Rates and Returns
Quantitative Methods
Learning Module 3: Rates of Return
Holding Period Return
\[ R = \frac{(P_1 - P_0) + I_1}{P_0} \tag{1} \]
Where:
- \(R\): holding period return
- \(P_1\): price at end of period \((t=1)\)
- \(P_0\): price at beginning of period \((t=0)\)
- \(I_1\): income received at end of period
View Markdown Source
## Holding Period Return
$$
R = \frac{(P_1 - P_0) + I_1}{P_0} \tag{1}
$$
Where:
* $R$: holding period return
* $P_1$: price at end of period $(t=1)$
* $P_0$: price at beginning of period $(t=0)$
* $I_1$: income received at end of periodArithmetic Mean Return
\[ \bar{R}_i = \frac{R_{i1} + R_{i2} + \cdots + R_{i,T-1} + R_{iT}}{T} = \frac{1}{T} \sum_{t=1}^{T} R_{it} \tag{2} \]
Where:
- \(\bar{R}_i\): arithmetic mean return for asset \(i\)
- \(R_{it}\): return in period \(t\) for asset \(i\)
- \(T\): total number of periods
View Markdown Source
## Arithmetic Mean Return
$$
\bar{R}_i = \frac{R_{i1} + R_{i2} + \cdots + R_{i,T-1} + R_{iT}}{T}
= \frac{1}{T} \sum_{t=1}^{T} R_{it} \tag{2}
$$
Where:
* $\bar{R}_i$: arithmetic mean return for asset $i$
* $R_{it}$: return in period $t$ for asset $i$
* $T$: total number of periodsGeometric Mean Return
\[ \bar{R}_{Gi} = \sqrt[T]{(1 + R_{i1}) \times (1 + R_{i2}) \times \cdots \times (1 + R_{i,T-1}) \times (1 + R_{iT})} - 1 \tag{3} \]
\[ \bar{R}_{Gi} = \sqrt[T]{\prod_{t=1}^{T} (1 + R_{t})} - 1 \]
Where:
- \(\bar{R}_{Gi}\): geometric mean return for asset \(i\)
- \(R_{it}\): return in period \(t\) for asset \(i\)
- \(T\): total number of periods
View Markdown Source
## Geometric Mean Return
$$
\bar{R}_{Gi}
= \sqrt[T]{(1 + R_{i1}) \times (1 + R_{i2}) \times \cdots \times (1 + R_{i,T-1}) \times (1 + R_{iT})} - 1
\tag{3}
$$
$$
\bar{R}_{Gi} = \sqrt[T]{\prod_{t=1}^{T} (1 + R_{t})} - 1
$$
Where:
* $\bar{R}_{Gi}$: geometric mean return for asset $i$
* $R_{it}$: return in period $t$ for asset $i$
* $T$: total number of periodsHarmonic Mean
\[ \bar{X}_H = \frac{n}{\sum_{i=1}^{n} (1/X_i)} \tag{4} \]
Where:
- \(\bar{X}_H\): harmonic mean
- \(X_i\): observation \(i\) (must be positive)
- \(n\): number of observations
View Markdown Source
## Harmonic Mean
$$
\bar{X}_H = \frac{n}{\sum_{i=1}^{n} (1/X_i)} \tag{4}
$$
Where:
* $\bar{X}_H$: harmonic mean
* $X_i$: observation $i$ (must be positive)
* $n$: number of observationsMoney-Weighted Return (Internal Rate of Return)
\[ \sum_{t=0}^{T} \frac{CF_t}{(1 + \text{IRR})^t} = 0 \tag{5} \]
Where:
- \(\text{IRR}\): internal rate of return (money-weighted return)
- \(CF_t\): cash flow at time \(t\)
- \(T\): number of periods
View Markdown Source
## Money-Weighted Return (Internal Rate of Return)
$$
\sum_{t=0}^{T} \frac{CF_t}{(1 + \text{IRR})^t} = 0 \tag{5}
$$
Where:
* $\text{IRR}$: internal rate of return (money-weighted return)
* $CF_t$: cash flow at time $t$
* $T$: number of periodsTime-Weighted Return
\[ R_{TW} = \left[(1 + R_1) \times (1 + R_2) \times \cdots \times (1 + R_N)\right]^{1/N} - 1 \tag{6} \]
Where:
- \(R_{TW}\): annualized time-weighted return
- \(R_i\): time-weighted return for year \(i\)
- \(N\): number of years
View Markdown Source
## Time-Weighted Return
$$
R_{TW} = \left[(1 + R_1) \times (1 + R_2) \times \cdots \times (1 + R_N)\right]^{1/N} - 1 \tag{6}
$$
Where:
* $R_{TW}$: annualized time-weighted return
* $R_i$: time-weighted return for year $i$
* $N$: number of yearsPresent Value with Non-Annual Compounding
\[ PV = FV_N \left(1 + \frac{R_s}{m}\right)^{-mN} \tag{7} \]
Where:
- \(PV\): present value
- \(FV_N\): future value at time \(N\)
- \(R_s\): quoted annual interest rate
- \(m\): number of compounding periods per year
- \(N\): number of years
View Markdown Source
## Present Value with Non-Annual Compounding
$$
PV = FV_N \left(1 + \frac{R_s}{m}\right)^{-mN} \tag{7}
$$
Where:
* $PV$: present value
* $FV_N$: future value at time $N$
* $R_s$: quoted annual interest rate
* $m$: number of compounding periods per year
* $N$: number of yearsAnnualized Return from Period Return
\[ R_{annual} = (1 + R_{period})^{c} - 1 \tag{8} \]
Where:
- \(R_{annual}\): annualized return
- \(R_{period}\): return for the period
- \(c\): number of periods in a year
View Markdown Source
## Annualized Return from Period Return
$$
