.e., daily, weekly and monthly, we have to compound by the number of periods in a year.

• daily single period: \(r_t^A = \left[\prod_{j=0}^{k-1}(1+r_{t-j})\right]^{\frac{1}{k}\cdot \color{red}{252}}-1\) for simple return, \(z_t^A = \frac{252}{k} \sum_{j=0}^{k-1} z_{t-j}\) for log return.

• weekly \(r_t^A = \left[\prod_{j=0}^{k-1}(1+r_{t-j})\right]^{\frac{1}{k}\cdot \color{red}{52}}-1\) for simple return, \(z_t^A = \frac{52}{k} \sum_{j=0}^{k-1} z_{t-j}\) for log return.

• monthly \(r_t^A = \left[\prod_{j=0}^{k-1}(1+r_{t-j})\right]^{\frac{1}{k}\cdot \color{red}{12}}-1\) for simple return, \(z_t^A = \frac{12}{k} \sum_{j=0}^{k-1} z_{t-j}\) for log return.

• quarterly \(r_t^A = \left[\prod_{j=0}^{k-1}(1+r_{t-j})\right]^{\frac{1}{k}\cdot \color{red}{4}}-1\) for simple return, \(z_t^A = \frac{4}{k} \sum_{j=0}^{k-1} z_{t-j}\) for log return.

Portfolio Returns

Consider a buy-and-hold portfolio invested in \(k\) different assets. The value at time \(t\) is

\[ V_t = \sum_{i=1}^k n_i P_{i,t} \] where \(n_i\) is the number of shares invested in asset \(i\).

Buy-and-hold portfolio: Portfolio weights from the initial portfolio are allowed to change over time as prices of the underlying assets change over time. In this case no rebalancing of the portfolio is done. \(n_i\) is constant for each asset \(i\) over the holding period. The weight increases if an asset’s price increases, and decreases otherwise. This strategy is passive and does not trade further except for the initial asset allocation.

Another trading strategy is rebalancing (monthly). This is to buy and trade at the end of each rebalance period such that your portfolio aligns with your target allocation. Regularly rebalancing a portfolio ensures that the investor maintains the desired risk and return characteristics. For instance, if the weight of a particular asset class has increased significantly due to strong performance, rebalancing involves selling a portion of that asset and reinvesting the proceeds in other assets to restore the desired portfolio weight. This is a rebalanced portfolio.

The simple one-period return of the portfolio is a weighted average of the returns of component stocks.

\[ r_{p,t} = \frac{V_t}{V_{t-1}}-1 = \sum_{i=1}^k w_{i,t} r_{i,t} \]

This result is useful. We can use the property to get the expected return and variance of the portfolio as:

\[ \begin{aligned} E[r_{p,t}] &= \sum_{i=1}^k w_{i,t} E[r_{i,t}] \\ \text{Var}[r_{p,t}] &= \sum_{i=1}^k\sum_{j=1}^k w_{i,t}\,w_{j,t}\,\text{Cov}(r_{i,t}, r_{j,t}) \end{aligned} \]

Proof. 2 Show \(r_{p,t} = \sum_{i=1}^k w_i r_{i,t}\).

\[ \begin{aligned} r_{p,t} &= \frac{V_t}{V_{t-1}} -1 \\ &= \frac{\sum_{i=1}^k n_iP_{i,t}}{\sum_{j=1}^k n_jP_{j,t-1}} -1 \\ &= \sum_{i=1}^k \frac{n_iP_{i,t-1}}{\sum_{j=1}^k n_jP_{j,t-1}} \cdot \frac{P_{i,t}}{P_{i,t-1}} - 1 \qquad \text{(multiply and divide by } P_{i, t-1} )\\ &= \sum_i w_{i,t} (r_{i,t}+1) -1 \\ &= \sum_i w_{i,t} r_{i,t} \qquad (w_{i,t} \text{ sums to 1}) \end{aligned} \] where \(w_{i,t}= \frac{n_iP_{i,t-1}}{\sum_{j=1}^k n_jP_{j,t-1}}\) is the weight of asset \(i\) at time \(t-1\).

\(\square\)

Note that an asset’s weight in month \(t\), \(w_{i,t}\), is decided by the ratio of the value of the asset to the portfolio value at the beginning of the period, i.e., at time \(t-1\).

This cross-sectional additivity does not apply to log returns. In stead we have:

\[ \begin{aligned} z_{p,t} &= \ln\,\left(\frac{V_t}{V_{t-1}}\right) \\ &= \ln \, \left(\frac{\sum_{i=1}^k n_iP_{i,t}}{\sum_{j=1}^k n_jP_{j,t-1}} \right) \\ &= \ln \, \left(\sum_{i=1}^k\frac{ n_iP_{i,t-1}}{\sum_{j=1}^k n_jP_{j,t-1}} \cdot \frac{P_{i,t}}{P_{i,t-1}} \right) \\ &= \ln \, \left(\sum_{i=1}^k w_{i,t} \frac{P_{i,t}}{P_{i,t-1}} \right) \\ &= \ln \, \left(\sum_{i=1}^k w_{i,t} \exp(z_{i,t}) \right) . \end{aligned} \] The log return of the portfolio is not a linear function for the log returns of the components.

Because the continuously compounded portfolio return is not a weighted average of the individual asset continuously compounded returns, the analysis of portfolios is typically performed using simple returns and not continuously compounded returns.

On the other hand, the log returns are additive when we consider the time series of returns:

\[ z_{p,t}(k) = \sum_{j=0}^{k-1} z_{p,t-j}. \] Given the expected values and the covariances of the subperiod returns, it is then easy to compute the expected value and the variance of the full period return.

In contrast, the time series additivity does not apply to simple returns.

\[ r_{p, t}(k) = \prod_{j=0}^{k-1} (1+r_{p,t-j}) -1 \]

---

Portfolio performance evaluation

We use a portfolio consisting of 2 assets as an example.

Return

Loop through assets to calculate monthly returns from daily price data.

