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Chapter 4: Problem 9

What rate of interest with continuous compounding is equivalent to \(15 \%\) per annum with monthly compounding?

Short Answer

Expert verified
Equivalent continuous compounding rate is approximately 14.934%.

Step by step solution

01

Understand the Problem

We need to find an equivalent interest rate with continuous compounding given a nominal interest rate of 15% per annum with monthly compounding.
02

Identify the Monthly Compounded Rate

The nominal interest rate given is 15% per annum with monthly compounding. We convert this to the effective annual rate using the formula:\[ r_{ ext{effective}} = \left(1 + \frac{r}{n}\right)^n - 1 \]where \( r = 0.15 \) and \( n = 12 \).
03

Calculate the Effective Annual Rate

Substitute the values into the formula:\[ r_{ ext{effective}} = \left(1 + \frac{0.15}{12}\right)^{12} - 1 \]Calculate:\[ r_{ ext{effective}} = \left(1 + 0.0125\right)^{12} - 1 \]\[ r_{ ext{effective}} = 1.0125^{12} - 1 \]\[ r_{ ext{effective}} = 1.16075 - 1 \]\[ r_{ ext{effective}} \approx 0.16075 \text{ or } 16.075\% \]
04

Use the Continuous Compounding Formula

Now that we have the effective annual rate, we find the equivalent rate for continuous compounding using the formula:\[ r_{ ext{continuous}} = \ln(1 + r_{ ext{effective}}) \]
05

Calculate the Continuous Compounding Rate

Substitute the effective annual rate into the formula:\[ r_{ ext{continuous}} = \ln(1 + 0.16075) \]\[ r_{ ext{continuous}} \approx \ln(1.16075) \]\[ r_{ ext{continuous}} \approx 0.14934 \]So, the rate is approximately \( 14.934\% \).

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Key Concepts

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

Interest Rate Conversion
Interest rate conversion is a fundamental concept in financial mathematics. It involves changing an interest rate from one compounding frequency to another without altering the financial value of the investment or loan.
This ensures you can compare different investment options or loans on an equivalent basis.
  • For example, converting a rate that compounds monthly into one that compounds continuously can help in understanding the true cost or return of financial products.
  • Typically, conversion involves understanding the relationships between nominal, effective, and continuous rates.
Without proper conversion, comparing rates from different financial contracts can be misleading.
Effective Annual Rate
The effective annual rate (EAR) expresses the true financial cost or benefit of an interest rate over a year, considering the impact of compounding. It gives a clearer picture of what you're actually earning or paying.
  • The formula for calculating EAR is: \[ r_{\text{effective}} = \left(1 + \frac{r}{n}\right)^n - 1 \]
  • Here, \( r \) is the nominal interest rate, and \( n \) is the compounding periods per year.
This rate allows investors and borrowers to make more informed financial decisions.
By converting a nominal rate to an EAR, you accurately reflect the effect of multiple compounding periods.
Nominal Interest Rate
The nominal interest rate is the reported or stated interest rate of a financial product before taking compounding into account. It is often expressed on an annual basis without considering how often the interest is applied.
  • For example, a nominal interest rate might be 15% per annum, but this doesn't tell the full story.
  • When compounded monthly, this rate will yield a higher effective annual rate due to the effect of earning interest on interest multiple times a year.
The nominal rate is important for discussion but does not fully capture the reality of financial growth or costs.
Financial Mathematics
Financial mathematics applies mathematical formulas and techniques to solve financial problems, including evaluating interest rates, investment growth, and loan repayments.
It's a key skill for anyone in finance or investing.
  • Understanding continuous compounding is part of financial mathematics, which assumes that interest is calculated and added at every possible moment.
  • This leads to the formula \( r_{\text{continuous}} = \ln(1 + r_{\text{effective}}) \), representing a more granular realistic rate of return.
Having a firm grasp of financial mathematics can empower individuals to make better-informed decisions when dealing with various financial products.

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Most popular questions from this chapter

The following table gives the prices of bonds: \begin{tabular}{cccc} \hline Bond principal \((\$)\) & Time to maturity \((\) years \()\) & Annual coupon \(^{*}\) \((8)\) & Bond price \((\$)\) \\ \hline 100 & \(0.50\) & \(0.0\) & 98 \\ 100 & \(1.00\) & \(0.0\) & 95 \\ 100 & \(1.50\) & \(6.2\) & 101 \\ 100 & \(2.00\) & \(8.0\) & 104 \\ \hline \end{tabular} * Haif the stated coupon is assumed to be paid every six months. (a) Calculate zero rates for maturities of 6 months, 12 months, 18 months, and 24 months. (b) What are the forward rates for the following periods: 6 months to 12 months, 12 months to 18 months, and 18 months to 24 months? (c) What are the 6 -month, 12 -month, 18 -month, and 24 -month par yields for bonds that provide semiannual coupon payments? (d) Estimate the price and yield of a 2 -year bond providing a semiannual coupon of \(7 \%\) per annum.

The cash prices of 6 -month and 1 -year Treasury bills are 94.0 and \(89.0 .\) A 1.5 -year bond that will pay coupons of \(\$ 4\) every 6 months currently sells for \(\$ 94.84\). A 2 -year bond that will pay coupons of \(\$ 5\) every 6 months currently sells for \(\$ 97.12\). Calculate the 6 -month, 1-year, 1.5-year, and 2-year zero rates.

A bank quotes an interest rate of \(14 \%\) per annum with quarterly compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?

Suppose that 6 -month, 12 -month, 18 -month, 24 -month, and 30 -month zero rates are, respectively, \(4 \%, 4.2 \%, 4.4 \%, 4.6 \%\), and \(4.8 \%\) per annum, with continuous compounding. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months and pays a coupon of \(4 \%\) per annum semiannually.

"When the zero curve is upward-sloping, the zero rate for a particular maturity is greater than the par yield for that maturity. When the zero curve is downward-sloping the reverse is true." Explain why this is so.
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