Quantitative Methods
Interest Rates and Return
Measurement
Interest Rates and Return Measurement
Interest Rates
Uses of interest rates
As required rates of return for investments
As discount rates to change future values to present values
As an opportunity cost of current consumption
Risk-free rates
A risk-free rate is a theoretical rate with no default risk
A real risk-free rate has no expected inflation or default risk
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Interest Rates and Return Measurement
Nominal Interest Rates
Nominal risk-free rate
(1 + nominal RFR) = (1 + real RFR) × (1 + expected inflation)
Approximation
Nominal risk-free rate = real risk-free rate + expected inflation
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Interest Rates and Return Measurement
Required Returns
Securities may have one or more types of risks. Each increases the
required return on the security:
Default risk—a borrower may not make the promised payments
Liquidity risk—receiving less than fair value; unable to sell
Maturity risk—longer-dated bonds have higher interest rate risk
Risk premiums
Each factor is associated with a risk premium required by investors
A required return is sum of the real risk-free rate plus the relevant
risk premiums
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Interest Rates and Return Measurement
Components of Interest Rates
Required nominal interest rate on a security =
real risk-free rate nominal risk-free rate
+ expected inflation
+ default risk premium
+ liquidity risk premium risk premiums
+ maturity risk premium
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Interest Rates and Return Measurement
Holding Period Returns
Investors may hold an investment for any chosen period. The HPR is
the return from the start to the end of this period.
Holding period return (HPR)
HPR no income = (end value / beginning value) – 1
HPR with income = [(end value + dividend) / beginning value] – 1
Example:
A stock is valued at $20 at T0, and $22 at T1 and pays a $1 dividend
over the period. Calculate the HPR.
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Interest Rates and Return Measurement
Holding Period Return Solution
HPR = [(end value + dividend) / beginning value] – 1
= [(22 + 1) / 20] – 1
= 15%
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Interest Rates and Return Measurement
Compounding Holding Period Returns
Holding period returns can be linked together over multiple
time periods:
HPR 3 yrs = (1 + HPRyr 1)(1 + HPRyr 2)(1 + HPRyr 3) – 1
Annualized returns
Annualized returns are commonly used rather than longer
time period returns
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Interest Rates and Return Measurement
Average Returns
The arithmetic mean return is the simple average of a series of
periodic returns:
Arithmetic mean = R1 R2 R3 ... Rn
N
Example:
For the last three years, the returns for Acme Corporation common
stock have been –9.34%, 23.45%, and 8.92%. Calculate the
arithmetic mean return over the three-year period.
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Interest Rates and Return Measurement
Arithmetic Mean Solution
Arithmetic mean:
R1 R2 R3 ... Rn
=
N
9.34 23.45 8.92
=
3
= 7.68%
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Interest Rates and Return Measurement
Geometric Mean
The geometric mean return is a compounded average return over
a period.
GM = n (1+ R1 ) × (1+ R2 )× (1+ R3 ) × ...× (1+ Rn ) – 1
Example:
For the last three years, the returns for Acme Corporation common
stock have been –9.34%, 23.45%, and 8.92%. Calculate the
compound annual rate of return over the three-year period.
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Interest Rates and Return Measurement
Geometric Mean Solution
GM = n (1 + R1 ) × (1 + R 2 ) × (1 + R3 ) × ...× (1 + Rn ) – 1
= 3 (1– 0.0934) × (1 + 0.2345) × (1 + 0.0892) – 1
3
= 1.21903 –1
= 6.825%
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Interest Rates and Return Measurement
Arithmetic vs. Geometric Mean
The geometric mean will always be less than or equal to the
arithmetic mean.
The greater dispersion of return observations, the greater the
difference between the means.
The only time the arithmetic and geometric means are equal is when
there is no variability in the observations.
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Interest Rates and Return Measurement
Harmonic Mean
Harmonic mean is used to find the average cost per share of stock
purchased over time, if each purchase is a constant dollar amount.
N N = number of purchases of equal dollar amount
Xharmonic = N
1
Xi = share price at time i
i=1 Xi
Example: An investor purchases $1,000 of mutual fund shares each
month. Over the last three months, the prices paid were $8, $9, and $10.
Calculate the average cost per share.
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Interest Rates and Return Measurement
Harmonic Mean Solution
N
Xharmonic = N
1
i=1 Xi
3
=
1 1 1
+ +
8 9 10
= $8.926 per share
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Interest Rates and Return Measurement
Harmonic Mean: Alternative Approach
Harmonic mean is the average cost per share of stock purchased
over time.
Shares purchased
Month 1: $1,000 / $8 = 125.00 shares
Month 2: $1,000 / $9 = 111.11 shares
Month 3: $1,000 / $10 = 100.00 shares
Total = 336.11 shares
Average price paid = $3,000 / 336.11 = $8.926 per share
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Interest Rates and Return Measurement
Harmonic, Arithmetic, and Geometric Means
What is the relationship between the means?
Arithmetic mean × harmonic mean = (geometric mean)2
For values that are not equal
Harmonic mean < geometric mean < arithmetic mean
Appropriate uses
Geometric mean: compound returns over multiple periods
Harmonic mean: average share cost from fixed money purchases
Arithmetic mean: include all values, even outliers 16
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Interest Rates and Return Measurement
Impact of Outliers
The arithmetic mean includes all observations, no matter how
extreme. This skews the mean in the direction of an outlier.
Adjusting for outliers
The trimmed mean or winsorized mean adjusts for outliers.
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