# calculate monthly return
data_return <- titlon_group %>%  
    group_modify( ~{
        xts(.$AdjustedPrice, order.by = .$Date) %>%
            to.monthly(indexAt="last", OHLC=FALSE) %>% 
            Return.calculate() %>% 
            data.frame() %>% 
            rownames_to_column(var="Date")
    })  %>% ungroup()
colnames(data_return)[3] <- "Return_monthly"
data_return

## # A tibble: 30,351 × 3
##    ISIN         Date       Return_monthly
##    <chr>        <chr>               <dbl>
##  1 AU000000CSS3 2021-05-31      NA       
##  2 AU000000CSS3 2021-06-30      -0.0147  
##  3 AU000000CSS3 2021-07-29       0.0746  
##  4 AU000000CSS3 2021-08-30       0.208   
##  5 AU000000CSS3 2021-09-30      -0.195   
##  6 AU000000CSS3 2021-10-29       0.114   
##  7 AU000000CSS3 2021-11-29      -0.000128
##  8 AU000000CSS3 2021-12-29      -0.0640  
##  9 AU000000CSS3 2022-01-24      -0.0193  
## 10 AU000000CSS3 2022-02-28       0.00349 
## # ℹ 30,341 more rows

## equity identifiers, ensure one-to-one mapping
select <- dplyr::select
unique_id <- titlon_data %>% 
    distinct(ISIN, .keep_all = TRUE) %>% 
    select(all_of(c("SecurityId", "CompanyId", "Symbol", "ISIN",
                    "Name", "Sector")))
unique_id

## # A tibble: 501 × 6
##    SecurityId CompanyId Symbol ISIN         Name               Sector           
##         <dbl>     <dbl> <chr>  <chr>        <chr>              <chr>            
##  1    1304857     12720 2020   BMG9156K1018 2020 Bulkers       Industrials      
##  2    1301972     12440 FIVEPG DK0060945467 5th Planet Games   Consumer Discret…
##  3         NA        NA AASB   NO0010672181 Aasen Sparebank    Financials       
##  4       6085      2017 ABG    NO0003021909 ABG Sundal Collier Financials       
##  5    1301198     12348 ABL    NO0010715394 ABL GROUP          Industrials      
##  6    1304655     12701 ADE    NO0010844038 Adevinta           Consumer Discret…
##  7    1304652     12701 ADEA   NO0010843998 Adevinta ser. A    Consumer Discret…
##  8         NA        NA ADS    CY0108052115 ADS Crude Carriers Industrials      
##  9    1251504     11273 AEGA   NO0010626559 Aega               Financials       
## 10         NA        NA AEGA   NO0012958539 AEGA               Financials       
## # ℹ 491 more rows

## add company info
data_return <- data_return %>% 
    left_join(unique_id, by="ISIN") %>% 
    select("ISIN", "Date",
           "Symbol", "Name", "Sector", 
           "Return_monthly")

## subset complete cases
start_date <- ymd("2015-01-01")
end_date <- ymd("2022-12-31")
data_return <- data_return %>% 
    mutate(Date = ymd(Date)) %>% 
    filter(between(Date, start_date, end_date))
ISIN_vec <- data_return %>% group_by(ISIN) %>% 
    tally(sort=TRUE) %>% 
    filter(n==96) %>% 
    pull(ISIN)
ISIN_vec %>% length() # 142 complete cases

## [1] 142

data_return <- data_return %>% filter(ISIN %in% ISIN_vec)    
data_return

## # A tibble: 13,632 × 6
##    ISIN         Date       Symbol Name   Sector Return_monthly
##    <chr>        <date>     <chr>  <chr>  <chr>           <dbl>
##  1 BMG0451H1170 2015-01-30 ARCHER Archer Energy       -0.240  
##  2 BMG0451H1170 2015-02-27 ARCHER Archer Energy       -0.228  
##  3 BMG0451H1170 2015-03-31 ARCHER Archer Energy        0.00844
##  4 BMG0451H1170 2015-04-30 ARCHER Archer Energy        0.167  
##  5 BMG0451H1170 2015-05-29 ARCHER Archer Energy       -0.0394 
##  6 BMG0451H1170 2015-06-30 ARCHER Archer Energy        0.00746
##  7 BMG0451H1170 2015-07-31 ARCHER Archer Energy       -0.215  
##  8 BMG0451H1170 2015-08-31 ARCHER Archer Energy       -0.401  
##  9 BMG0451H1170 2015-09-30 ARCHER Archer Energy       -0.130  
## 10 BMG0451H1170 2015-10-30 ARCHER Archer Energy       -0.0769 
## # ℹ 13,622 more rows

Get sample data for two assets and create an equally-weighted portfolio of them.

k <- 2 # number of asset
sample_data <- data_return %>% 
    filter(ISIN %in% c("BMG0451H1170", "NO0003079709"))
sample_data

## # A tibble: 192 × 6
##    ISIN         Date       Symbol Name   Sector Return_monthly
##    <chr>        <date>     <chr>  <chr>  <chr>           <dbl>
##  1 BMG0451H1170 2015-01-30 ARCHER Archer Energy       -0.240  
##  2 BMG0451H1170 2015-02-27 ARCHER Archer Energy       -0.228  
##  3 BMG0451H1170 2015-03-31 ARCHER Archer Energy        0.00844
##  4 BMG0451H1170 2015-04-30 ARCHER Archer Energy        0.167  
##  5 BMG0451H1170 2015-05-29 ARCHER Archer Energy       -0.0394 
##  6 BMG0451H1170 2015-06-30 ARCHER Archer Energy        0.00746
##  7 BMG0451H1170 2015-07-31 ARCHER Archer Energy       -0.215  
##  8 BMG0451H1170 2015-08-31 ARCHER Archer Energy       -0.401  
##  9 BMG0451H1170 2015-09-30 ARCHER Archer Energy       -0.130  
## 10 BMG0451H1170 2015-10-30 ARCHER Archer Energy       -0.0769 
## # ℹ 182 more rows

ret_mat <- sample_data %>% pull(Return_monthly) %>% 
    matrix(ncol=k)
ret_mat %>% dim()

## [1] 96  2

wts <- rep(1/k, k) 
wts

## [1] 0.5 0.5

Calculate portfolio return manually (most concise). It is straightforward to calculate rebalanced, as the begin of period weights are always the same. But it is a bit cumbersome to calculate buy-and-hold as we need to update the weight for each period.

Here shows an example of rebalancing monthly.

# option 1, matrix representation
ptf_ret <- ret_mat %*% matrix(wts, ncol=1)
names(ptf_ret) <- "port_ret_manual"
ptf_ret %>% head(10)

##               [,1]
##  [1,] -0.122990026
##  [2,] -0.043001570
##  [3,]  0.022356632
##  [4,]  0.247278035
##  [5,]  0.003363661
##  [6,] -0.036708697
##  [7,] -0.007408891
##  [8,] -0.253805031
##  [9,] -0.061228133
## [10,]  0.081909129

Alternatively, we can use PerformanceAnalytics package. It provides many functions convenient for performance evaluation. We need to convert the data to xts before providing it as the input for PerformanceAnalytics functions.

