Lecture Notes: Returns, Yields, Annualization, and Real vs Nominal Rates

Returns, Yields, and the Yearly View of Investments

Why we study returns instead of prices
- Returns measure how a sum of money grows over time, independent of the initial amount invested, enabling cross-asset comparison.
- Statistically, returns model more easily than prices (prices tend to follow a random walk with drift, making estimation harder).
- In asset pricing, we focus on returns (the payoff growth) and use them for valuation.
Key terms: return, yield, gross vs net
- Return: the overall growth from investment, can be gross (before expenses? and including cash flows) or net (actual growth).
- Yield: the return an investment is expected to earn over a period (forward-looking, often quoted for bonds); typically used for fixed income and maturities.
- Gross return: $Rg = rac{PT + ext{CF}{total}}{P0}$ where $P0$ is the initial price, $PT$ is the terminal price, and $ext{CF}_{total}$ are all cash flows (dividends, coupons) during the holding period.
- Net return (holding period return): $R{HP} = rac{PT - P0 + ext{CF}{total}}{P0} = Rg - 1$
- Relationship: the capital gain component plus the income component sum to the total return over the period.
Bond cash flows: decomposition into capital gain and income
- Capital gain component: $ext{CapGain} = rac{PT - P0}{P_0}$
- Income component (coupon/dividends): $ext{Coupon} = rac{ ext{CF}{total}}{P0}$
- Total gross return equals the sum of these components: $R_g = ext{CapGain} + ext{Coupon}$
- Example structure from lecture:
- Initial price $P_0 = 285$
- End price $P_T = 300$
- Cash flow (coupon) $CF_{total} = 15$
- CapGain = $(300 - 285)/285 = 15/285 \approx 0.0526$ (5.26%)
- Coupon = $15/285 \approx 0.0526$ (5.26%)
- Gross return: $Rg = (PT + CF{total})/P0 = (300 + 15)/285 = 315/285 \approx 1.1053$
- Net return (HPR): $R{HP} = Rg - 1 = 0.1053 \approx 10.53\%$
- The two components add up to the total return: CapGain + Coupon ≈ 5.26% + 5.26% ≈ 10.53%
Practical terminology and notes on yields
- Yields are broadly the forward-looking measure used for quoted prices (e.g., bond yields when you see price quotes in media).
- Returns are often backward-looking (what you actually earned), but in many instruments, yields and returns are used interchangeably in casual speech.
- Always verify whether a quoted figure is a yield (forward-looking) or a realized return (historical).
- A basis point (bp) is 1/100 of a percentage point: 1 bp = 0.01% = 0.0001 in decimal terms.
- Reason for using basis points: avoids confusion when discussing small changes (e.g., +75 basis points vs +0.75 percentage points).
Basis points example
- If the Fed raises rates by 75 basis points, that is an increase of $0.75\%$ , making the new rate 5.75% if the prior rate was 5.00%.
- Quick mental check: changes in basis points translate directly to percentage-point changes in rates.
Holding Period Return (HPR) in detail
- Definition: the total return earned over the holding period, taking into account price movement and all income streams within that period.
- Formula (alternative notation):
- $ext{HPR} = \frac{PT - P0 + \sum \, CFt}{P0}$ where $CF_t$ are cash flows received during the holding period.
- This is often stated in terms of the end price and cash flows minus the initial price, all divided by the initial price.
- Note: when talking about a single period, HPR is the period’s total return; annualization then translates this return into a yearly rate.
Annualization: converting returns across different periods
- Why: investments are often held for different lengths; annualization allows apples-to-apples comparison.
- Two common notions:
- Simple annualization (APR, annual percentage rate): assumes no reinvestment of intermediate cash flows within the period. If you have a per-period rate $rp$ and $n$ periods per year, APR ≈ $n \cdot rp$ .
- Effective annual rate (EAR): accounts for compounding within the year. If per-period rate is $rp$ and there are $n$ periods per year, EAR = \left(1 + rp\right)^n - 1.
- Example patterns:
- A monthly rate of 1% implies EAR = ( (1 + 0.01)^{12} - 1 = 12.68\% ) approximately.
- A daily rate of 0.06% (0.0006) implies EAR = ( (1 + 0.0006)^{365} - 1 \approx 0.224 ) or about 22.4% (the lecture had a miscalculation noted; the standard calculation is as shown).
- Simple vs compounded: when you reinvest, compounding increases the effective return over time, which is captured by EAR or continuous compounding.
- Two-step relation for average period returns:
- If you know the overall holding period return over t years, the per-year rate with compounding satisfies:
 $(1 + \text{HPR})^{1/t} - 1$
 which gives the annualized rate that would compound to the observed HPR over t years.
