what normal annual interst rate has an effective annual yield of 7.8% under continuous compounding?
We calculate the amount after t=1 year at rate r under continuous compounding, and equate it to 1.078:
1.078=1*ert
take natural log on both sides:
ln(1.078)=r*1
r=ln(1.078)
=0.07511
or 7.511% p.a.
thanks
To find the normal annual interest rate with an effective annual yield of 7.8% under continuous compounding, we can use the formula:
A = P * e^(rt),
where:
A = final amount
P = principal amount
e = Euler's number (approximately 2.71828)
r = interest rate
t = time in years
For continuous compounding, the interest is compounded continuously throughout the year. Therefore, the formula can be rewritten as:
A = P * e^(r).
In this case, we have A = P * 1.078, as the effective annual yield is given as 7.8%.
Dividing both sides of the equation by P, we get:
1.078 = e^r.
To solve for r, we take the natural logarithm (ln) of both sides:
ln(1.078) = ln(e^r).
Using the logarithmic property ln(e^r) = r, we have:
r = ln(1.078).
Using a calculator, we find that ln(1.078) ≈ 0.07668.
Therefore, the normal annual interest rate with an effective annual yield of 7.8% under continuous compounding is approximately 7.668%.
To find the normal annual interest rate that has an effective annual yield of 7.8% under continuous compounding, we can use the formula for continuous compounding:
A = P*e^(rt)
Where:
A is the future value
P is the principal amount
r is the interest rate
t is the time
In this case, we want to find the interest rate (r) that corresponds to an effective annual yield of 7.8%. We can rewrite the formula as:
r = (1/t) * ln(A/P)
Since we are looking for the interest rate that has an effective annual yield of 7.8%, we set A/P equal to 1 + (7.8/100) = 1.078.
Now, substitute the values into the formula:
r = (1/1) * ln(1.078)
Use a scientific calculator to find the natural logarithm (ln) of 1.078:
r ≈ 0.0757
Therefore, the normal annual interest rate that has an effective annual yield of 7.8% under continuous compounding is approximately 7.57%.