## using PerformanceAnalytics::Return.portfolio
sample_data <- sample_data %>%      # standardize date
    mutate(yrmon=as.yearmon(Date))
sample_data <- sample_data %>%  
    mutate(Date=as.Date(yrmon, frac=1))
return_xts <- sample_data %>% pull(Return_monthly) %>% 
    matrix(ncol=k) %>% 
    xts(order.by = sample_data$Date %>% unique())
colnames(return_xts) <- sample_data$Symbol %>% unique()
return_xts %>% str()

## An xts object on 2015-01-31 / 2022-12-31 containing: 
##   Data:    double [96, 2]
##   Columns: ARCHER, KIT
##   Index:   Date [96] (TZ: "UTC")

return_xts %>% 
    data.frame %>% 
    knitr::kable(digits = 5, caption = "Component asset monthly returns") %>%
    kable_styling(bootstrap_options = c("striped", "hover"), full_width = F, latex_options="scale_down") %>% 
    scroll_box(height = "500px")

Table 2: Component asset monthly returns

	ARCHER	KIT
2015-01-31	-0.24010	-0.00588
2015-02-28	-0.22801	0.14201
2015-03-31	0.00844	0.03627
2015-04-30	0.16736	0.32719
2015-05-31	-0.03943	0.04615
2015-06-30	0.00746	-0.08088
2015-07-31	-0.21481	0.20000
2015-08-31	-0.40094	-0.10667
2015-09-30	-0.12992	0.00746
2015-10-31	-0.07692	0.24074
2015-11-30	-0.16176	0.06567
2015-12-31	-0.27485	0.09244
2016-01-31	-0.27419	0.05128
2016-02-29	-0.40667	-0.02683
2016-03-31	0.80899	0.08772
2016-04-30	0.46998	0.07126
2016-05-31	-0.09296	0.00899
2016-06-30	-0.16149	-0.01114
2016-07-31	-0.11111	0.26126
2016-08-31	-0.17292	-0.03036
2016-09-30	0.20907	0.01289
2016-10-31	0.40000	0.01273
2016-11-30	-0.04167	0.08618
2016-12-31	0.95652	-0.00331
2017-01-31	0.30159	0.10448
2017-02-28	-0.29878	0.08108
2017-03-31	0.18261	-0.09444
2017-04-30	-0.04412	0.09618
2017-05-31	-0.12692	0.04499
2017-06-30	-0.13833	0.10000
2017-07-31	0.18098	0.11742
2017-08-31	-0.14545	-0.07684
2017-09-30	0.21581	-0.01469
2017-10-31	-0.18833	-0.11553
2017-11-30	-0.13860	0.01124
2017-12-31	0.19785	-0.02917
2018-01-31	0.04876	0.18026
2018-02-28	-0.12334	0.03636
2018-03-31	-0.12662	0.01754
2018-04-30	0.16481	0.10216
2018-05-31	0.15319	0.07301
2018-06-30	-0.01292	0.01134
2018-07-31	-0.14486	-0.03466
2018-08-31	-0.21749	0.03485
2018-09-30	0.06145	-0.00306
2018-10-31	-0.07895	-0.12590
2018-11-30	-0.12286	0.04684
2018-12-31	-0.29235	-0.02685
2019-01-31	0.08631	-0.01149
2019-02-28	0.17797	0.03488
2019-03-31	-0.11151	-0.02247
2019-04-30	0.12348	0.10345
2019-05-31	-0.28649	-0.04996
2019-06-30	0.14394	0.05263
2019-07-31	0.02980	0.11087
2019-08-31	-0.10825	-0.07045
2019-09-30	-0.05529	-0.02526
2019-10-31	-0.11832	0.00864
2019-11-30	-0.16883	0.00321
2019-12-31	0.10417	0.17395
2020-01-31	-0.09119	-0.07454
2020-02-29	-0.01557	0.03144
2020-03-31	-0.36801	-0.18286
2020-04-30	-0.07119	0.25874
2020-05-31	0.23653	0.08889
2020-06-30	0.04116	0.00510
2020-07-31	-0.02791	0.26903
2020-08-31	0.13636	0.09066
2020-09-30	-0.10947	0.11736
2020-10-31	0.01182	-0.03241
2020-11-30	0.44626	-0.04593
2020-12-31	-0.00646	0.03899
2021-01-31	0.36098	-0.10265
2021-02-28	-0.06810	0.04551
2021-03-31	0.11923	0.29411
2021-04-30	0.19817	-0.07088
2021-05-31	-0.13193	0.08707
2021-06-30	0.02423	-0.09108
2021-07-31	-0.05699	-0.02115
2021-08-31	-0.09920	-0.01440
2021-09-30	0.24051	-0.05219
2021-10-31	-0.01020	0.00906
2021-11-30	-0.21237	0.07016
2021-12-31	0.00000	0.22789
2022-01-31	0.11257	-0.04025
2022-02-28	-0.23294	-0.08168
2022-03-31	0.01227	-0.03125
2022-04-30	-0.10606	-0.06995
2022-05-31	0.36441	0.02489
2022-06-30	-0.13043	-0.07392
2022-07-31	0.01429	0.18871
2022-08-31	0.08310	-0.03837
2022-09-30	-0.14174	-0.06135
2022-10-31	0.11515	0.23805
2022-11-30	0.00136	0.01073
2022-12-31	-0.06649	0.19108

Return.portfolio can do buy-and-hold and rebalanced strategies rather easily. We do not need to calculate the weight on our own in case of buy-and-hold.

# buy and hold ptf
ptf_bh <- Return.portfolio(R = return_xts, weights = wts, verbose = TRUE)
# rebalanced ptf
ptf_rebal <- Return.portfolio(R = return_xts, weights = wts, rebalance_on="month", verbose = TRUE)

Plot the portfolio return time series.

plot_data <- ptf_bh$returns %>% 
    merge(ptf_rebal$returns) %>% 
    setNames(c("buy-and-hold", "rebalance"))
plot(plot_data, multi.panel=TRUE, main="Portfolio monthly return")

By setting verbose = TRUE, it allows us to check intermediary calculations, such as contributions, weight for each asset.

Here we check the end of period weight.

# plot end of period weights
eop_weight_bh <- ptf_bh$EOP.Weight
eop_weight_rebal <- ptf_rebal$EOP.Weight
par(mfrow = c(2, 1), mar = c(2, 4, 2, 2))
plot.xts(eop_weight_bh$KIT)
plot.xts(eop_weight_rebal$KIT)

Fig. 3: Weight of KIT under buy-and-hold and rebalancing.

We see that the weight of KIT basically skyrockets and dominates the portfolio, taking up more than \(99\%\) of the portfolio. This is due to its strong performance.

Now we check the relative performance of the two assets.

## check relative performance by calculating the equity curve
equity_curve_xts <- return_xts %>% 
    apply(2, function(col) cumprod(1 + col))
equity_curve_xts <- xts(equity_curve_xts, index(return_xts))
colnames(equity_curve_xts) <- colnames(return_xts)
plot(equity_curve_xts, multi.panel=TRUE, yaxis.same=FALSE)

Q: what does the begin of period weight look like?

