Quant Ch-1

1 Introduction

• Time value of money concept: prefer money today over the same amount in the future because of its additional value today.

• Example framing: If you have $100$ today vs $100$ in 3 years, you’d typically prefer $100$ today due to time value of money.

• Indifference example: Indifferent between $100$ today and $110$ in one year implies PV = $100$, FV = $110$, and I = 10% (I = (FV − PV)/PV).

• Purpose of this reading: meaning and interpretation of interest rates, how to calculate and interpret different return measures, and how to compare them.

2 Interest Rates and Time Value of Money

• Interpretations of interest rates (three views):

  1. Required rate of return

  2. Discount rate

  3. Opportunity cost

• Example: Lend $900$ today and receive $990$ after one year. If you value that payoff, the implied rate is 10%.

2.1 Required Rate of Return

• If you lend $900$ today and get $990$ after one year, the required return is 10%.

• Simple calculation:
  $\text{Rate} = \frac{FV - PV}{PV} = \frac{990 - 900}{900} = 0.10 = 10\%$

2.2 Discount Rate

• Discounting $990$ back to present value at 10% yields $900$:
  $PV = \frac{FV}{1 + i} = \frac{990}{1.10} = 900.$

• Therefore, 10% can be viewed as a discount rate as well.

2.3 Opportunity Cost

• If instead you spent the $900$ on something else, you forgone earning 10% interest.

• Therefore, 10% is the opportunity cost of not lending the money.

2.4 Determinants of Interest Rates

• Interest rate = Real risk-free rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium
  $\text{Interest Rate} = r_{\text{real}} + \text{Inflation Premium} + \text{Default Premium} + \text{Liquidity Premium} + \text{Maturity Premium}$

2.4.1 Real risk-free rate

• Rate on a security with no risk and very high liquidity; assumes no inflation.

2.4.2 Inflation Premium

• Compensation for expected inflation in the upcoming period.

2.4.3 Default Risk Premium

• Extra return investors demand to compensate for default risk (e.g., lending to a riskier borrower).

• Example framing: Lending $100$ to A vs B; B has higher default risk, so higher return (premium).

2.4.4 Liquidity Premium

• Compensates for difficulty in converting an investment to cash quickly at fair value.

• Example: investment C is highly liquid vs D is illiquid; D requires higher return (liquidity premium).

2.4.5 Maturity Premium

• Longer-maturity securities have higher sensitivity of price to interest-rate changes; investors demand compensation.

• Example: Security E has 1-year maturity vs F has 4 years; F carries higher maturity risk.

2.5 Nominal risk-free rate

• Relationship to real rate and inflation:
  $r<em>{\text{nominal}} = r</em>{\text{real}} + \text{Inflation premium}$

• If real risk-free rate is 3% and inflation premium is 2%, then the nominal risk-free rate is 5%.

• Exam note: When the term risk-free is used without clarification, it refers to the nominal risk-free rate.

3 Rates of Return

• A financial asset’s total return consists of two components: income and capital appreciation.

3.1 Holding Period Return (HPR)

• HPR is the return earned over a specified holding period.

• Holding period can range from days to years.

• Formula (one-time cash flow at end): $\text{HPR} = \frac{P<em>1 - P</em>0 + D<em>1}{P</em>0}$ where

  • $P_0$ = initial investment

  • $P_1$ = ending price

  • $D_1$ = cash flow at end (e.g., dividend)

3.1.1 HPR Example

• Buy stock for $50$. Six months later, price is $53$ and dividend is $2$.

• $\text{HPR} = \frac{53 + 2 - 50}{50} = \frac{5}{50} = 0.10 = 10\%$

3.2 Multiple Holding Periods Return

• If returning over multiple years, the holding period return is:
  $R = [(1 + R<em>1)(1 + R</em>2)(1 + R_3)] - 1$

• $R<em>1, R</em>2, R_3$ are the annual returns in each period.

3.2.1 Example

• Mutual fund annual returns: 20%, -8%, -1%.

• $R = [(1 + 0.20)(1 - 0.08)(1 - 0.01)] - 1 = 0.0929 = 9.296\%$.

3.3 Arithmetic or Mean Return

• Arithmetic mean = simple average of observations:
  $\bar{X} = \frac{\sum<em>{i=1}^{n} X</em>i}{n} = \frac{1}{n} \left( \text{sum of } X_i \right)$

• Example: returns 20%, -8%, -1% → mean = $\frac{20 - 8 - 1}{3} = 3.66\%$

• Drawback: sensitive to outliers (extreme values can skew the mean).

