Math Problem Statement

Consider the given nominal rate (14 5/8%) compounded daily. (a) Find the annual yield corresponding to the given nominal rate. (b) Interpret the annual yield corresponding to the given nominal rate and find the equivalent simple interest rate.

Solution

The image you uploaded contains a problem regarding a nominal interest rate with daily compounding. Let's break it down:

  1. Given nominal rate: 1458%14 \frac{5}{8} \% compounded daily.

    • Convert 1458%14 \frac{5}{8} \% to a decimal:
      1458=14+58=14+0.625=14.625%14 \frac{5}{8} = 14 + \frac{5}{8} = 14 + 0.625 = 14.625\% Therefore, the nominal interest rate is 14.625%.
  2. Part (a): Find the annual yield corresponding to the given nominal rate.
    The formula for annual yield (Effective Annual Rate, EAR) is: EAR=(1+rn)n1\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 Where:

    • rr is the nominal interest rate as a decimal (14.625% or 0.14625),
    • nn is the number of compounding periods per year (365 for daily compounding).
  3. Part (b): Interpret the annual yield corresponding to the given nominal rate.
    The question asks for the simple interest equivalent, which refers to the nominal interest rate expressed in terms of a simple annual interest rate. This can be compared with the compounded interest, which we find in part (a).

Let's proceed to calculate these values for both parts (a) and (b). I will compute the effective annual yield (EAR) and then compare it with the nominal rate.The annual yield (Effective Annual Rate, EAR) corresponding to the nominal rate of 1458%14 \frac{5}{8} \% compounded daily is approximately 15.75%.

Next, I'll compute the simple interest equivalent as requested in part (b), which is simply the nominal rate provided, but interpreted as the equivalent simple interest rate.The given compound rate is equivalent to 14.63% simple interest.

To summarize:

  • (a) The annual yield (effective annual rate) is approximately 15.75%.
  • (b) The equivalent simple interest rate is 14.63%.

Would you like more details or explanations on these calculations?

Here are five related questions:

  1. How does the effective annual rate (EAR) differ from nominal interest rates?
  2. What happens to the EAR if the number of compounding periods increases?
  3. How is simple interest different from compound interest?
  4. Can you use the EAR formula for monthly or quarterly compounding rates?
  5. Why is the EAR typically higher than the nominal rate with daily compounding?

Tip: Always remember, the more frequently interest is compounded, the higher the effective annual yield will be.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Interest Rates
Compound Interest
Effective Annual Rate
Simple Interest

Formulas

EAR = (1 + r/n)^n - 1
r = nominal interest rate (as a decimal)
n = number of compounding periods per year (daily compounding = 365)

Theorems

Compound Interest Theorem

Suitable Grade Level

College level (Business/Finance/Mathematics)