A: a horizontal line at \(y=50\%\).

bop_weight_rebal <- ptf_rebal$BOP.Weight
plot(bop_weight_rebal$KIT)

---

Risk-adjusted returns

Excess return and Sharpe ratio.

## excess return
f_name <- "data/NO_Rf-OSEBX_2015-2023_monthly.csv"
rfr <- read_csv(f_name)
rfr <- rfr %>% mutate(yrmon=as.yearmon(Date))
ptf_ret3 <- ptf_rebal$returns %>% 
    data.frame(row.names = index(ptf_bh$returns)) %>% 
    as_tibble(rownames="Date") %>% 
    mutate(yrmon=as.yearmon(Date))
bind_cols(ptf_ret, ptf_ret3) # cross validate hand and computer calculations

## # A tibble: 96 × 4
##        ...1 Date       portfolio.returns yrmon    
##       <dbl> <chr>                  <dbl> <yearmon>
##  1 -0.123   2015-01-31          -0.123   Jan 2015 
##  2 -0.0430  2015-02-28          -0.0430  Feb 2015 
##  3  0.0224  2015-03-31           0.0224  Mar 2015 
##  4  0.247   2015-04-30           0.247   Apr 2015 
##  5  0.00336 2015-05-31           0.00336 May 2015 
##  6 -0.0367  2015-06-30          -0.0367  Jun 2015 
##  7 -0.00741 2015-07-31          -0.00741 Jul 2015 
##  8 -0.254   2015-08-31          -0.254   Aug 2015 
##  9 -0.0612  2015-09-30          -0.0612  Sep 2015 
## 10  0.0819  2015-10-31           0.0819  Oct 2015 
## # ℹ 86 more rows

Merge with risk free rate.

colnames(ptf_ret3)[2] <- "port_ret"
ptf_ret3 <- ptf_ret3 %>% 
    left_join(rfr[,-1], by="yrmon") %>% 
    mutate(excess_ret = port_ret-rf_1month)
ptf_ret3

## # A tibble: 96 × 7
##    Date     port_ret yrmon mkt_return_monthly rf_1month rf_annualized excess_ret
##    <chr>       <dbl> <yea>              <dbl>     <dbl>         <dbl>      <dbl>
##  1 2015-01… -0.123   Jan …            0.0350    0.0012        0.0145    -0.124  
##  2 2015-02… -0.0430  Feb …            0.0326    0.00113       0.0136    -0.0441 
##  3 2015-03…  0.0224  Mar …            0.00578   0.00127       0.0153     0.0211 
##  4 2015-04…  0.247   Apr …            0.0326    0.00123       0.0149     0.246  
##  5 2015-05…  0.00336 May …            0.00988   0.00117       0.0141     0.00219
##  6 2015-06… -0.0367  Jun …           -0.0257    0.00102       0.0123    -0.0377 
##  7 2015-07… -0.00741 Jul …            0.0156    0.00097       0.0117    -0.00838
##  8 2015-08… -0.254   Aug …           -0.0702    0.00095       0.0115    -0.255  
##  9 2015-09… -0.0612  Sep …           -0.0207    0.00083       0.0100    -0.0621 
## 10 2015-10…  0.0819  Oct …            0.0575    0.0008        0.00964    0.0811 
## # ℹ 86 more rows

Monthly mean and volatility

• arithmetic mean (AM) portfolio return \(\bar{r}_{p, 0:T}\)

\[ \text{Arithmetic } \bar{r}_{p, 0:T} = \frac{\sum_{t=1}^T r_{p,t}}{T} \]

• geometric mean (GM) portfolio return

\[ \text{Geometric } \bar{r}_{p, 0:T} = \left[\prod_{t=1}^T (1+r_{p,t}) \right] ^{\frac{1}{T}}-1 \] Written as \(\bar{r}_p\) in short, representing the arithmetic/geometric mean return of the portfolio from time \(0\) to time \(T\).

• Arithmetic average sometimes cannot precisely reflect historical gains and losses. Suppose the original capital of \(\$100\) is invested over a two-month period with 10% return in the first month and \(10\%\) loss in the second month.

  • The arithmetic average return is \((10\% - 10\%)/2=0\%\), which implies that there is no change in the \(\$100\) invested. However, the actual gain/loss for the investment grows from \(\$100\) to \(\$110\) in the first month, and drops 10% from \(\$110\) to \(\$99\) in the second month, indicating an actual loss of \(\$1\).

  • The geometric return is \((1.1\times0.9)^{0.5}-1\approx -0.5\%\), indicating a loss of \(-0.5\%\) per month. The geometric return that incorporates the compounding effect of growth is a better indication of historical performance.

• Geometric mean is smaller than or equal to arithmetic mean and the equality holds if and only if \(r_{p,1}=r_{p,2}=\cdots=r_{p,T}\). This can be proven by using Jensen’s inequality that says, for any concave function \(g(x)\),

\[ E[g(x)] \le g(E[x]). \] And \(\ln(x)\) is a concave function.

• Common practice is to report geometric mean portfolio return.

# arithmetic mean monthly returns
mean(ptf_ret3$port_ret)

## [1] 0.01681626

# geometric mean
mean.geometric(ptf_ret3$port_ret) 

## [1] 0.009155608

prod(1+ptf_ret3$port_ret)^(1/96)-1

## [1] 0.009155608

# mean excess return
mean(ptf_ret3$excess_ret) 

## [1] 0.01599762

mean.geometric(ptf_ret3$excess_ret)

## [1] 0.008326127

# standard deviation
sd(ptf_ret3$port_ret)

## [1] 0.1278693

Portfolio return and standard deviation are usually published as annualized.

• annualized arithmetic return assuming \(r_{p,t}\) being monthly return

\[ r_{p}^A = \frac{12}{T}\sum_{t=1}^T r_{p,t} = 12\times \bar{r}_p \]

• annualized geometric return

\[ r_{p}^A = \left[\prod_{t=1}^T (1+r_{p,t}) \right]^{\frac{12}{T}}-1 = (1+\bar{r}_p)^{12}-1 \]

• annualized standard deviation.
  Note that in Finance, the standard deviation of returns is usually called volatility.

\[ \begin{aligned} \sigma_{p}^A &= \sqrt{12} \sigma_p \\ \sigma_p^2 &= \frac{1}{T-1} \sum_{t=1}^T (r_{p,t}-\bar{r}_p)^2 \end{aligned} \]

It is possible to use Return.annualized, StdDev.annualized from the PerformanceAnalytics package to get the annualized statistics, but we need to convert the portfolio return as an xts object.

