The effective yield (or effective annual interest rate) for...

The effective yield (or effective annual interest rate) for an investment is the simple interest rate that would yield at the end of one year the same amount as is yielded by the compounded rate that is actually applied. Approximate, to the nearest $0.01 \%$, the effective yield corresponding to an interest rate of $\boldsymbol{r} \%$ per year compounded (a) quarterly and (b) continuously. $$r=3$$

Key Concepts

Quarterly Compounding

Quarterly compounding refers to the application of compound interest four times per year. With each quarter, the interest is calculated on the current balance, and the process results in a higher effective annual rate compared to simple interest applied once a year.

Continuous Compounding

Continuous compounding is a theoretical concept where interest is compounded an infinite number of times per year, leading to the exponential growth governed by the mathematical constant e. It represents the limit of compound interest as the compounding frequency increases without bound.

Compound Interest

Compound interest is the process by which interest is earned on both the initial principal and the accumulated interest from previous periods. This results in exponential growth of the investment, as interest is calculated on an increasingly larger base over time.

Effective Annual Interest Rate (EAR)

The effective annual interest rate is the rate that, when applied as simple interest over one year, yields the same accumulated value as an interest rate that is compounded more frequently. It allows for the comparison of different interest rate compounding methods on a common basis over one year.

Transcript

00:01 In this problem, we need to determine the effective yield.

00:03 Now in the first problem, we need to assume that the interest is compounded quarterly.

00:08 Now if we consider the effective yield to be capital r, then the amount obtained after one year on simple interest should be equal to the amount obtained after one year on compound interest.

00:19 Now if p is the amount invested, then the amount obtained after t years on simple interest is p plus p times capital r times t.

00:29 This should be equal to the compound interest, which is equal to p times 1 plus r by n to the power n t, where smaller is the compound interest rate and n is the number of times that the interest is compounded in a year.

00:42 If we cancel out p from both sides, we will get 1 plus r t is equal to 1 plus r by n to the power n t.

00:50 Now we are considering a time of 1 year, so the value of t is 1, and thus the left -hand side becomes 1 plus r.

00:57 On the right hand side, we have 1 plus small r, the value which is given to be 3, and in percentage form this is 3 by 100, and the value of n will be 4 since the interest is compounded quarterly.

01:09 And in the exponent we have n t, so that's 4 times 1, which is just 4.

01:13 So the value of capital r is 1 plus 3 by 100 divided by 4, whole to the power 4 minus 1, and the value of this is approximately equal to 3 .03%.

01:27 So that is the effective yield in this case.

01:30 Now in the second problem, we need to assume that the interest is compounded continuously...

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