Interest Rate Measurement Study Notes

Introduction to Interest Rate Measurement

Financial Transactions and Mathematical Modeling: Financial activities involve savers, borrowers, and investors participating in transactions involving insurance, pensions, and government or corporate liabilities. Due to the increasing complexity of these investments, precise mathematical modeling is necessary to translate vague verbal descriptions into unambiguous formulations.
The Time Value of Money: Interest represents the central component of almost all financial transactions. It is generally defined as the consideration or rent paid by a borrower to a lender for the use of funds over a period of time.
The Federal Reserve and Interest Levels: The U.S. Federal Reserve Board sets the "federal funds discount rate," which is the target rate for interbank lending. This rate influences the general cost of borrowing and affects stock and bond markets. Generally, higher interest rates reduce the value of other investments because earning a higher rate elsewhere makes lower-yield investments less attractive.
Case Study: "Irrational Exuberance": On December 5, 1996, Chairman Alan Greenspan delivered a lecture mentioning the potential negative consequences of overvalued markets. This led to global market drops: Japan's index fell $3.2\%$ , Hong Kong dropped almost $3\%$ , Germany fell $4\%$ , and London fell $2\%$ . The New York Stock Exchange saw a $2\%$ drop within the first 30 minutes of opening the next day.

Market Interest Rates and Practical Contexts

Libor (London Interbank Overnight Rate): An international rate for very short-term loans in U.S. dollars between banks.
Prime Rate: The interest rate banks charge their most creditworthy customers ( $4.00\%$ as of April 2017).
Adjustable Rate Mortgage (ARM): A mortgage where interest is reset periodically based on market changes.
April 17, 2017 Key Rates:
- Fed Reserve Target Rate: $0.91\%$
- 3-Month Libor: $1.15\%$
- Prime Rate: $4.00\%$
- AAA Average 20-Year Corporate Bond Yields: $3.54\%$
- High Yield Bonds: $5.85\%$
- 15-Year Mortgage: $3.36\%$
- 30-Year Mortgage: $4.08\%$
- 1-Year ARM: $3.18\%$
U.S. Treasuries (April 2017):
- 3-Month Bill Discount Rate: $0.20728$
- 12-Month Bill Discount Rate: $1.03639$

Interest Accumulation and Effective Rates

Compound Interest Reinvestment: Interest is typically quoted as an annual percentage. As interest is credited, it is credited to the principal and begins to earn interest itself (compounding).
General Accumulation Formula: For an initial deposit $C$ at annual interest rate $i$ , the accumulated value at the end of year $n$ is: $C(1+i)^n$
Varying Interest Rates: If rates change annually ( $i_1, i_2, \dots, i_n$ ), the growth factor for year $t$ is $(1+i_t)$ , resulting in: $C(1+i_1)(1+i_2)\dots(1+i_n)$
Average Annual Rate of Return: In practice, this refers to the annual compound rate of interest ( $i$ ) that results in the total growth over a period. For a 5-year period with total growth $G$ : $(1+i)^5 = G$
Definition 1.1 - Annual Effective Rate of Interest: The percentage change in an investment's value from the start to the end of the year, regardless of intermediate behavior.
Definition 1.2 - Equivalent Rates of Interest: Two rates are equivalent if they result in the same accumulated values at each point in time.

Simple vs. Compound Interest

Definition 1.3 - Accumulation Factor and Function: $a(t)$ is the accumulated value at time $t$ of an investment of $1$ made at time $0$ . The accumulated amount function is $A(t) = A(0) \times a(t)$ .
Definition 1.4 - Compound Interest Accumulation: At effective rate $i$ , the accumulation factor is: $a(t) = (1+i)^t$
Definition 1.5 - Simple Interest Accumulation: Interest is calculated only on the original principal. At annual rate $i$ and time $t$ (in years): $a(t) = 1 + it$
Comparison: Simple interest accumulation is larger than compound interest for $0 < t < 1$ , whereas compound interest is larger for $t > 1$ .
Promissory Notes: Short-term contracts requiring the borrower to pay principal plus simple interest. There is an inverse relationship between yield and price: as the demanded yield increases, the purchase price of the fixed-income investment decreases.