# convert to xts
ptf_xts <- xts(ptf_ret3[,-c(1,3)], order.by=ymd(ptf_ret3$Date))
ptf_xts %>% 
    data.frame %>% 
    knitr::kable(digits = 5, caption = "Portfolio return in `xts`") %>%
    kable_styling(bootstrap_options = c("striped", "hover"), full_width = F, latex_options="scale_down") %>% 
    scroll_box(height = "500px")

Table 3: Portfolio return in xts

	port_ret	mkt_return_monthly	rf_1month	rf_annualized	excess_ret
2015-01-31	-0.12299	0.03498	0.00120	0.01450	-0.12419
2015-02-28	-0.04300	0.03262	0.00113	0.01364	-0.04413
2015-03-31	0.02236	0.00578	0.00127	0.01535	0.02109
2015-04-30	0.24728	0.03256	0.00123	0.01486	0.24605
2015-05-31	0.00336	0.00988	0.00117	0.01413	0.00219
2015-06-30	-0.03671	-0.02566	0.00102	0.01231	-0.03773
2015-07-31	-0.00741	0.01561	0.00097	0.01170	-0.00838
2015-08-31	-0.25381	-0.07016	0.00095	0.01146	-0.25476
2015-09-30	-0.06123	-0.02072	0.00083	0.01001	-0.06206
2015-10-31	0.08191	0.05749	0.00080	0.00964	0.08111
2015-11-30	-0.04805	0.02198	0.00108	0.01304	-0.04913
2015-12-31	-0.09121	-0.02942	0.00087	0.01049	-0.09208
2016-01-31	-0.11146	-0.08083	0.00085	0.01025	-0.11231
2016-02-29	-0.21675	0.02063	0.00080	0.00964	-0.21755
2016-03-31	0.44835	0.00917	0.00070	0.00843	0.44765
2016-04-30	0.27062	0.04938	0.00069	0.00831	0.26993
2016-05-31	-0.04198	0.01819	0.00068	0.00819	-0.04266
2016-06-30	-0.08631	-0.02341	0.00073	0.00880	-0.08704
2016-07-31	0.07508	0.01621	0.00068	0.00819	0.07440
2016-08-31	-0.10164	0.01028	0.00073	0.00880	-0.10237
2016-09-30	0.11098	0.00608	0.00082	0.00988	0.11016
2016-10-31	0.20636	0.02491	0.00078	0.00940	0.20558
2016-11-30	0.02225	0.02888	0.00107	0.01292	0.02118
2016-12-31	0.47661	0.04148	0.00087	0.01049	0.47574
2017-01-31	0.20303	0.01353	0.00059	0.00710	0.20244
2017-02-28	-0.10885	-0.00411	0.00076	0.00916	-0.10961
2017-03-31	0.04408	-0.00351	0.00068	0.00819	0.04340
2017-04-30	0.02603	0.01426	0.00069	0.00831	0.02534
2017-05-31	-0.04096	0.01818	0.00072	0.00867	-0.04168
2017-06-30	-0.01916	-0.01656	0.00060	0.00722	-0.01976
2017-07-31	0.14920	0.04857	0.00056	0.00674	0.14864
2017-08-31	-0.11115	0.01005	0.00057	0.00686	-0.11172
2017-09-30	0.10056	0.05842	0.00056	0.00674	0.10000
2017-10-31	-0.15193	0.03047	0.00052	0.00626	-0.15245
2017-11-30	-0.06368	-0.01254	0.00056	0.00674	-0.06424
2017-12-31	0.08434	0.02211	0.00056	0.00674	0.08378
2018-01-31	0.11451	-0.00422	0.00063	0.00759	0.11388
2018-02-28	-0.04349	0.01080	0.00078	0.00940	-0.04427
2018-03-31	-0.05454	-0.01763	0.00079	0.00952	-0.05533
2018-04-30	0.13348	0.06785	0.00076	0.00916	0.13272
2018-05-31	0.11310	0.01809	0.00068	0.00819	0.11242
2018-06-30	-0.00079	0.00413	0.00064	0.00771	-0.00143
2018-07-31	-0.08976	0.01963	0.00065	0.00783	-0.09041
2018-08-31	-0.09132	0.01148	0.00068	0.00819	-0.09200
2018-09-30	0.02920	0.03482	0.00082	0.00988	0.02838
2018-10-31	-0.10242	-0.05180	0.00082	0.00988	-0.10324
2018-11-30	-0.03801	-0.03224	0.00095	0.01146	-0.03896
2018-12-31	-0.15960	-0.07145	0.00089	0.01073	-0.16049
2019-01-31	0.03741	0.04484	0.00084	0.01013	0.03657
2019-02-28	0.10642	0.03588	0.00086	0.01037	0.10556
2019-03-31	-0.06699	-0.00251	0.00104	0.01255	-0.06803
2019-04-30	0.11346	0.02062	0.00103	0.01243	0.11243
2019-05-31	-0.16822	-0.03272	0.00105	0.01267	-0.16927
2019-06-30	0.09829	0.01472	0.00114	0.01377	0.09715
2019-07-31	0.07034	-0.00635	0.00114	0.01377	0.06920
2019-08-31	-0.08935	0.00250	0.00117	0.01413	-0.09052
2019-09-30	-0.04028	0.02939	0.00130	0.01571	-0.04158
2019-10-31	-0.05484	0.01291	0.00140	0.01693	-0.05624
2019-11-30	-0.08281	0.00490	0.00143	0.01730	-0.08424
2019-12-31	0.13906	0.03213	0.00141	0.01705	0.13765
2020-01-31	-0.08287	-0.01894	0.00137	0.01656	-0.08424
2020-02-29	0.00793	-0.09143	0.00135	0.01632	0.00658
2020-03-31	-0.27544	-0.14830	0.00052	0.00626	-0.27596
2020-04-30	0.09378	0.09614	0.00023	0.00276	0.09355
2020-05-31	0.16271	0.02794	0.00008	0.00096	0.16263
2020-06-30	0.02313	-0.00195	0.00018	0.00216	0.02295
2020-07-31	0.12056	0.03900	0.00014	0.00168	0.12042
2020-08-31	0.11351	0.03998	0.00008	0.00096	0.11343
2020-09-30	0.00395	-0.00369	0.00020	0.00240	0.00375
2020-10-31	-0.01029	-0.05168	0.00036	0.00433	-0.01065
2020-11-30	0.20017	0.14601	0.00023	0.00276	0.19994
2020-12-31	0.01627	0.04684	0.00029	0.00349	0.01598
2021-01-31	0.12916	-0.00726	0.00026	0.00312	0.12890
2021-02-28	-0.01129	0.04155	0.00026	0.00312	-0.01155
2021-03-31	0.20667	0.05143	0.00020	0.00240	0.20647
2021-04-30	0.06364	0.01661	0.00017	0.00204	0.06347
2021-05-31	-0.02243	0.03382	0.00015	0.00180	-0.02258
2021-06-30	-0.03342	0.00728	0.00011	0.00132	-0.03353
2021-07-31	-0.03907	0.01188	0.00012	0.00144	-0.03919
2021-08-31	-0.05680	0.00668	0.00017	0.00204	-0.05697
2021-09-30	0.09416	0.01879	0.00037	0.00445	0.09379
2021-10-31	-0.00057	0.02532	0.00040	0.00481	-0.00097
2021-11-30	-0.07111	-0.00965	0.00057	0.00686	-0.07168
2021-12-31	0.11394	0.01707	0.00067	0.00807	0.11327
2022-01-31	0.03616	-0.02212	0.00073	0.00880	0.03543
2022-02-28	-0.15731	0.02325	0.00074	0.00892	-0.15805
2022-03-31	-0.00949	0.04923	0.00087	0.01049	-0.01036
2022-04-30	-0.08801	-0.01707	0.00081	0.00976	-0.08882
2022-05-31	0.19465	0.03837	0.00083	0.01001	0.19382
2022-06-30	-0.10218	-0.09097	0.00114	0.01377	-0.10332
2022-07-31	0.10150	0.07084	0.00135	0.01632	0.10015
2022-08-31	0.02237	-0.00360	0.00169	0.02047	0.02068
2022-09-30	-0.10155	-0.11683	0.00225	0.02734	-0.10380
2022-10-31	0.17660	0.06589	0.00228	0.02771	0.17432
2022-11-30	0.00604	0.03855	0.00270	0.03289	0.00334
2022-12-31	0.06230	-0.02602	0.00253	0.03079	0.05977

