CFA Problem 1 — Expected Risk Premium

Given $100,000 to invest, with the following outcomes:

Action Probability Expected Return
Invest in equities 0.6 $50,000
Invest in equities 0.4 −$30,000
Invest in T-bill 1.0 $5,000 
# Equity expected return
p_up   <- 0.6;  r_up   <- 50000
p_down <- 0.4;  r_down <- -30000

E_equity <- p_up * r_up + p_down * r_down
cat("Expected return of equities: $", E_equity, "\n")
## Expected return of equities: $ 18000
# Risk-free return
rf <- 5000
cat("Risk-free (T-bill) return:   $", rf, "\n")
## Risk-free (T-bill) return:   $ 5000
# Risk premium
risk_premium <- E_equity - rf
cat("Expected Risk Premium:       $", risk_premium, "\n")
## Expected Risk Premium:       $ 13000

The expected risk premium is $1.3^{4}, which represents the additional compensation an investor expects for bearing the uncertainty of equities over the guaranteed T-bill return.

Chapter 5, Problem 12 — Six Fama-French Portfolios (1930–2018)

We download the value-weighted monthly returns for the 6 portfolios formed on size × book-to-market (2 × 3) from Ken French’s data library, then split the sample in half and compare summary statistics.

# ── packages ──────────────────────────────────────────────────────────────────
if (!requireNamespace("moments", quietly = TRUE)) install.packages("moments")
library(moments)   # skewness / kurtosis

# ── 1. Read data ───────────────────────────────────────────────────────────────
# Download: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
# File: "6 Portfolios Formed on Size and Book-to-Market (Value Weight Monthly Returns)"
# After downloading, unzip and place the .txt file in your working directory.
# The file is named something like: "6_Portfolios_2x3.CSV"

# For illustration we read a locally saved CSV (skip header rows as needed).
# Adjust the filename and skip= argument to match your downloaded file.

# --- REPLACE THIS PATH with your actual file location ---
# ff6 <- read.csv("6_Portfolios_2x3.CSV", skip = 15, header = TRUE,
#                 stringsAsFactors = FALSE)

# ── Simulated stand-in (remove once you have the real data) ───────────────────
set.seed(42)
dates   <- seq(as.Date("1930-01-01"), as.Date("2018-12-01"), by = "month")
n       <- length(dates)
portnames <- c("SL","SM","SH","BL","BM","BH")   # Small/Big × Low/Mid/High BtM
ff6 <- as.data.frame(matrix(rnorm(n * 6, mean = 0.8, sd = 5), nrow = n))
colnames(ff6) <- portnames
ff6 <- cbind(Date = format(dates, "%Y%m"), ff6)

# ── 2. Filter Jan 1930 – Dec 2018 ─────────────────────────────────────────────
ff6$Date <- as.integer(ff6$Date)
ff6 <- ff6[ff6$Date >= 193001 & ff6$Date <= 201812, ]

# ── 3. Split in half ──────────────────────────────────────────────────────────
mid     <- floor(nrow(ff6) / 2)
first_h <- ff6[1:mid, ]
second_h <- ff6[(mid + 1):nrow(ff6), ]

cat("First half: ", format(min(ff6$Date[1:mid])), "–",
    format(max(ff6$Date[1:mid])), " (", mid, " months)\n")
## First half:  193001 – 197406  ( 534  months)
cat("Second half:", format(min(ff6$Date[(mid+1):nrow(ff6)])), "–",
    format(max(ff6$Date[(mid+1):nrow(ff6)])), " (",
    nrow(ff6) - mid, " months)\n\n")
## Second half: 197407 – 201812  ( 534  months)
# ── 4. Summary statistics function ───────────────────────────────────────────
port_stats <- function(df, label) {
  nums <- df[, portnames]
  out  <- data.frame(
    Portfolio = portnames,
    Mean      = round(colMeans(nums), 4),
    SD        = round(apply(nums, 2, sd), 4),
    Skewness  = round(apply(nums, 2, skewness), 4),
    Kurtosis  = round(apply(nums, 2, kurtosis), 4)
  )
  cat("\n===", label, "===\n")
  print(out, row.names = FALSE)
  invisible(out)
}

s1 <- port_stats(first_h,  "First Half")
## 
## === First Half ===
##  Portfolio   Mean     SD Skewness Kurtosis
##         SL 0.6858 4.9475   0.0843   3.0556
##         SM 0.7131 4.8596  -0.0043   2.9676
##         SH 1.0516 5.1589  -0.0812   3.1717
##         BL 0.8424 5.0819   0.0850   2.8306
##         BM 1.0334 5.0181  -0.0620   3.0097
##         BH 0.8412 5.1857   0.0971   2.6880
s2 <- port_stats(second_h, "Second Half")
## 
## === Second Half ===
##  Portfolio   Mean     SD Skewness Kurtosis
##         SL 0.5898 5.1024  -0.1171   3.2372
##         SM 0.8123 4.9110   0.0503   2.9193
##         SH 0.4629 5.1083   0.0143   2.9551
##         BL 0.2898 5.0656  -0.0444   2.9041
##         BM 0.9703 5.0862  -0.1285   2.9285
##         BH 0.8284 5.0330  -0.1395   2.8291

Comparison Table

Split-Halves Statistics for 6 FF Portfolios
Portfolio Mean_H1 SD_H1 Skewness_H1 Kurtosis_H1 Mean_H2 SD_H2 Skewness_H2 Kurtosis_H2
BH 0.8412 5.1857 0.0971 2.6880 0.8284 5.0330 -0.1395 2.8291
BL 0.8424 5.0819 0.0850 2.8306 0.2898 5.0656 -0.0444 2.9041
BM 1.0334 5.0181 -0.0620 3.0097 0.9703 5.0862 -0.1285 2.9285
SH 1.0516 5.1589 -0.0812 3.1717 0.4629 5.1083 0.0143 2.9551
SL 0.6858 4.9475 0.0843 3.0556 0.5898 5.1024 -0.1171 3.2372
SM 0.7131 4.8596 -0.0043 2.9676 0.8123 4.9110 0.0503 2.9193

Interpretation

  • If both halves share very similar means, SDs, skewness, and kurtosis, the returns are consistent with being drawn from the same distribution.
  • Pronounced differences (especially in the tails — skewness / kurtosis) suggest structural changes over time (e.g., the Great Depression in the first half vs. post-war expansion in the second).
  • Equity return distributions are typically negatively skewed and leptokurtic (fat tails), and this may differ between the two sub-periods.

Note: Replace the simulated data block with the actual Ken French data file before knitting and publishing to RPubs.