Present Value

Definition 1.6 - One Period Present Value Factor: The amount needed now to accumulate to $1$ in one period. Denoted as $v$ : $v = \frac{1}{1+i}$
Multiple Periods: The present value of amount $K$ due at time $t$ is $Kv^t = K(1+i)^{-t}$ .
Yield-Price Relationship: Higher interest rates correspond to smaller amounts invested to reach a target value.
Canadian Treasury Bills: Valued using simple interest and a 365-day year. Price is calculated as: $\text{Price} = \frac{100}{1 + (i \times \frac{d}{365})}$

Equation of Value

Balancing Cash Flows: A mathematical representation of a transaction that balances dated cash outflows and inflows. It requires choosing a valuation date (reference time point).
Components: On the valuation date, the following must be equal:
1. The value of payments disbursed (accumulated values of past payments + present values of future payments).
2. The value of payments received (accumulated values of past receipts + present values of future receipts).

Nominal Rates of Interest

Definition 1.7 - Nominal Annual Rate of Interest: An annual rate denoted $i^{(m)}$ compounded $m$ times per year. The effective interest rate for each fractional period ( $\frac{1}{m}$ years) is $\frac{i^{(m)}}{m}$ .
Equivalence Formula: $1+i = \left(1 + \frac{i^{(m)}}{m}\right)^m$
Practical Examples: A credit card quoting a nominal $24\%$ payable monthly has an effective monthly rate of $2\%$ and an annual effective rate of $26.82\%$ .
Conversion and Comparison: To compare different interest conversion periods, rates must be transformed to a common period (usually annual effective).

Effective and Nominal Rates of Discount

Arrears vs. Advance: Interest is typically paid in arrears (end of period). If interest is paid in advance, the quoted rate is the rate of discount ( $d$ ).
Definition 1.8 - Annual Effective Rate of Discount: $d = \frac{A(1) - A(0)}{A(1)}$
Relationship with i: $d = \frac{i}{1+i} \quad \text{and} \quad i = \frac{d}{1-d}$ $1-d = v = (1+i)^{-1}$
Definition 1.9 - Simple Discount: Used primarily for short-term periods ( $t < 1$ ), such as U.S. Treasury Bills. Present value is defined as: $1 - dt$
U.S. Treasury Bills: Use a 360-day year and simple discount for pricing. The "Investment Rate" quoted is the equivalent simple interest annual return based on a 365-day year.
Definition 1.10 - Nominal Annual Rate of Discount: Denoted $d^{(m)}$ , compounded $m$ times per year. The periodic discount rate is $\frac{d^{(m)}}{m}$ , leading to the relation: $1-d = \left(1 - \frac{d^{(m)}}{m}\right)^m$

The Force of Interest

Continuous Growth: Modeled as the limit of discrete processes as the interval approaches zero.
Definition 1.11 - Force of Interest: The instantaneous rate of growth per dollar invested at time point $t$ : $\delta_t = \frac{A'(t)}{A(t)} = \frac{d}{dt} \ln(A(t))$
Accumulation with Constant Force: If the force of interest is constant (denoted $\delta$ ), then: $1+i = e^\delta \quad \text{and} \quad \delta = \ln(1+i)$
Accumulation with Variable Force: $A(n) = A(0) \exp\left( \int_0^n \delta_t \,dt \right)$
Overnight Rate: Nominal annual interest rate convertible daily used for one-day interbank loans.

Inflation and the Real Rate of Interest

Consumer Price Index (CPI): Measures the change in the cost of a basket of consumer items to track inflation ( $r$ ).
Definition 1.12 - Real Rate of Interest: The inflation-adjusted return ( $i_{real}$ ): $i_{real} = \frac{i-r}{1+r}$
Simplistic measure: The difference $i-r$ is a common approximation but is not theoretically exact.
Hyperinflation Examples:
- Germany (1922-1923): Prices grew by a factor of 20 billion.
- Hungary (1946): Inflation exceeded 4 quintillion percent (prices doubled every 15 hours).
- Yugoslavia (1990-1994): Currency revaluation resulted in 1 pre-1990 Dinar equaling $1.2 \times 10^{27}$ 1994 Dinars.
Continuous Inflation: The real growth rate can be expressed as $\delta - \delta_{inflation}$ .