# Compute the annualized arithmetic mean
Return.annualized(ptf_xts$port_ret, geometric = FALSE)

##                    port_ret
## Annualized Return 0.2017952

mean(ptf_ret3$port_ret)*12

## [1] 0.2017952

# Compute the annualized geometric mean
Return.annualized(ptf_xts$port_ret, geometric = TRUE)

##                    port_ret
## Annualized Return 0.1155721

prod(1+ptf_ret3$port_ret)^(12/96)-1

## [1] 0.1155721

# Compute the annualized standard deviation
StdDev.annualized(ptf_xts$port_ret)

##                                port_ret
## Annualized Standard Deviation 0.4429521

sd(ptf_ret3$port_ret)*sqrt(12)

## [1] 0.4429521

Histogram of portfolio returns.

# histogram to check normality
chart.Histogram(ptf_xts$port_ret, methods = c("add.density", "add.normal"))

Skewness and Kurtosis

Skewness \(\mu_3\) is the third (normalized) moment about the mean.
It measures the asymmetry of the distribution, with symmetric distribution having \(\mu_3=0\).

• \(\mu_3>0\) means positively skewed, i.e., long right tail.
• \(\mu_3<0\) means negatively skewed, i.e., long left tail.

\[ \mu_3 = \frac{1}{T} \frac{\sum_{t=1}^T (r_{p,t}-\bar{r}_p)^3}{\sigma_p^3} \]

Fig. 4: Diagram of Skewness.

# skewness
skewness(ptf_xts$port_ret)

## [1] 0.7779903

Kurtosis \(K_p\) is the fourth (normalized) moment about the mean.
It measures the tail behavior in comparison with the normal distribution.

\[ \mu_4 = \frac{1}{T} \frac{\sum_{t=1}^T (r_{p,t}-\bar{r}_p)^4}{\sigma_p^4} \]

The kurtosis for any normal distribution is three. For this reason, we subtract three from \(\mu_4\) to get the “excess kurtosis”.

• \(\mu_4-3>0\) indicate a heavy-tailed distribution with higher chances of outliers.
  • Note that we cannot infer from kurtosis the shape of the peak. \(\mu_4-3>0\) can associate with either a falttened (e.g., \(t\)-distribution, see Fig. 5) or a pointy peak (Laplace distribution, see Fig. 6).
• \(\mu_4-3<0\) indicate a light-tailed distribution with lower chances of outliers.

Fig. 5: Examples of heavy-tailed distributions.

\(t\)-distribution has higher kurtosis than normal distributions.

• Meaning that \(t\)-distribution has a higher probability of obtaining values that are far from the mean than a normal distribution.
• It is less peaked in the center and higher in the tails than normal distribution.
• As the degree of freedom increases, \(t\)-distribution approximates to normal distribution, kurtosis decreases and approximates to 3.

set.seed(125)
rnorm(1000) %>% moments::kurtosis()

## [1] 3.037806

rt(1000, df=1) %>% moments::kurtosis()

## [1] 503.3928

rt(1000, df=2) %>% moments::kurtosis()

## [1] 61.25994

rt(1000, df=10) %>% moments::kurtosis()

## [1] 4.303517

rt(1000, df=30) %>% moments::kurtosis()

## [1] 3.646411

Fig. 6: Laplace distribution. The dotted green curve shows a normal distribution. The blue curve shows a Laplace distribution with kurtosis of 6.54. On the far left and right sides of the distribution—the tails—the space below the Laplace distribution curve (blue) is slightly thicker than the space below the normal distribution curve (green). This is an example of a heavy-tailed distribution yet with a sharper peak.

Distinguish from standard deviation/variance.

Standard deviation and kurtosis are both measures of the variability of a distribution, but they are not directly related.
Two distributions with identical means and standard deviations can have very different shapes, and kurtosis is one of the measures of that difference. It looks at how much of the ‘weight’ of the distribution (recall that the total weight, or the area under the curve, is 1) is sitting in the tails as opposed to the middle of the distribution.

• Standard deviation is useful for measuring the spread.
• Kurtosis focuses on detecting outliers.

PerformanceAnalytics::kurtosis(x) returns excess kurtosis by default. Need to specify method="moment" in order to get the original kurtosis.

moments::kurtosis(x) returns the original kurtosis. Need to be compared with 3 on your own.

