Find the annual rate, r that produces an effective annual yield of 6.3% when compounded continuosly.
To find the annual rate, r, that produces an effective annual yield of 6.3% when compounded continuously, we can use the formula for continuous compounding:
A = P * e^(rt)
Where:
A = final amount (compounded continuously)
P = principal amount (initial investment)
e = mathematical constant approximately equal to 2.71828
r = annual interest rate (unknown)
t = time (in years)
The effective annual yield formula is:
A = P * (1 + r/n)^(n*t)
Where:
n = number of compounding periods per year (for continuous compounding, n approaches infinity)
By comparing both formulas, we can conclude that:
e^(rt) = (1 + r/n)^(n*t)
Let's substitute the given values. We want an effective annual yield of 6.3%, which means:
A = P * (1 + 0.063)
n approaches infinity for continuous compounding, therefore:
e^(rt) = (1 + 0.063)^(rt)
To find the annual rate, we rearrange the equation:
e^(rt) = 1 + 0.063
e^(rt) - 1 = 0.063
Now, take the natural logarithm (ln) of both sides:
ln(e^(rt) - 1) = ln(0.063)
Using logarithmic properties, we can simplify the equation:
rt*ln(e) = ln(0.063)
rt = ln(0.063)
Finally, divide both sides by t:
r = ln(0.063) / t
Since the time (t) is not given in the question, we cannot calculate the specific value of r.
To find the annual rate, r, that produces an effective annual yield of 6.3% when compounded continuously, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A is the final amount (effective annual yield)
P is the principal amount (initial investment)
e is the base of natural logarithm (approximately 2.71828)
r is the annual interest rate
t is the time in years
In this case, we are given the effective annual yield, which means A = 1 (since it represents 100% or the initial amount plus the interest earned). Plugging in the values, we get:
1 = P * e^(rt)
We are looking for the annual rate, r, so we need to isolate it. Divide both sides of the equation by P:
1/P = e^(rt)
Next, take the natural logarithm (ln) of both sides of the equation to remove the exponential function:
ln(1/P) = ln(e^(rt))
Using the property of logarithms, we can bring down the exponent:
ln(1/P) = rt * ln(e)
The natural logarithm of e (ln(e)) is equal to 1, so the equation simplifies to:
ln(1/P) = rt
Finally, we solve for r by dividing both sides by t:
r = ln(1/P) / t
Now, we can substitute the given values to find the annual rate:
r = ln(1/P) / t = ln(1/1) / t = 0 / t = 0
Therefore, the annual rate, r, that produces an effective annual yield of 6.3% when compounded continuously is 0%. This means there is no interest being earned or added on to the initial investment.
let that rate be i
1+i = e^(.063)
1+i = 1.0650268
i = .0650268 = 6.50268 %