Question

Suppose that the following spot rates are provided by central London banks (LIBOR, the London Interbank Offer Rate, is the rate at which money can be deposited; LIBID, the London Interbank Bid Rate, is the rate at which money can be borrowed): \(\begin{array}{lcc}\text { Rate } & \text { LIBOR } & \text { LIBID } \\ 1 \text { month } & 8.41 \% & 8.59 \% \\ 2 \text { months } & 8.44 \% & 8.64 \% \\\ 3 \text { months } & 9.01 \% & 9.23 \% \\ 6 \text { months } & 9.35 \% & 9.54 \%\end{array}\) As a bank manager acting for a customer who wishes to arrange a loan of \(\$ 100,000\) in a month's time for a period of 5 months, what rate could you offer and how would you construct the loan? Suppose that another institution offers the possibility of making a deposit for 4 months, starting 2 months from now, at a rate of 10.23\%. Does this present an arbitrage op: portunity? All rates stated in this exercise are contimuous compounding rates.

Short answer

Offer the loan at a 9.35% rate and check for arbitrage by comparing borrowing and depositing rates.

Step-by-step solution

Identify Relevant Rates

Given the need to arrange a loan starting in 1 month for a period of 5 months, identify the relevant 1-month and 6-month rates from the provided table. LIBOR for 1 month is 8.41% and for 6 months is 9.35%. The 2-month and 4-month deposit starting 2 months from now has a rate of 10.23%.

Calculate Future Value of Initial Loan

Using the 1-month LIBOR of 8.41%, calculate the amount for one month later. Use the continuous compounding formula: Future Value = Principal \times e^{(Rate \times Time)} \[ Future Value = 100,000 \times e^{(0.0841 \times (1/12))} \] Calculate the result.

Construct the Loan Using Available Rates

Since the loan is for 5 months starting after the initial 1-month period, use the 6-month LIBOR rate for the remaining 5 months. Calculate how much will be repaid in 5 months using this rate starting from the future value calculated in Step 2.

Check for Arbitrage with the Additional Deposit

Analyze the given opportunity to deposit for 4 months at 10.23%. Compare the returns and costs involved between borrowing and lending after 2 months to identify if there's an arbitrage opportunity. Calculate the amounts that can be borrowed and deposited to see if there is a risk-free profit.

Summarize the Findings

Summarize the calculations and determine which rates to use for the loan administration. Conclude if the arbitrage opportunity is present based on calculated values.

Key concepts

LIBOR

The London Interbank Offer Rate, commonly known as LIBOR, is a key benchmark interest rate at which major global banks lend to one another in the international interbank market for short-term loans. LIBOR is pivotal in financial engineering because it serves as a reference rate for various financial instruments including loans, derivatives, and mortgages. In the given exercise, we see different LIBOR rates for various durations—1 month, 2 months, 3 months, and 6 months. These rates indicate the cost at which banks would deposit money at different time frames. The LIBOR rate reflects the borrowing cost for banks and thus influences lending rates for customers.
For example, when dealing with an interest rate of 8.41% for a 1-month period, the rate is understood to be continuously compounded. Continuous compounding means that interest is calculated and added to the principal infinitely many times in a period, leading to the formula: \[A = P \times e^{rt}\] where A is the future value, P is the principal amount, r is the rate, and t is the time period. Effectively, LIBOR is central to our understanding of short-term interest rates and plays a crucial role in managing banking operations and customer loans.

Arbitrage

Arbitrage is the practice of taking advantage of a price difference between two or more markets, generating a profit with minimal risk. It is a critical concept in financial markets, especially for identifying opportunities where the current market does not reflect the fair value. Arbitrage ensures that prices do not deviate significantly from fair value over time. In the given exercise, you might find an arbitrage opportunity by comparing the returns on a loan versus a deposit.
For instance, if the LIBOR rate for a 6-month deposit is different from an alternative rate offered for a 4-month deposit starting 2 months from now, we consider if the differences in these rates create a risk-free profit scenario. By borrowing at a lower rate using one set of terms, and simultaneously lending at a higher rate using different terms, profit can be made due to the discrepancies in interest rates and deposit terms. This process ensures efficient markets where rates adjust accordingly to eliminate such risk-free profit opportunities.

Continuous Compounding

Continuous compounding is a fundamental concept in financial mathematics. Unlike simple or regular compounding, where interest is added at specific intervals (daily, monthly, yearly), continuous compounding assumes that interest is added constantly at every possible instant. The formula for calculating future value with continuous compounding is: \[A = P \times e^{rt}\] here, `e` is the base of the natural logarithm (approximately equal to 2.71828). `P` is the principal amount, `r` is the annual interest rate, and `t` is the time in years.
In the context of the exercise, continuous compounding helps in accurately calculating the exact amount needed at future dates. This precision is crucial when dealing with large amounts of money and helps in minimizing errors in financial calculations. Continuous compounding is used to determine how much a present value will grow over time given a specific interest rate, providing more accurate and finer results than other compounding methods.