# kurtosis
kurtosis(ptf_xts$port_ret)

## [1] 1.69699

# summary statistic
returns_statistics <- table.Stats(ptf_xts$port_ret)
returns_statistics

##                 port_ret
## Observations     96.0000
## NAs               0.0000
## Minimum          -0.2754
## Quartile 1       -0.0680
## Median            0.0014
## Arithmetic Mean   0.0168
## Geometric Mean    0.0092
## Quartile 3        0.1008
## Maximum           0.4766
## SE Mean           0.0131
## LCL Mean (0.95)  -0.0091
## UCL Mean (0.95)   0.0427
## Variance          0.0164
## Stdev             0.1279
## Skewness          0.7780
## Kurtosis          1.6970

Sharpe ratio

The Sharpe ratio was first introduced by Sharpe (1966) as the (arithmetic) average portfolio excess return over the sample period divided by the standard deviation of the portfolio return over that period. Sharpe ratio measure the portfolio’s excess return per unit of standard deviation.

\[ \begin{aligned} \text{Sharpe Ratio} &= S_p = \frac{\bar{r}_p-\bar{r}_f}{\sigma_p} \\ \text{Annualized Sharpe Ratio} &= S_p^A = \frac{\bar{r}_p-\bar{r}_f}{\sigma_p} \times \sqrt{12} \end{aligned} \] where \(\bar{r}_f\) is the (arithmetic) average return of the risk free rate for the time period under evaluation.

Note that consistency is required between portfolio and risk free rate for

• time period.
• frequency, i.e., monthly or yearly.

Also note that

• The common practice is to use arithmetic mean in the numerator. This is referred to as “Modified Sharpe” ratio.

• When the numerator uses the geometric mean, the ratio is referred to as “Geometric Sharpe” ratio to distinguish. (Not commonly used.)

# monthly Sharpe ratio
with(ptf_ret3, mean(excess_ret)/sd(port_ret)) 

## [1] 0.1251092

Alternatively, we can use the PerformanceAnalytics::SharpeRatio function.
Again in order to use it, we must provide the portfolio return as an xts object.

# use arithmetic return
SharpeRatio(R=ptf_xts$port_ret, Rf=ptf_xts$rf_1month)

##                                   port_ret
## StdDev Sharpe (Rf=0.1%, p=95%): 0.12510917
## VaR Sharpe (Rf=0.1%, p=95%):    0.10094484
## ES Sharpe (Rf=0.1%, p=95%):     0.08362338

# Compute the annualized Sharpe ratio
with(ptf_ret3, mean(excess_ret)/sd(port_ret))*sqrt(12)

## [1] 0.4333909

ann_sharpe <- Return.annualized(ptf_xts$excess_ret, geometric = FALSE)/StdDev.annualized(ptf_xts$port_ret)
ann_sharpe

##                   excess_ret
## Annualized Return  0.4333909

# SharpeRatio.annualized use geometric return as default
SharpeRatio.annualized(R=ptf_xts$port_ret, Rf=ptf_xts$rf_1month, geometric=FALSE)

##                                  port_ret
## Annualized Sharpe Ratio (Rf=1%) 0.4333909

# Geometric Sharpe is the default
SharpeRatio.annualized(R=ptf_xts$port_ret, Rf=ptf_xts$rf_1month, geometric=TRUE)

##                                  port_ret
## Annualized Sharpe Ratio (Rf=1%) 0.2361843

# verify with hand calculation
Return.annualized(ptf_xts$excess_ret, geometric = TRUE)/StdDev.annualized(ptf_xts$port_ret)

##                   excess_ret
## Annualized Return  0.2361843

Information ratio

While the Sharpe ratio measures the excess return as the difference between the return of the portfolio and the risk-free rate, the information ratio, on the other hand, measures the portfolio performance against a comparable benchmark rather than the risk-free rate.

The portfolio return in excess of the return of the benchmark is known as the active return, and the variability of the active return is known as the active risk or tracking error of the portfolio.

The information ratio is the active return per unit of active risk.

\[ \begin{aligned} \text{Information Ratio} &= \text{IR}_p = \frac{\bar{r}_p-\bar{r}_m}{\sigma_{p-m}} \\ \text{Annualized Informatione Ratio} &= \text{IR}_p^A = \frac{\bar{r}_p-\bar{r}_m}{\sigma_{p-m}} \times \sqrt{12} \end{aligned} \] where

• \(\bar{r}_m\) is the average return for the benchmark; and
• \(\sigma_{p-m}\) is the standard deviation for the active return, i.e., return of portfolio in excess of the benchmark return.

Beta coefficient-adjusted measures

Sharpe and Information ratios use standard deviation as the measure of portfolio risk. An alternative risk measure is the beta coefficient based on the CAPM model.

Recall that the beta coefficient can be estimated using the follow regression

\[ r_{i,\color{red}{t}} - r_{f,\color{red}{t}} = \alpha_i + \beta_i (r_{m,\color{red}{t}}-r_{f,\color{red}{t}}) + \varepsilon_{i,\color{red}{t}} . \] The OLS \(\beta_i\) estimate can be expressed as

\[ \hat{\beta}_i = \frac{\text{Cov}(R_{m}^e, R_{i}^e)}{\text{Var}(R_{i}^e)} \] where

• \(R^e_i=r_i-r_f\) stands for the excess return on asset \(i\), and
• \(R^e_m=r_m-r_f\) stands for the excess return on the market index.

Treynor ratio is a variant of Sharpe ratio. It substitute the standard deviation in Sharpe ratio with the beta coefficient.

The Treynor ratio is computed as

\[ \text{Treynor Ratio}_p = \frac{\bar{r}_p-\bar{r}_f}{\beta_p} \] Note that

• the only relevant risk in the computation of the Treynor ratio is systematic risk. It assumes that the portfolio is fully diversified. Appropriate for diversified equity funds, the element of unsystematic risk would be negligible.

• Sharpe ratio assumes that the relevant risk is total risk (systematic and unsystematic risk), and it measures excess return per unit of total risk. It is appropriate for the portfolio that is less diversified.

• Jensen’s alpha also assumes that the relevant risk is systematic risk.

Jensen’s alpha measures the average return on the portfolio in excess of what predicted by the CAPM, given the portfilio’s beta and the average market return.

\[ \alpha_p = \bar{r}_p - \left[\bar{r}_f + \beta_p(\bar{r}_m-\bar{r}_f)\right] \]

Exercise

Given the following information about the return on a portfolio, the market index and risk free returns.

Year	Portfolio	Market index	Risk free rate
1	14%	12%	7%
2	10	7	7.5
3	19	20	7.7
4	-8	-2	7.5
5	23	12	8.5
6	28	23	8
7	20	17	7.3
8	14	20	7
9	-9	-5	7.5
10	19	16	8
Average	13%	12%	7.6%
Standard deviation	12.39%	9.43%	0.47%
Cov(\(r_p\), \(r_m\))	0.0107

 

1. Calculate the beta.

\[ \beta_p = \frac{\text{Cov}(r_p, r_m)}{\text{Var}(r_m)} = \frac{0.0107}{9.43\%^2} = 1.203 \]

2. Calculate the Sharpe ratio.

\[ S_p = \frac{\bar{r}_p-\bar{r}_f}{\sigma_p} = \frac{(13-7.6)\%}{12.39\%} = 0.436 \]

3. Calculate the Treynor ratio.

\[ T_p = \frac{\bar{r}_p-\bar{r}_f}{\beta_p} = \frac{(13-7.6)\%}{1.203} = 0.0449 \]

4. Calculate Jensen’s Alpha.

\[ \alpha_p = \bar{r}_p - \left[\bar{r}_f + \beta_p(\bar{r}_m-\bar{r}_f)\right] = 13\% - 7.6\% - 1.203\times (12-7.6)\% = 0.107\% \]

 

M-squared

The M-squared (\(M^2\)) measure is first introduced by Franco Modigliani. \(M^2\) provides a direct comparison between the leverage-adjusted portfolio and the market portfolio.