- Continuous compounding (limit case):
- If the per-year continuous rate is $r{cc}$ , the accumulation over time t is $e^{r{cc} t}$ , so the holding-period return is $\text{HPR} = e^{r_{cc} t} - 1$ .
- Equivalently, $r{cc} = \frac{\ln(1 + \text{HPR})}{t}$ and $\text{HPR} = e^{r{cc} t} - 1$ for any t.
- Practical tips from the lecture:
- For short-term periods, quote as APR (simple annualization).
- For longer-term or for comparisons that include reinvestment effects, prefer EAR.
- When converting between APR and EAR, know which convention your data uses (per-period rate vs nominal rate with a given compounding frequency).
Converting between APR and EAR (and related conversions)
- From APR to EAR (assuming compounding m times per year):
- If per-period rate is $r_p = \text{APR}/m$ , then EAR = $\left(1 + \frac{\text{APR}}{m}\right)^m - 1.$
- From EAR to APR (one common form):
- If EAR is known and compounding frequency m is known, then APR = m\left( (1 + \text{EAR})^{1/m} - 1 \right).
- Continuous compounding as a limiting case: if the nominal rate is allowed to compound continuously, then $\text{EAR} = e^{r{cc}} - 1$ and $r{cc} = \ln(1 + \text{EAR})$ .
- Worked example from lecture (illustrative):
- A loan quotes an APR of 5% with monthly compounding.
- Monthly rate = $0.05/12$ ; EAR = $\left(1 + \frac{0.05}{12}\right)^{12} - 1\approx 0.05116\ (5.116\%)$ .
- Another example mentioned: converting a quoted APR to an effective rate with different compounding, and the idea that as compounding frequency increases, the effective return approaches the continuous-compounding limit.
Real vs nominal rates and inflation (Fisher relation)
- Real rate vs nominal rate:
- Nominal return: measured in current dollars (includes inflation effects).
- Real return: reflects true increase in purchasing power after accounting for inflation.
- The Fisher equation (basic intuition):
- Approximate form: $i \approx r + \pi$ where $i$ is the nominal rate, $r$ is the real rate, and $\pi$ is expected inflation.
- Exact form (commonly used): $1 + i = (1 + r)(1 + \pi)$ , so
  $r = \frac{1 + i}{1 + \pi} - 1.$
- Relation used in the lecture (illustrative numbers):
- Example: nominal return = 5%, inflation = 4%.
- Approximate real return: $r \approx 5\% - 4\% = 1\%$ .
- Exact real return: $r = \frac{1 + 0.05}{1 + 0.04} - 1 = \frac{1.05}{1.04} - 1 \approx 0.009615 \approx 0.962\%.$
- Purchasing power and the macro rationale: investors care about real purchasing power, not just dollar nominal gains; inflation erodes purchasing power, so real returns are critical for saving and investment decisions.
Inflation signals and macro context
- Interest rates are closely tied to macro fundamentals (especially central bank policy and inflation expectations).
- Treasury yields and other rate benchmarks influence asset prices, risk premia, and loanable funds dynamics (supply and demand for funds).
- The spread between nominal rates and inflation informs investors about expected real returns.
Practical implications and common mistakes (quick tips)
- Always clarify whether a number is an APR, EAR, or a simple period rate.
- When asked for returns, remember to state whether you’re providing gross return (before subtracting initial investment) or net return (HPR).
- Round returns to the nearest basis point when reporting results (as instructed in the course).
- If you’re decomposing total return on bonds, verify both the capital gain component and the coupon component; their sum should match the total return.
- For long horizons, use EAR or continuous compounding to capture compounding effects rather than the simple APR.
Quick glossary (key terms from the lecture)
- Return: the growth of an investment over a period; can be gross or net.
- Gross return: total payoff including price end and cash flows, relative to initial investment.
- Net return / HPR: realized return over the holding period; equals gross return minus 1.
- Yield: forward-looking measure of return for an investment, often quoted for bonds; can be used interchangeably with return in some contexts.
- Capital gain: increase in price from initial to end of holding period, expressed as a percentage of the initial price.
- Coupon yield: cash flow received during the holding period, expressed as a percentage of the initial price.
- Basis point (bp): one hundredth of a percentage point; 1 bp = 0.01%.
- APR (annual percentage rate): simple annualized rate, typically used for short-term or straightforward annualization.
- EAR (effective annual rate): annualized rate that accounts for intra-year compounding.
- HPR (holding period return): the total return earned over the holding period.
- P_0: initial investment price.
- P_T: terminal price at the end of the holding period.
- CF_t: cash flows received during the holding period (coupons, dividends).
- rp, r{EAR}, r_{cc}: per-period rate, effective annual rate, and continuous-compounding rate, respectively.
- i (nominal), r (real), \pi (inflation): notations used for nominal return, real return, and inflation in the Fisher framework.