QM.1: Rates and Returns

Interest Rates: Three Interpretations

An interest rate can be viewed in three equivalent ways:

1. Required rate of return — the minimum return an investor demands for parting with capital
2. Discount rate — the rate used to find the present value of future cash flows
3. Opportunity cost — the return foregone by choosing one investment over another

Interest rates are composed of several building blocks:

$r = r_f + \text{Inflation premium} + \text{Default risk premium} + \text{Liquidity premium} + \text{Maturity premium}$

Where $r_f$ is the real risk-free rate. The nominal risk-free rate approximates $r_f + \text{Inflation premium}$.

Holding Period Return

The simplest return measure is the holding period return (HPR), which captures total return over the entire investment period:

$R = \frac{P_1 - P_0 + D_1}{P_0} = \frac{P_1 + D_1}{P_0} - 1$

Where $P_0$ is the beginning price, $P_1$ is the ending price, and $D_1$ is any income received. The HPR makes no assumptions about the length of the holding period — it could be a day, a month, or ten years.

Arithmetic vs Geometric Mean Return

When summarizing returns over multiple periods, the choice of mean matters enormously.

Arithmetic Mean

The simple average of periodic returns:

$\bar{R}_A = \frac{R_1 + R_2 + \cdots + R_n}{n}$

The arithmetic mean is the best unbiased estimator of the expected single-period return. It is always greater than or equal to the geometric mean.

Geometric Mean

The compound average return that equates starting and ending wealth:

$\bar{R}_G = \left[\prod_{t=1}^{n}(1+R_t)\right]^{1/n} - 1$

The geometric mean is the better measure of actual investment performance over multiple periods because it accounts for compounding.

Example

An investment returns +50% in Year 1 and -50% in Year 2:

• Arithmetic mean: $(50\% + (-50\%)) / 2 = 0\%$
• Geometric mean: $[(1.50)(0.50)]^{1/2} - 1 = (0.75)^{0.5} - 1 = -13.4\%$

Starting with $100: $100 × 1.50 × 0.50 = $75. You lost money, so the geometric mean correctly shows a negative return, while the arithmetic mean misleadingly suggests breakeven.

Money-Weighted vs Time-Weighted Returns

These two methods answer different questions about portfolio performance.

Money-Weighted Return (MWR)

The MWR is the internal rate of return (IRR) of all cash flows into and out of the portfolio. It solves:

$\sum_{t=0}^{n} \frac{CF_t}{(1+r)^t} = 0$

The MWR is affected by the timing and size of cash flows. If an investor adds money before a strong period, the MWR exceeds the TWR (and vice versa). It measures the investor's actual experience.

Time-Weighted Return (TWR)

The TWR chains sub-period returns, removing the impact of cash flow timing:

$R_{TWR} = \left[\prod_{t=1}^{n}(1+R_t)\right] - 1$

Each sub-period return is calculated between external cash flows. The TWR is the preferred method for evaluating manager performance because the manager typically does not control when clients add or withdraw money.

When to Use Which

Other Return Measures

Annualized Return

Converts a holding period return to an annual equivalent:

$R_{annual} = (1 + R_{period})^{1/n} - 1$

where $n$ is the number of years. For a period less than one year (say 90 days): $R_{annual} = (1 + R_{90d})^{365/90} - 1$.

Continuously Compounded Return

$r_{cc} = \ln(1 + R) = \ln\left(\frac{P_1}{P_0}\right)$

Continuously compounded returns are additive over time: $r_{cc,total} = r_{cc,1} + r_{cc,2} + \cdots + r_{cc,n}$, which makes them convenient for mathematical modeling.

Gross vs Net Returns

• Gross return: Return before deducting management fees; used to evaluate the manager's investment skill
• Net return: Return after management fees; what the investor actually earns

Real vs Nominal Returns

$1 + R_{real} = \frac{1 + R_{nominal}}{1 + \pi}$

where $\pi$ is the inflation rate. Approximately: $R_{real} \approx R_{nominal} - \pi$.

After-Tax Return

$R_{after-tax} = R_{pre-tax} \times (1 - t)$

where $t$ is the marginal tax rate.

Leveraged Return

If an investor uses leverage with equity $E$ and borrowed funds $B$ at borrowing cost $r_B$:

$R_{leveraged} = R_{portfolio} + \frac{B}{E}(R_{portfolio} - r_B)$

Leverage magnifies both gains and losses.