A managed portfolio \(p\) is mixed with a position in the risk free asset to make the “leverage-adjusted” (or “adjusted” in short) portfolio have the same volatility as the market. Suppose the managed portfolio p has a total variability equal to \(1.5\times \sigma_m\). The “adjusted” portfolio \(p^*\) is found by investing a weight \(w\) in \(p\) and a weight \((1 − w)\) in the risk free asset, such that the portfolio has the same standard deviation as the market:

\[ \begin{aligned} r_{p^*} &= w\, r_p + (1-w)\, r_f \\ \sigma_{p^*} &= w\sigma_p + (1-w)\sigma_{r_f} = w\sigma_p + (1-w)\cdot 0 = w\sigma_p \end{aligned} \] Let

\[ \sigma_{p^*} = \sigma_m , \] we have

\[ w\sigma_p = \sigma_m \Rightarrow w=\frac{\sigma_m}{\sigma_p} = \frac{\sigma_m}{1.5\sigma_m} = \frac{2}{3} \] By investing two thirds in \(p\) and one third in the risk free asset \(r_f\), achieve the same volatility as the market.

Since \(P^∗\) and \(m\) have the same volatility, we may compare their returns simply by calculating the difference:

\[ M_p^2 = r_{p^*} - r_m . \]

Downside risk measures

When the return distribution is asymmetric (skewed), investors use additional risk measures that focus on describing the potential losses.

• The Semi-Deviation is the calculation of the variability of returns below the mean return. \[ \begin{aligned} \text{Semi } \sigma_p &= \sqrt{\frac{1}{n}\textstyle \sum_{t=1}^T \big[\min(r_{p,t}-\bar{r}_p, 0)\big]^2 } \\ \text{Annualized Semi } \sigma_p^A &= \sqrt{\frac{1}{n}\textstyle \sum_{t=1}^T \big[\min(r_{p,t}-\bar{r}_p, 0)\big]^2 } \times \sqrt{12} \end{aligned} \] where \(n\) is either the number of observations of the entire series or the number of observations in the subset of the series falling below the average.

• Value-at-Risk (VaR) measures the potential loss in value of a risky asset or portfolio over a defined period for a given confidence interval. It corresponds to a probability \(p\), which is a confidence level.

  A \(p\) VaR means that the probability of a loss greater than VaR is (at most) \((p)\) while the probability of a loss less than VaR is (at least) \(1-p\).

  • For example, a one-day \(5\%\) VaR of \(\$1\) million implies the portfolio has a \(5\%\) chance that the value of the asset drops more than \(\$1\) million over one day. In other words, there is a \(95\%\) probability that the asset makes a profit or lose less than \(\$1\) million over a day.

  • Different conventions are used for \(p\). You can tell by the magnitude whether it refers to a confidence level or a significance level. For instance, sometimes you see \(95\%\) VaR (confidence level) and \(5\%\) VaR (significance level).

  • To obtain a sample estimate of \(5\%\) VaR, we sort the observations from high to low. The VaR is the return at the \(5\)th percentile of the empirical sample distribution.

  • More formally, \[ \text{VaR}(\alpha) = -F^{-1}(\alpha) \] where \(F(\cdot)\) is the cdf of the portfolio return \(r_p\).

• The expected shortfall (ES) is the expected value of the loss, given the loss is greater than the VaR.

\[ \text{ES}(\alpha) = E[r_p \vert r_p \leq F^{-1}(\alpha)] \] - VaR is the most optimistic measure of worst-case scenarios as it takes the smallest loss of all these cases, while ES informs the magnitudes of potential losses given that the loss is greater than the VaR.

## downside risk measures
# Calculate the SemiDeviation
SemiDeviation(ptf_xts$port_ret)

##                  port_ret
## Semi-Deviation 0.08114918

sd(ptf_xts$port_ret)

## [1] 0.1278693

# Calculate the value at risk, p is the confidence level
VaR(ptf_xts$port_ret, p=.95) # 95% VaR

##       port_ret
## VaR -0.1584788

VaR(ptf_xts$port_ret, p=.99) # 99% VaR

##       port_ret
## VaR -0.2278215

# Calculate the expected shortfall
ES(ptf_xts$port_ret, p=.95)

##      port_ret
## ES -0.1913056

ES(ptf_xts$port_ret, p=.99)

##      port_ret
## ES -0.2278215

The other popular downside risk estimate, the maximum drawdown (MDD) of the portfolio, measures the largest loss from peak to trough over the examination period.

The maximum drawdown of the portfolio indicates the maximum possible loss investors have ever experienced over the examination period.

\[ MDD = \frac{LP-PV}{PV} \]

• where \(LP\) is the lowest value after peak value, this is also called “Trough Value”;
• \(PV\) is the peak value.

# Table of drawdowns
maxDrawdown(ptf_xts$port_ret)

## [1] 0.5840139

table.Drawdowns(ptf_xts$port_ret)

##         From     Trough         To   Depth Length To Trough Recovery
## 1 2018-06-30 2020-03-31 2021-03-31 -0.5840     34        22       12
## 2 2015-06-30 2016-02-29 2016-12-31 -0.5637     19         9       10
## 3 2022-02-28 2022-04-30 2022-12-31 -0.2388     11         3        8
## 4 2017-08-31 2017-11-30 2018-05-31 -0.2232     10         4        6
## 5 2015-01-31 2015-02-28 2015-04-30 -0.1607      4         2        2

# Plot of drawdowns
chart.Drawdown(ptf_xts$port_ret)

## rolling annualized mean and volatility
chart.RollingPerformance(R = ptf_xts$port_ret, width = 12, FUN = "Return.annualized")

chart.RollingPerformance(R = ptf_xts$port_ret, width = 12, FUN = "StdDev.annualized")

chart.RollingPerformance(R = ptf_xts$port_ret, width = 12, FUN = "SharpeRatio.annualized", Rf=ptf_xts$excess_ret)

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References

Sharpe, William F. 1966. “Mutual Fund Performance.” The Journal of Business 39: 119–38. http://www.jstor.org/stable/2351741.