R_{annual} = (1 + R_{period})^{c} - 1 \tag{8}
$$
Where:
* $R_{annual}$: annualized return
* $R_{period}$: return for the period
* $c$: number of periods in a yearConverting Returns to Weekly
\[ R_{weekly} = (1 + R_{daily})^{5} - 1; \quad R_{weekly} = (1 + R_{annual})^{1/52} - 1 \tag{9} \]
Where:
- \(R_{weekly}\): weekly return
- \(R_{daily}\): daily return
- \(R_{annual}\): annual return
View Markdown Source
## Converting Returns to Weekly
$$
R_{weekly} = (1 + R_{daily})^{5} - 1; \quad
R_{weekly} = (1 + R_{annual})^{1/52} - 1 \tag{9}
$$
Where:
* $R_{weekly}$: weekly return
* $R_{daily}$: daily return
* $R_{annual}$: annual returnContinuously Compounded Return (associated with a holding period)
\[ r_{t,t+1} = \ln(P_{t+1}/P_t) = \ln(1 + R_{t,t+1}) \tag{10} \]
Where:
- here we are using \(r\) to refer specifically to continuously compounded returns
- \(r_{t,t+1}\): continuously compounded return from \(t\) to \(t+1\)
- \(P_{t+1}\): price at time \(t+1\)
- \(P_t\): price at time \(t\)
- \(R_{t,t+1}\):
View Markdown Source
## Continuously Compounded Return (associated with a holding period)
$$
r_{t,t+1} = \ln(P_{t+1}/P_t) = \ln(1 + R_{t,t+1}) \tag{10}
$$
Where:
* here we are using $r$ to refer specifically to continuously compounded returns
* $r_{t,t+1}$: continuously compounded return from $t$ to $t+1$
* $P_{t+1}$: price at time $t+1$
* $P_t$: price at time $t$
* $R_{t,t+1}$:Continuously Compounded Return (Multi-Period)
\[ r_{0,T} = \ln(P_T/P_0) \tag{11} \]
Where:
- \(r_{0,T}\): continuously compounded return from time \(0\) to \(T\)
- \(P_T\): price at time \(T\)
- \(P_0\): price at time \(0\)
View Markdown Source
## Continuously Compounded Return (Multi-Period)
$$
r_{0,T} = \ln(P_T/P_0) \tag{11}
$$
Where:
* $r_{0,T}$: continuously compounded return from time $0$ to $T$
* $P_T$: price at time $T$
* $P_0$: price at time $0$Price Relatives Product
\[ P_T/P_0 = (P_T/P_{T-1})(P_{T-1}/P_{T-2}) \ldots (P_1/P_0) \tag{12} \]
Where:
- \(P_T/P_0\): product of price relatives
- \(P_T\): price at time \(T\)
- \(P_0\): price at time \(0\)
- \(P_t\): price at time \(t\)
View Markdown Source
## Price Relatives Product
$$
P_T/P_0 = (P_T/P_{T-1})(P_{T-1}/P_{T-2}) \ldots (P_1/P_0) \tag{12}
$$
Where:
* $P_T/P_0$: product of price relatives
* $P_T$: price at time $T$
* $P_0$: price at time $0$
* $P_t$: price at time $t$Sum of Continuously Compounded Returns
\[ r_{0,T} = r_{T-1,T} + r_{T-2,T-1} + \ldots + r_{0,1} \tag{13} \]
Where:
- \(r_{0,T}\): continuously compounded return from time \(0\) to \(T\)
- \(r_{T,T+1}\):
View Markdown Source
## Sum of Continuously Compounded Returns
$$
r_{0,T} = r_{T-1,T} + r_{T-2,T-1} + \ldots + r_{0,1} \tag{13}
$$
Where:
* $r_{0,T}$: continuously compounded return from time $0$ to $T$
* $r_{T,T+1}$: Real Return
\[ (1 + \text{real return}) = (1 + \text{real risk-free rate})(1 + \text{risk premium}) \tag{14} \]
Where:
- \(1 + \text{real return}\): the real risk-free return and the risk premium combined
- \(\text{real risk-free rate}\):
- \(\text{risk premium}\):
View Markdown Source
## Real Return
$$
(1 + \text{real return}) = (1 + \text{real risk-free rate})(1 + \text{risk premium}) \tag{14}
$$
Where:
* $1 + \text{real return}$: the real risk-free return and the risk premium combined
* $\text{real risk-free rate}$:
* $\text{risk premium}$: Leveraged Return
\[ R_L = \frac{\text{Portfolio return}}{\text{Portfolio equity}} = \frac{[R_p \times (V_E + V_B) - (V_B \times r_D)]}{V_E} = R_p + \frac{V_B}{V_E}(R_p - r_D) \tag{15} \]
\[ R_L = R_p + \frac{V_B}{V_E}(R_p - r_D) \]
Where:
- \(R_L\): leveraged portfolio return
- \(R_p\): total investment return on leveraged portfolio
- \(V_B\): debt or borrowed funds
- \(V_E\): equity of the portfolio
- \(r_D\): borrowing cost on debt
View Markdown Source
## Leveraged Return
$$
R_L
= \frac{\text{Portfolio return}}{\text{Portfolio equity}}
= \frac{[R_p \times (V_E + V_B) - (V_B \times r_D)]}{V_E}
= R_p + \frac{V_B}{V_E}(R_p - r_D)
\tag{15}
$$
$$
R_L = R_p + \frac{V_B}{V_E}(R_p - r_D)
$$
Where:
* $R_L$: leveraged portfolio return
* $R_p$: total investment return on leveraged portfolio
* $V_B$: debt or borrowed funds
* $V_E$: equity of the portfolio
* $r_D$: borrowing cost on debt