3.3.1 Outliers

• Options for dealing with outliers:

  • Option 1: Do nothing; use data as is.

  • Option 2: Delete all outliers.

  • Option 3: Replace outliers with other values.

  • Trimmed mean: exclude a stated percentage of lowest and highest values, then compute mean.

  • Winsorized mean: replace a stated percentage of extreme values with specified low/high values, then compute mean.

3.4 Geometric Mean Return

• Geometric mean (compounded average) is the nth root of the product of $(1 + R<em>i)$ minus 1: $GM = \left( (1 + R</em>1)(1 + R<em>2) \ldots (1 + R</em>n) \right)^{1/n} - 1 \quad \text{or more simply} \quad GM = \left( \frac{P<em>t}{P</em>0} \right)^{1/n} - 1$

• Common application: average return over multiple periods with compounding.

3.4.1 Geometric Mean Example

• Returns: 10%, 8%, -5%, 2%

• $1 + R$ sequence: 1.10, 1.08, 0.95, 1.02

• $GM: (1.10 \times 1.08 \times 0.95 \times 1.02)^{1/4} - 1 = 3.58\%$

• Interpretation: $1$ invested grows to $1.151$ over four periods, matching the compounded growth at 3.58%.

3.4.2 Geometric Mean Return with Compounding

• An alternative formula for log-mean growth:
  $\text{ln-mean } \hat{X}<em>G = \frac{1}{n} \sum</em>{i=1}^n \ln(X_i)$

• Note: This formula is less testable.

• Example: P/E ratios 10, 15, 14, 13.
  $\text{GM} = e^{(\ln10 + \ln15 + \ln14 + \ln13)/4} \approx 12.807$

3.5 Using Geometric and Arithmetic Means

• Geometric mean is appropriate for measuring past performance over multiple periods.

• Arithmetic mean is appropriate for forecasting single-period returns.

3.5.1 Using Geometric Mean

• Example

  • Portfolio two-year returns: 100% and -50%.

  • GM = $[(1 + 1.0)(1 - 0.5)]^{1/2} - 1 = 0\%$

3.5.2 Using Arithmetic Mean

• Example

  • Next year potential returns: 100% or -50%

  • Expected return (arithmetic mean) = $(100 + (-50))/2 = 25\%$

3.6 The Harmonic Mean

• Harmonic mean is a weighted mean where weight is inversely proportional to magnitude:
  $X<em>H = \frac{n}{\frac{1}{X</em>1} + \frac{1}{X<em>2} + \ldots + \frac{1}{X</em>n}}$

• Used to find average purchase price for equal periodic investments.

3.6.1 Harmonic Mean Example

• Monthly purchases: $10,$15, $20$; invest $1,000$ each month.

• Harmonic mean of prices: three over open parenthesis one over ten plus one over fifteen plus one over twenty close parenthesis (calculate) → price $\approx \$13.85$ per share.

• Alternative intuitive method: total money spent / total shares purchased.

• Money spent: $3,000$; Shares: 216.67; Average price = $3,000/216.67 \approx \$13.85$.

3.7 Comparison of AM, GM and HM

• AM $\times$ HM = GM$\text{\textasciicircum}2$.

• If returns are constant over time: AM = GM = HM.

• If returns vary: AM > GM > HM.

3.7.1 Which mean to use?

• Arithmetic mean: single-period or cross-sectional data.

• Trimmed mean / Winsorized mean: data with extreme outliers.

• Geometric mean: time-series data.

• Harmonic mean: average purchase price for equal periodic investments.

4 Money-Weighted and Time-Weighted Return

4.1 Net Present Value (NPV)

• Formula: $\text{NPV} = CF<em>0 + \frac{CF</em>1}{(1 + R)^1} + \frac{CF<em>2}{(1 + R)^2} + \ldots + \frac{CF</em>N}{(1 + R)^N}$ where

  • $CF_0$ = initial investment (usually outflow)

  • $CF_t$ = net cash flow at time t

  • $R$ = discount rate

• Alternatively: $NPV = \text{PV of inflows} - \text{PV of outflows}$.

4.2 Internal Rate of Return (IRR)

• Definition: the discount rate that makes NPV equal to zero.

