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Chapter 3: Problem 8

An individual receives \(\$ 1,100\) in 1 year in return for an investment of \(\$ 1,000\) now. What is the percentage retum per annum with (a) annual compounding? (b) semiannual compounding? (c) monthly compounding? (d) continuous compounding?

Short Answer

Expert verified
(a) 10%, (b) 9.762%, (c) 9.5688%, (d) 9.531%

Step by step solution

01

Identify the Known Variables

We invest \(P = \\(1000\) and get \(F = \\)1100\) after 1 year. We need to determine the interest rate \(r\) that leads to this final amount using various compounding methods.
02

Calculate Annual Compounding Rate

For annual compounding, use the formula:\[ F = P(1+r)^n \]where \(n = 1\) year. Rearrange to solve for \(r\):\[ 1100 = 1000(1+r) \]\[ 1+r = \frac{1100}{1000} = 1.1 \]\[ r = 1.1 - 1 = 0.1 \]The percentage return is \(r \times 100\% = 10\%\).
03

Calculate Semiannual Compounding Rate

For semiannual compounding, use the formula:\[ F = P\left(1 + \frac{r}{2}\right)^{2n} \]Plug in \(n = 1\):\[ 1100 = 1000\left(1 + \frac{r}{2}\right)^2 \]\[ 1.1 = \left(1 + \frac{r}{2}\right)^2 \]Take the square root of both sides:\[ 1 + \frac{r}{2} = \sqrt{1.1} \approx 1.04881 \]\[ \frac{r}{2} = 1.04881 - 1 = 0.04881 \]\[ r = 0.04881 \times 2 \approx 0.09762 \]The percentage return is \(r \times 100\% \approx 9.762\%\).
04

Calculate Monthly Compounding Rate

For monthly compounding, use the formula:\[ F = P\left(1 + \frac{r}{12}\right)^{12n} \]For \(n = 1\):\[ 1100 = 1000\left(1 + \frac{r}{12}\right)^{12} \]\[ 1.1 = \left(1 + \frac{r}{12}\right)^{12} \]Take the 12th root of both sides:\[ 1 + \frac{r}{12} = 1.1^{\frac{1}{12}} \approx 1.007974 \]\[ \frac{r}{12} = 1.007974 - 1 = 0.007974 \]\[ r = 0.007974 \times 12 = 0.095688 \]The percentage return is \(r \times 100\% \approx 9.5688\%\).
05

Calculate Continuous Compounding Rate

For continuous compounding, use the formula:\[ F = Pe^{rn} \]For \(n = 1\):\[ 1100 = 1000e^r \]\[ e^r = \frac{1100}{1000} = 1.1 \]Take the natural logarithm of both sides:\[ r = \ln(1.1) \approx 0.09531 \]The percentage return is \(r \times 100\% \approx 9.531\%\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Financial Mathematics
In financial mathematics, one of the most fundamental concepts is understanding how investments grow over time. Compounding interest, a key part of financial mathematics, allows investors to see how their initial investments can increase as returns are reinvested over time. This is accomplished through various methods of compounding, such as annual, semiannual, monthly, or continuous, each resulting in different growth outcomes for the investment.

Understanding these compounding methods can help investors make better decisions. For example, a higher frequency of compounding typically results in more accumulated interest, and thus, a larger return on investment. Financial mathematics helps individuals and businesses forecast the future value of investments, enabling them to plan effectively.

When working through financial mathematics problems, it's crucial to identify known variables like the principal amount, interest rate, and compounding frequency. Mastery of these concepts underpins successful investment planning and financial decision-making.
Investment Analysis
Investment analysis involves evaluating an investment for its potential profitability and risk. Understanding compounding interest is critical to this process, as it affects the returns on investment over time. Through investment analysis, investors can assess the trade-offs between different investment opportunities and make informed financial decisions.

When analyzing investments, it is important to consider the compounding frequency's impact on the total return. An example can be investing $1,000 today in hopes of receiving $1,100 after one year. Depending on how often the interest is compounded, the effective interest rate, and thus the return, will vary. Investment analysis often involves comparing multiple scenarios to identify which compounding frequency enhances returns.

A thorough investment analysis also takes into account other factors such as market conditions, inflation rates, and other potential risks. With this comprehensive approach, investors are better able to optimize their portfolios and achieve their financial goals.
Interest Rate Calculation
Calculating interest rates is a fundamental skill necessary for evaluating investments and loans. The interest rate determines how much an investment grows over time. It can be expressed with various compounding frequencies such as annual, semiannual, monthly, or continuous.

For annual compounding, the formula takes the form: \[ F = P(1+r)^n \] Rearranging this formula allows us to solve for the interest rate \( r \). If compounding is more frequent, other adjustments are made, such as dividing the rate by the number of periods for semiannual or monthly compounding:
- Semiannual: \[ F = P \left(1 + \frac{r}{2}\right)^{2n} \]
- Monthly: \[ F = P \left(1 + \frac{r}{12}\right)^{12n} \] For continuous compounding, we switch to the natural exponential function \( e \) with the formula:
\[ F = Pe^{rn} \]
Calculating these rates gives insight into the return on investments, helping investors understand what percentage increase their investment will yield over a specified period. Mastery in interest rate calculation enables more accurate forecasting of future financial performance.

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Most popular questions from this chapter

Suppose that \(F_{1}\) and \(F_{2}\) are two futures contracts on the same commodity with maturity dates of \(t_{1}\) and \(t_{2}\) and \(t_{2}>t_{1} .\) Prove that \(F_{2} \leq\left(F_{1}+U\right) e^{r\left(t_{2}-t_{1}\right)}\) where \(r\) is the risk-free interest rate (assumed to be constant) between \(t_{1}\) and \(t_{2}\) and \(U\) is the cost of storing the commodity between times \(t_{1}\) and \(t_{2}\) discounted to time \(t_{1}\) at the risk-free rate. For the purposes of this problem, assume that a futures contract is the same as a forward contract.

Suppose that you enter into a 6 -month forward contract on a non-dividend- paying stock when the stock price is \(\$ 30\) and the risk-free interest rate (with continuous compounding) is \(12 \%\) per annum. What is the forward price?

Assume that the risk-free interest rate is \(9 \%\) per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, the dividend yield is \(5 \%\) per annum. In other months. it is \(2 \%\) per annum. Suppose that the value of the index on July 31,1992 is 300 . What is the futures price for a contract deliverable on December \(31,1992 ?\)

When a known cash outflow in a foreign currency is hedged by a company using a forward contract, there is no foreign exchange risk. When it is hedged using futures contracts, the marking to market process does leave the company exposed to some risk. Explain the nature of this risk. In particular, consider whether the company is better off using a futures contract or a forward contract when (a) the value of the foreign currency falls rapidly during the life of the contract (b) the value of the foreign currency rises rapidly during the life of the contract (c) the value of the foreign currency first rises and then falls back to its initial level (d) the value of the foreign currency first falls and then rises back to its initial level Assume that the forward price equals the futures price.

The 2 -month interest rates in Switzerland and the United States with continuous compounding are \(3 \%\) and \(8 \%\) per annum, respectively. The spot price of the Swiss franc is \(\$ 0.6500 .\) The futures price for a contract deliverable in 2 months is \(\$ 0.6600\). What arbitrage opportunities does this create?
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