CFA Level 1 : Quantitative Methods

Rates and Returns

• 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 

• In capital budgeting decisions, the opportunity cost of capital is often represented by the discount rate. It’s the rate you could earn on an alternative investment of similar risk. Therefore, when evaluating a project, if the project’s return is less than the opportunity cost (discount rate), it may not be a good investment.  

•  Discount Rate: Used to calculate the present value of future cash flows, reflecting the time value of money and risk.

• U.S. Treasury Inflation-Protected Securities (TIPS) can be seen as approximating the real risk-free rate since they adjust for inflation and are backed by the U.S. government.

 Inflation Premium

• Interpretation: Because inflation erodes the value of money over time, investors demand compensation for this loss. The nominal interest rate includes an inflation premium to ensure that the real return (after adjusting for inflation) meets their required rate of return. 

• Example: If inflation is expected to be 2% per year, the interest rate on a bond might include a 2% inflation premium to compensate for this expected loss in purchasing power.

• Nominal interest: Nominal interest refers to the interest rate that is stated or advertised by a financial institution, without taking into account the effects of inflation. It represents the percentage of interest that a borrower is required to pay, or that an investor earns, on the principal amount of a loan or investment.

For example, if a bank offers a savings account with a nominal interest rate of 5% per year, it means that the bank will pay 5% of the account balance in interest over the course of a year. However, this rate does not reflect the actual purchasing power of the interest earned, as it does not account for inflation, which can erode the real value of the returns.

In summary:

• Nominal Interest Rate: The stated or advertised rate of interest.

• Real Interest Rate: Adjusts the nominal rate for inflation, giving a more accurate picture of the actual purchasing power of the interest earned or paid.

Default Risk Premium

• Definition: The default risk premium compensates investors for the risk that the borrower might fail to make the promised payments (interest or principal) on time or at all.

• Interpretation: The higher the risk of default, the higher the premium investors will demand. This premium varies based on the creditworthiness of the borrower; bonds issued by corporations typically have higher default risk premiums than government bonds.

• Example: A corporate bond issued by a company with a low credit rating might offer a higher interest rate than a U.S. Treasury bond to compensate investors for the higher risk of default.

Liquidity Premium

• Definition: The liquidity premium compensates investors for the risk of not being able to quickly sell the investment at its fair value.

• Interpretation: Assets that are less liquid (harder to sell quickly without a significant price concession) typically offer higher interest rates to compensate investors for this risk.

• Example: A small company's bond might offer a liquidity premium because the bond market for such bonds is less active, making it more difficult to sell without affecting the price.

Maturity (Term) Premium

• Definition: The maturity premium compensates investors for the risk associated with holding a longer-term investment, which is more sensitive to changes in interest rates.

• Interpretation: Longer-term bonds are exposed to more significant interest rate risk (the risk that rising interest rates will reduce the bond’s market value), and therefore investors demand a higher interest rate as compensation. The longer the maturity, the higher the maturity premium tends to be.

• Example: A 30-year bond typically offers a higher interest rate than a 10-year bond due to the greater uncertainty and interest rate risk over the longer period.

Putting It All Together: The Nominal Interest Rate

The nominal interest rate on any given investment can be expressed as:

Each component adds up to ensure that investors are adequately compensated for the time value of money, loss of purchasing power, and various risks associated with the investment.

Example Calculation:

Suppose you're considering investing in a corporate bond. The components of the interest rate might be:

• Real Risk-Free Rate: 2%

• Inflation Premium: 3% (reflecting expected inflation)

• Default Risk Premium: 1.5% (reflecting the company's credit risk)

• Liquidity Premium: 0.5% (reflecting the bond's relative illiquidity)

• Maturity Premium: 1% (reflecting the bond’s 10-year maturity)

The nominal interest rate (yield) on the bond would be:

This 8% yield compensates the investor for the various risks associated with the bond, as well as the time value of money and expected inflation.

Money-Weighted Rate of Return (MWRR)

Definition:

The Money-Weighted Rate of Return (MWRR), also known as the internal rate of return (IRR), takes into account the timing and size of cash flows into and out of the portfolio. It is essentially the return that equates the present value of cash inflows with the present value of cash outflows. MWRR is sensitive to the timing of cash flows, making it useful for investors who control cash inflows and outflows.

Calculation:

MWRR is calculated by solving for the rate of return (r) in the following equation:

Where:

Advantages:

• Reflects Investor Control: MWRR reflects the impact of the timing of contributions and withdrawals, which is under the investor’s control.

• Useful for Individual Investors: Particularly relevant for individual investors who make frequent deposits and withdrawals.