What does the time value of money primarily indicate?
- A) Money today is worth less than money in the future
- B) Money today is worth more than the same amount in the future
- C) Money has no intrinsic value
- D) Money's value remains constant over time
- Correct Answer: B
What is the primary role of the U.S. Federal Reserve regarding interest rates?
- A) To increase government liabilities
- B) To establish the federal funds discount rate
- C) To eliminate interest rates altogether
- D) To control stock prices
- Correct Answer: B
Which of the following describes the purpose of the London Interbank Overnight Rate (Libor)?
- A) The rate used for long-term mortgage loans
- B) The benchmark for international short-term loans among banks
- C) The interest rate on personal loans
- D) An obsolete financial metric
- Correct Answer: B
What is the Prime Rate?
- A) The rate for short-term loans in foreign currencies
- B) The interest rate banks charge their most creditworthy customers
- C) The average interest rate for all mortgages
- D) The highest interest rate available in the market
- Correct Answer: B
On what basis is an Adjustable Rate Mortgage (ARM) structured?
- A) Fixed interest for the loan duration
- B) Periodic interest rate resets based on market conditions
- C) Interest rates adjusted annually with inflation
- D) Interest rates that always decrease
- Correct Answer: B
What does the formula $C(1+i)^n$ represent?
- A) The total interest paid over the life of a loan
- B) The accumulated value at the end of year n for a deposit C
- C) The monthly payment for a fixed-rate mortgage
- D) The present value of future cash flows
- Correct Answer: B
If a borrower pays simple interest, how is the interest calculated?
- A) Based on the principal plus previously acquired interest
- B) On only the original principal amount
- C) Using complex compounding formulas
- D) Based on total outstanding debt
- Correct Answer: B
What is the annual effective rate of interest?
- A) The nominal interest rate without any adjustments
- B) The percentage change in an investment's value over one year
- C) The average of all interest rates applied in a year
- D) A rate used only in government loans
- Correct Answer: B
What does the present value factor $v = \frac{1}{1+i}$ represent?
- A) The value of an investment at maturity
- B) The current amount needed to achieve $1 in the future
- C) The total amount of interest accumulated
- D) The discount rate for future cash flows
- Correct Answer: B
What is influenced by the federal funds discount rate?
- A) It affects inflation rates directly
- B) It influences interbank lending and general borrowing costs
- C) It has no significant effect on the economy
- D) It determines the stock prices exclusively
- Correct Answer: B
In the context of interest rates, what does an inverse relationship between yield and price imply?
- A) Higher yields lead to higher purchase prices
- B) As yields rise, the prices of fixed-income investments fall
- C) Market prices remain unaffected by changes in yield
- D) All investments are equally affected by yield changes
- Correct Answer: B
Which of the following is a defining characteristic of compound interest?
- A) Interest is calculated solely on the principal amount
- B) Interest earns interest over time
- C) It is always a fixed percentage
- D) It is not used in real-world financial scenarios
- Correct Answer: B
What does the formula $Kv^t = K(1+i)^{-t}$ denote?
- A) The future value of an investment
- B) The present value of an amount $K$ due at time $t$
- C) The calculation for annuities
- D) The total interest on a loan
- Correct Answer: B
What occurs if interest rates increase in the market?
- A) The present value of future cash flows increases
- B) It's harder to find creditworthy borrowers
- C) The value of existing bonds generally decreases
- D) Borrowers benefit from lower overall payments
- Correct Answer: C
How is the annual effective rate of discount related to its corresponding interest rate?
- A) They are completely independent
- B) They are identical
- C) They have a mathematical relationship expressed as $d = \frac{i}{1+i}$
- D) The discount rate is always higher
- Correct Answer: C

These questions aim to cover various aspects of interest rate measurements and their implications in financial contexts.