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Rates and Returns

Quantitative Methods

Interest Rates and Time Value of Money

Learning Outcome Statement:

interpret interest rates as required rates of return, discount rates, or opportunity costs and explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk

Summary:

The learning outcome focuses on understanding interest rates in terms of required rates of return, discount rates, and opportunity costs. It explains how interest rates are composed of a real risk-free rate and additional premiums that compensate for various risks such as inflation, default, liquidity, and maturity. The time value of money is emphasized, illustrating the importance of comparing cash flows at different times using interest rates to establish equivalency.

Key Concepts:

Real Risk-Free Interest Rate

This is the theoretical rate of return of an investment with zero risk, assuming no inflation. It reflects the time preference of individuals for current versus future consumption.

Inflation Premium

Compensates investors for expected inflation over the maturity of the debt, reflecting the anticipated decrease in purchasing power of money.

Default Risk Premium

Compensates investors for the risk that the borrower may fail to make a promised payment at the contracted time and amount.

Liquidity Premium

Compensates investors for the potential loss relative to an investment’s fair value if the investment needs to be converted to cash quickly.

Maturity Premium

Compensates investors for the increased sensitivity of the market value of debt to changes in market interest rates as maturity extends.

Formulas:

Interest Rate Composition

r = \text{Real risk-free interest rate} + \text{Inflation premium} + \text{Default risk premium} + \text{Liquidity premium} + \text{Maturity premium}

This formula represents how the total interest rate is composed of the real risk-free rate and additional premiums that account for various risks associated with the investment.

Variables:

$r$ :: Total interest rate

Units: percentage

Nominal Risk-Free Rate Approximation

\text{Nominal risk-free rate} = \text{Real risk-free rate} + \text{Inflation premium}

This formula approximates the nominal risk-free rate by summing the real risk-free rate and the inflation premium.

Variables:

$Nominal risk-free rate$ :: The interest rate on short-term government debt, representing the risk-free rate over a short time horizon.

Units: percentage

Rates of Return

Learning Outcome Statement:

calculate and interpret different approaches to return measurement over time and describe their appropriate uses

Summary:

This LOS covers various methods to measure and interpret returns over time, focusing on holding period return, arithmetic mean return, geometric mean return, and the harmonic mean. Each method has specific applications and implications for financial analysis and portfolio management.

Key Concepts:

Holding Period Return

The return earned from holding an asset for a specified period. It includes both capital gains and any income received during the period.

Arithmetic Mean Return

A simple average of returns over multiple periods. It is easy to compute and useful for understanding average single-period returns.

Geometric Mean Return

Provides a measure of average compound return per period. It is more accurate for understanding the growth of an investment portfolio over time.

Harmonic Mean

Used for averaging ratios or rates, particularly useful when dealing with variables that are inversely proportional to what they measure (e.g., P/E ratios).

Formulas:

Holding Period Return

R = \frac{(P_1 - P_0) + I_1}{P_0}

Calculates the total return including price changes and income received.

Variables:

$R$ :: Holding period return
$P_1$ :: Price at the end of the period
$P_0$ :: Price at the beginning of the period
$I_1$ :: Income received during the period

Units: percentage or decimal

Arithmetic Mean Return

\overline{R_i} = \frac{1}{T} \sum_{t=1}^{T} R_{it}

Calculates the average return over multiple periods.

Variables:

$R_i$ :: Arithmetic mean return for asset i
$T$ :: Total number of periods
$R_{it}$ :: Return of asset i in period t

Units: percentage or decimal

Geometric Mean Return

\overline{R_{Gi}} = \sqrt[T]{\prod_{t=1}^{T} (1 + R_t)} - 1

Calculates the compound average return over multiple periods.

Variables:

$R_{Gi}$ :: Geometric mean return for asset i
$T$ :: Total number of periods
$R_t$ :: Return in period t

Units: percentage or decimal

Harmonic Mean

\overline{X_H} = \frac{n}{\sum_{i=1}^{n} \frac{1}{X_i}}

Calculates the harmonic mean of a set of observations, particularly useful for rates and ratios.

Variables:

$X_H$ :: Harmonic mean
$n$ :: Number of observations
$X_i$ :: Value of the ith observation

Units: units depend on the variable X

Money-Weighted and Time-Weighted Return

Learning Outcome Statement:

compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures

Summary:

This LOS explores the differences between money-weighted and time-weighted rates of return, explaining how each is calculated and their implications for evaluating investment performance. Money-weighted return, similar to the internal rate of return (IRR), considers the timing and amount of cash flows, making it sensitive to when investments are made. Time-weighted return, on the other hand, removes the effects of cash flows and measures the compound rate of growth of an initial investment, providing a measure that is not influenced by the investor's specific actions.

Key Concepts:

Money-Weighted Return

The money-weighted return measures the rate of return on a portfolio by considering the size and timing of cash flows, similar to the internal rate of return (IRR) for a series of cash flows. It reflects the actual return earned by the investor based on the specific amounts invested and withdrawn.

Time-Weighted Return

The time-weighted return measures the compound growth rate of an initial investment over a period, independent of any investor-specific cash flows. This method segments the investment period into intervals around each cash flow and calculates the growth rate of the investment as if no cash flows occurred.