Disadvantages:

Cash Flow Sensitivity: The MWRR can be heavily influenced by the timing of cash flows, which might not accurately reflect the performance of the underlying investments.

Time-Weighted Rate of Return (TWRR)

Definition:

The Time-Weighted Rate of Return (TWRR) measures the compounded growth rate of one unit of currency invested in the portfolio over a period of time. It is not affected by the timing or size of cash flows, making it a better measure of the portfolio manager’s performance.

Calculation:

TWRR is calculated by breaking down the total period into sub-periods between cash flows, calculating the return for each sub-period, and then chaining these returns together:  

Calculate the return for each sub-period: 

Where: 

Advantages:

• Reflects Portfolio Performance: TWRR is considered a pure measure of portfolio performance, as it is unaffected by the timing of cash flows.

• Standard for Comparison: Often used by mutual funds and professional money managers to report performance, allowing for consistent comparisons across different portfolios.

Disadvantages:

• Doesn’t Reflect Investor Experience: Since it ignores cash flows, TWRR might not reflect the actual experience of an investor who is adding or withdrawing funds.
  

Evaluating Portfolio Performance

1. MWRR vs. TWRR:

   • Investor’s Perspective: MWRR is more relevant if you want to evaluate how well you have managed your investments considering the timing of your contributions and withdrawals. It shows the actual return you have earned based on your cash flow decisions.

   • Manager’s Perspective: TWRR is better suited for evaluating the performance of the portfolio manager or the underlying investments themselves, independent of cash flow timing.

2. Scenario Analysis:

   • Positive Cash Flows During High Returns: If you invest more money when the portfolio is doing well (high returns), the MWRR will be higher than the TWRR because the impact of the high returns is amplified by the larger investment.

   • Negative Cash Flows During Low Returns: Conversely, if you withdraw money during periods of poor performance, MWRR will typically be lower than TWRR, as the negative returns are magnified by the larger cash outflows.

Portfolio Evaluation:

• MWRR: If you control the cash flows, and those flows are substantial relative to the portfolio, MWRR gives a more accurate picture of your personal investment performance.

• TWRR: If you're evaluating a portfolio manager or comparing the performance of different investment strategies where you have no control over cash flows, TWRR is the more appropriate measure.

Annualized Return Measures

Definition:Annualized return, also known as compound annual growth rate (CAGR), represents the geometric average annual return of an investment over a specified period, assuming that the returns are reinvested. It allows investors to compare the performance of investments over different time periods on an equivalent annual basis.

Calculation:

The formula for calculating the annualized return is:

Where:

• Ending Value is the final value of the investment.

• Beginning Value is the initial value of the investment.

• n is the number of years.

Interpretation:

• Positive Annualized Return: Indicates that the investment grew over the period.

• Negative Annualized Return: Indicates that the investment decreased in value.

Appropriate Uses:

• Long-Term Investments: Annualized return is most appropriate for evaluating investments held over multiple years, providing a smooth average annual rate of return.

• Performance Comparisons: Useful for comparing the performance of different investments or portfolios over different time periods.

Continuously Compounded Returns

Definition:

Continuously compounded returns refer to the return on an investment assuming that it is compounded an infinite number of times per period. This type of return is calculated

using the natural logarithm of the ratio of the ending value to the beginning value of the investment.

Calculation:

The formula for calculating continuously compounded returns is:

Where:

• ln is the natural logarithm.

• Ending Value and Beginning Value are the final and initial values of the investment, respectively.

Interpretation:

• Positive Return: The investment's value has increased.

• Negative Return: The investment's value has decreased.

Appropriate Uses:

• Theoretical Models: Continuously compounded returns are commonly used in financial models, such as the Black-Scholes option pricing model, due to their mathematical properties.

• Short-Term Investments: In the context of short-term trading, continuously compounded returns can provide a more precise measure of return due to frequent compounding.

• Risk Management: This measure is also useful in risk management practices, particularly in calculating Value at Risk (VaR) and in portfolio optimization.
  

Comparison and Contextual Use

• Annualized Returns vs. Continuously Compounded Returns:

• Annualized Returns are more intuitive for most investors, as they relate directly to the concept of annual growth and are easier to compare across different investments and periods.

• Continuously Compounded Returns are often more relevant in academic and professional financial settings, especially where precise calculations and theoretical models are necessary.

• When to Use Each:

  • Annualized Returns are best for evaluating and comparing the long-term performance of investments, such as mutual funds or retirement portfolios.

  • Continuously Compounded Returns are preferable in contexts requiring precise, theoretically sound calculations, particularly in financial modeling, options pricing, and quantitative finance.