• Equation:
  $0 = \text{NPV} = CF<em>0 + \frac{CF</em>1}{(1 + \text{IRR})^1} + \frac{CF<em>2}{(1 + \text{IRR})^2} + \ldots + \frac{CF</em>N}{(1 + \text{IRR})^N}$

• IRR is a single number representing the return generated by the investment.

4.2.1 IRR – Example

• Simple case: $CF<em>0 = -100$, $CF</em>1 = 110$ (one year later).

• IRR = 10% since $0 = -100 + 110/(1+\text{IRR})$ implies IRR = 0.10.

4.2.1 IRR – Example (TI calculator steps and result)

• Example details (larger cash flows): IRR $\approx$ 13.11% (from steps using a financial calculator).

4.3 Money-Weighted Rate of Return

• Definition: IRR of an investment, i.e., the money-weighted return equals IRR.

4.3.1 Money-Weighted Return – Example

• Cash flows example: buy at $20$, year 1 dividend $0.50$ and buy another share for $22.50$, year 2 sell both shares for $23.50$ plus dividends $0.50$ per share.

• Result: money-weighted return $\approx$ 9.39% (calculated via IRR).

4.4 Time-Weighted Rate of Return (TWRR)

• Definition: measures the compound growth rate of $1$ initially invested, ignoring cash flows timing/amount.

• Steps to compute TWRR:

  1. Break period into sub-periods around cash flows.

  2. Calculate HPR for each sub-period.

  3. Link/compound sub-period returns to obtain annual rate for the period.

  4. If >1 year, take the geometric mean of annual returns for the overall TWRR.

4.4.1 Time-Weighted Return – Example 1

• Cash flows occur at start/end of year; compute HPRs for each period and then combine.

• Year 1 HPR = 15.00%; Year 2 HPR = 6.67%; TWRR over 2 years =
  $(1.15 \times 1.067)^{1/2} - 1 = 0.1077 = 10.77\%$

4.4.2 Time-Weighted Return – Example 2

• Quarterly cash flows: compute sub-period returns, multiply to get annual factors, then take the geometric mean for multi-year periods.

• Example result: TWRR = 16.18% for the two-year period.

4.5 Money-Weighted v/s Time-Weighted Returns

• Money-weighted return is affected by the timing and size of cash flows.

• Time-weighted return is not affected by cash flows and is appropriate when the manager does not control flows.

• Use MWPR when the manager controls timing/amount of investments; use TWRR when flows are external.

5 Annualized Return

5.1 Annual Compounding

• FV of a single cash flow with annual compounding:
  $FV_N = PV (1 + r)^N$

• $PV$ = present value, $r$ = periodic rate, $N$ = number of periods.

• Example: $PV = 100$, $r = 0.10$, $N = 2$ → $FV_2 = 100(1.1)^2 = 121$.

• Difference with simple interest: $121$ vs $120$; the extra $1$ is interest-on-interest.

5.1.1 Annual Compounding – Example

• $PV = 5$ million, $r = 5\%$, $N = 2.5$ years, with compounding for 2.5 years.

• $FV_{2.5} = 5 \times (1 + 0.05)^{2.5} = 5.649 \text{ million.}$

5.1.2 FV Calculation using a Financial Calculator

• Common TI BA II Plus keys: N, I/Y, PV, PMT, FV, CPT.

• Signs: if PV is negative, FV is positive; cash inflows are positive, outflows are negative.

• Setup tips: set decimals to 9 for accuracy; CF registers step-by-step for cash flows.

5.1.2 FV Calculation – Calculator Example

• Invest $100$ today at 10% for 5 years; FV $\approx -161.05$ (calculator display shows negative because of conventions).

5.2 Non-Annual Compounding

• General formula for non-annual compounding:
  $FV<em>N = PV \left(1 + \frac{r</em>s}{m} \right)^{mN}$

• $r_s$ = stated annual rate (decimal)

• $m$ = compounding periods per year

• $N$ = number of years

5.2.1 Non-Annual Compounding – Example 1

• $PV = 80,000$, stated annual rate $r_s = 10\%$, $m = 4$, $N = 3$.

• $FV_{12} = 80,000 \left(1 + \frac{0.10}{4} \right)^{12} = 107,591.$

5.2.2 Non-Annual Compounding – Example 2

• $PV = 3,000,000$, $r_s = 4\%$, daily compounding ($m = 365$).

