1

The Time Value of Money and Interest Rates

At the heart of finance lies the Time Value of Money (TVM): a dollar today is worth more than a dollar tomorrow. Why? Because money can be invested to earn a return. This principle allows us to compare cash flows that occur at different times.

The tool that makes this comparison possible is the interest rate. But what exactly is an interest rate? It can be interpreted in three powerful ways:

Three Interpretations of an Interest Rate

  • Required Return: The minimum return an investor demands to take on the risk of an investment. If an asset doesn't meet this threshold, it's not worth investing in.
  • Discount Rate: The rate used to calculate the present value of future cash flows. It "discounts" future money back to today's terms.
  • Opportunity Cost: The return you give up by choosing one investment over another. For example, if you spend $1,000 on a vacation instead of investing it at 6%, your opportunity cost is $60 per year.
2

What Makes Up an Interest Rate?

A quoted interest rate isn't just one number — it's a sum of several risk premiums. Think of it as a "risk buffet" that investors are compensated for bearing.

Nominal Interest Rate =
Real Risk-Free Rate + Inflation Premium +
Default Risk Premium + Liquidity Premium + Maturity Premium

Breaking Down the Components

  • Real Risk-Free Rate: The theoretical return on a zero-risk, zero-inflation investment. It reflects the pure time value of money.
  • Inflation Premium: Compensation for the expected erosion of purchasing power over time. If inflation is expected to be 3%, investors demand at least 3% extra.
  • Default Risk Premium: Extra return demanded for the risk that a borrower might fail to repay (e.g., corporate bonds vs. government bonds).
  • Liquidity Premium: Compensation for difficulty in selling an asset quickly at fair value. Illiquid assets (e.g., private equity) carry higher premiums.
  • Maturity Premium: Extra return for holding longer-term bonds, which are more sensitive to interest rate changes and economic uncertainty.

Example: Yield on a 10-Year Corporate Bond

Suppose the real risk-free rate is 1%, expected inflation is 2.5%, default risk is 1.2%, liquidity premium is 0.3%, and maturity premium is 1.0%.

Total Yield = 1% + 2.5% + 1.2% + 0.3% + 1.0% = 6.0%

3

Measuring Investment Returns

An investment's return comes from two sources:

  • Income: Dividends, interest, or coupons.
  • Capital Gain/Loss: Change in the asset's price.

Holding Period Return (HPR)

The Holding Period Return (HPR) is the total return earned over a specific investment period.

HPR = (P₁ - P₀ + I₁) / P₀

Where:

  • P₀: Beginning price
  • P₁: Ending price
  • I₁: Income received (dividend or interest)

Example: HPR Calculation

You buy a stock for $50. It pays a $2 dividend and you sell it for $55.

HPR = (55 - 50 + 2) / 50 = 7 / 50 = 14%

Averaging Returns Over Time

When returns vary across periods, we use different averages to summarize performance.

Arithmetic Mean Return

The simple average of periodic returns. Best for forecasting next period's expected return.

Arithmetic Mean = (R₁ + R₂ + ... + Rₙ) / n

Example: Arithmetic Mean

Returns: Year 1: 10%, Year 2: -5%, Year 3: 15%

Arithmetic Mean = (10% - 5% + 15%) / 3 = 20% / 3 = 6.67%

Geometric Mean Return

The compound growth rate over multiple periods. Most accurate for measuring long-term "buy-and-hold" performance.

Geometric Mean = [(1 + R₁) × (1 + R₂) × ... × (1 + Rₙ)]^(1/n) - 1

Example: Geometric Mean

Same returns: 10%, -5%, 15%

(1.10) × (0.95) × (1.15) = 1.20175

(1.20175)^(1/3) ≈ 1.063

1.063 - 1 = 6.3%

✅ Note: Geometric mean < Arithmetic mean when returns vary.

Harmonic Mean Return

Used when averaging rates (e.g., cost-averaging $100/month into a stock).

Harmonic Mean = n / [1/R₁ + 1/R₂ + ... + 1/Rₙ]

Always ≤ Geometric Mean ≤ Arithmetic Mean when returns vary.

Robust Averaging: Dealing with Outliers

  • Trimmed Mean: Remove top and bottom 5% of returns, then compute the average.
  • Winsorized Mean: Replace extreme values with nearest non-extreme values, then average.
4

Money-Weighted vs. Time-Weighted Return

When an investor makes deposits or withdrawals, performance measurement gets tricky. Two methods solve this:

Metric Money-Weighted Return (MWR) Time-Weighted Return (TWR)
Definition Compound growth rate of all funds in the portfolio (like IRR). Compound growth rate of $1 invested at the start.
Cash Flows Considers timing and size of contributions/withdrawals. Ignores cash flows — focuses on portfolio manager's performance.
Use Case Evaluates investor's overall experience (personal performance). Evaluates portfolio manager's skill (industry standard).
Sensitivity High: Large inflows before gains inflate MWR. Low: Removes investor behavior from the equation.
Limitations Not comparable across investors due to different cash flows. Requires valuing portfolio at every cash flow (costly).

Example: Why TWR is Preferred for Managers

Two clients have the same fund. One adds money just before a big market rise; the other withdraws. Their MWRs will differ wildly — but the manager's performance (TWR) is the same.

✅ So TWR isolates the manager's skill from investor timing.

5

Annualized and Continuously Compounded Returns

To compare investments fairly, we annualize returns — converting them to an equivalent one-year basis.

Annualized Return (Periodic Compounding)

Converts periodic returns (monthly, quarterly) to an annual rate assuming reinvestment.

Annualized Return = (1 + Periodic Return)^c - 1
Where c = number of periods per year

Example: Annualizing a Quarterly Return

Quarterly return = 3% → Annualized Return = (1.03)⁴ - 1 = 1.1255 - 1 = 12.55%

⚠️ Assumption: You can reinvest at 3% each quarter — may not be realistic.

Continuously Compounded Return

Assumes compounding happens infinitely often — a powerful tool in finance and derivatives pricing.

Continuous Return = ln(Ending Price / Beginning Price)
For multiple periods: Total Return = ln(Pₜ / P₀)

Where ln = natural logarithm

✅ Key fact: Continuously compounded return is always < HPR for the same period.

Example: Continuous Compounding

Stock rises from $100 to $110 → HPR = 10%

Continuous Return = ln(110 / 100) = ln(1.10) ≈ 0.0953 = 9.53%

6

Adjusted Return Measures

To get a true picture of investment performance, we adjust returns for fees, taxes, and inflation.

Gross vs. Net Return

  • Gross Return: Return before fees and expenses. Measures pure investment performance.
  • Net Return: Return after all fees. What the investor actually keeps.

Pre-Tax vs. After-Tax Return

  • Different returns (dividends, interest, capital gains) are taxed at different rates.
  • After-Tax Return: Net return after all taxes are paid.
  • Tax-efficient strategies (e.g., tax-loss harvesting, holding tax-advantaged accounts) improve after-tax returns.

Nominal vs. Real Return

Adjusts for inflation — shows changes in purchasing power.

1 + Real Return = (1 + Nominal Return) / (1 + Inflation Rate)

Approximation: Real Return ≈ Nominal Return - Inflation

Example: Real Return

Nominal return = 8%, Inflation = 3%

1 + Real Return = 1.08 / 1.03 ≈ 1.0485 → Real Return ≈ 4.85%

Leveraged Return

Using borrowed money or derivatives to amplify exposure.

  • Increases both potential gains and losses.
  • Common in hedge funds, futures trading, and margin investing.
  • High risk: Can lead to large losses or margin calls.
Progress:
Chapter 1 of 11