Formulas:

Money-Weighted Return (IRR)

\sum_{t=0}^{T} \frac{CF_t}{(1 + IRR)^t} = 0

This formula calculates the IRR where the sum of the present values of all cash flows (both inflows and outflows) equals zero, representing the rate of return that exactly discounts the future value of cash flows to the initial investment.

Variables:

$T$ :: total number of periods
$CF_t$ :: cash flow at time t
$IRR$ :: internal rate of return or money-weighted rate of return

Units: percentage

Time-Weighted Return

R_{TW} = [(1 + r_1) \times (1 + r_2) \times \ldots \times (1 + r_N)]^{1/N} - 1

This formula links the holding period returns over each subperiod to calculate the overall time-weighted return, providing a measure of the compound growth rate of the investment independent of any cash flows.

Variables:

$r_i$ :: holding period return for subperiod i
$N$ :: number of subperiods

Units: percentage

Annualized Return

Learning Outcome Statement:

calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses

Summary:

The concept of annualized return involves converting returns calculated for periods shorter or longer than one year into an annual basis to facilitate comparison. This includes handling non-annual compounding by adjusting the present value formula for different compounding frequencies, and calculating continuously compounded returns using natural logarithms.

Key Concepts:

Non-annual Compounding

Interest may be compounded on a semiannual, quarterly, monthly, or daily basis. The present value formula is adjusted by using the periodic interest rate and the number of compounding periods per year.

Annualizing Returns

To annualize a return for a period shorter than one year, the return is compounded by the number of periods in a year. For periods longer than a year, the return is raised to the power of the fraction of a year the period represents.

Continuously Compounded Returns

Continuously compounded returns are calculated using the natural logarithm of one plus the holding period return, or the natural logarithm of the ending price over the beginning price.

Formulas:

Present Value with Non-annual Compounding

PV = \frac{FV}{(1 + \frac{R_s}{m})^{mN}}

This formula calculates the present value of a future sum considering multiple compounding periods per year.

Variables:

$PV$ :: Present Value
$FV$ :: Future Value
$R_s$ :: Quoted annual interest rate
$m$ :: Number of compounding periods per year
$N$ :: Number of years

Units: currency

Annualized Return

R_{annual} = (1 + R_{period})^c - 1

This formula is used to convert a return for any period shorter than one year into an annualized return by compounding it over the number of periods in a year.

Variables:

$R_{annual}$ :: Annualized return
$R_{period}$ :: Return for the period
$c$ :: Number of periods in a year

Units: percentage

Continuously Compounded Return

r_{t,t+1} = \ln\left(\frac{P_{t+1}}{P_t}\right) = \ln(1 + R_{t,t+1})

This formula calculates the continuously compounded return for a single period based on the price at the beginning and end of the period.

Variables:

$r_{t,t+1}$ :: Continuously compounded return from time t to t+1
$P_{t+1}$ :: Price at time t+1
$P_t$ :: Price at time t
$R_{t,t+1}$ :: Holding period return from time t to t+1

Units: logarithmic

Other Major Return Measures and Their Applications

Learning Outcome Statement:

calculate and interpret major return measures and describe their appropriate uses

Summary:

This LOS covers various return measures including gross and net returns, pre-tax and after-tax nominal returns, real returns, and leveraged returns. It discusses the implications of fees, taxes, inflation, and leverage on investment returns and provides methodologies for calculating these returns. The content emphasizes the importance of understanding different return measures for evaluating investment performance across different scenarios and asset classes.

Key Concepts:

Gross and Net Return

Gross return is the total return on an investment before any expenses are deducted. Net return is the return after all expenses, including management and administrative fees, are deducted. Gross return evaluates investment skill without the influence of external costs, while net return reflects the actual return received by investors.

Pre-Tax and After-Tax Nominal Return

Pre-tax nominal returns are calculated without adjustments for taxes. After-tax returns account for taxes paid on dividends, interest, and realized capital gains. The choice between these depends on the investor's focus on either raw performance or actual take-home returns.

Real Returns

Real returns adjust nominal returns for the effects of inflation, providing a measure of the actual increase in purchasing power derived from an investment. They are crucial for comparing returns across different time periods or economic environments.

Leveraged Return

Leveraged returns reflect the performance of investments made with borrowed funds. They show the amplified gains or losses due to using leverage, calculated based on the proportion of own funds to borrowed funds and the cost of borrowing.

Formulas:

After-tax real return

\left(1 + r_{\text{after-tax}}\right) / \left(1 + \text{inflation rate}\right) - 1

This formula calculates the real return after accounting for taxes and inflation, providing the investor's actual increase in purchasing power.

Variables:

$r_{\text{after-tax}}$ :: after-tax return
$\text{inflation rate}$ :: rate of inflation

Units: percentage

Leveraged Return

R_L = R_p + \frac{V_B}{V_E} (R_p - r_D)

This formula calculates the return on a leveraged portfolio by considering the total investment return, the proportion of borrowed funds to equity, and the cost of borrowing.

Variables:

$R_L$ :: return on the leveraged portfolio
$R_p$ :: portfolio return
$V_E$ :: equity of the portfolio
$V_B$ :: borrowed funds
$r_D$ :: borrowing cost

Units: percentage

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