• $FV_1 = 3,000,000 \left(1 + \frac{0.04}{365} \right)^{365} \approx 3,122,000.$

5.3 Annualizing Returns

• Purpose: convert returns from periods shorter or longer than a year to an annualized rate for easy comparison.

• Formula:
  $\text{Annualized return} = (1 + r_{\text{period}})^{c} - 1$
  where $c$ = number of periods in a year.

5.3.1 Annualizing Returns – Example

• ETF time-inception returns: ETF1, ETF2, ETF3 with different days/weeks/months since inception.

• Calculations (illustrative):

  • ETF1: about 11.93% annualized

  • ETF2: about 12.05% annualized (highest)

  • ETF3: about 11.32% annualized

5.4 Continuously Compounded Returns

• Concept: as compounding frequency increases to infinity, return approaches continuous compounding.

• Continuously compounded return for a holding period:
  $R<em>c = \ln(1 + R</em>t) = \ln \left( \frac{P<em>t}{P</em>0} \right)$

5.4.1 Continuously Compounded Returns – Examples

• If holding period return $R<em>t = 4\%$, then $R</em>c = \ln(1 + 0.04) = 0.03922 = 3.922\%.$

• If a stock is bought at $P<em>0 = 30$ and $P</em>1 = 34.50$, then $R_c = \ln \left(\frac{34.50}{30}\right) = 0.13976 = 13.976\%.$

6 Other Major Return Measures and Their Applications

6.1 Gross Return

• Return earned by an asset manager before fees and taxes.

• Measures the manager’s investment skill.

6.2 Net Return

• Return after all managerial/administrative expenses have been deducted.

• Example: manager returns 20% gross, charges 2% fee; net return = 18%.

6.3 Pre-tax and After-tax Nominal Return

• Pre-tax nominal return: returns before taxes or inflation adjustments (default assumption).

• After-tax nominal return: return after taxes.

• Example: gross/net return 18% pre-tax; tax rate 33.33% → after-tax nominal return $\approx$ 12.0006%.

6.4 Real Return

• Real return = return after taxes and inflation.

• Formula relation:
  $(1 + r) = (1 + r<em>{\text{real}})(1 + \pi)$ where $r$ = nominal rate, $r</em>{\text{real}}$ = real rate, $\pi$ = inflation rate.

• Calculation tip: use exact formula when close choices; otherwise, approximate: nominal $\approx$ real + inflation.

6.5 Leveraged Return

• Return earned on investor’s own money after accounting for interest paid on borrowed funds.

Summary (Learning Objectives and Key Takeaways)

• LO: Interpret interest rates as required rate of return, discount rate, or opportunity cost; express

The formulas provided in sections 4.1 to 4.4, such as for Net Present Value (NPV) and Internal Rate of Return (IRR), are standard mathematical representations used in finance. They are presented using LaTeX to ensure accurate display of complex financial equations as requested by the output format. Concepts like the Time-Weighted Rate of Return (TWRR) in section 4.4 are defined through a series of computational steps rather than a single overarching formula, as they involve breaking down periods and compounding holding period returns.

4 Money-Weighted and Time-Weighted Return

4.1 Net Present Value (NPV)

• Formula: $\text{NPV} = CF_0 + \frac{CF_1}{(1 + R)^1} + \frac{CF_2}{(1 + R)^2} + \ldots + \frac{CF_N}{(1 + R)^N}$

  • Where $CF_0$ is the initial investment, $CF_t$ represents net cash flow at time t, and $R$ is the discount rate.

4.2 Internal Rate of Return (IRR)

• Definition: the discount rate that makes NPV equal to zero.

• Equation: $0 = \text{NPV} = CF_0 + \frac{CF_1}{(1 + \text{IRR})^1} + \frac{CF_2}{(1 + \text{IRR})^2} + \ldots + \frac{CF_N}{(1 + \text{IRR})^N}$

  • IRR is a single number representing the return generated by the investment.

4.3 Money-Weighted Rate of Return

• Definition: IRR of an investment, i.e., the money-weighted return equals IRR. This means its calculation directly involves the IRR formula.

4.4 Time-Weighted Rate of Return (TWRR)

• Definition: measures the compound growth rate of $1$ initially invested, ignoring cash flows timing/amount.

• Steps to compute TWRR:

  1. Break period into sub-periods around cash flows.

  2. Calculate HPR for each sub-period.

  3. Link/compound sub-period returns to obtain annual rate for the period.

  4. If >1 year, take the geometric mean of annual returns for the overall TWRR.