You would like to have $3000 in four years for a special vacation by making a lump-sum investment in an account that pays 9.5% compounded semiannually. How much should you deposit now ?
D [1 + (.095 / 2)]^(4 * 2) = 3000
D * 1.0475^8 = 3000 ... D = 3000 / 1.0475^8
To calculate how much you should deposit now, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value (the desired $3000)
P = the principal amount (the amount you should deposit now)
r = annual interest rate (9.5% or 0.095)
n = number of times the interest is compounded per year (semiannually, so n = 2)
t = number of years (4)
Plugging in the given values, the equation becomes:
3000 = P(1 + 0.095/2)^(2 * 4)
Simplifying:
3000 = P(1 + 0.0475)^8
3000 = P(1.0475)^8
3000 = P(1.43009375)
Now, we need to solve for P. Divide both sides of the equation by 1.43009375:
3000/1.43009375 = P
P ≈ 2096.22
So, you should deposit approximately $2096.22 now to have $3000 in four years, assuming a 9.5% annual interest rate compounded semiannually.
To calculate the amount you should deposit now to have $3000 in four years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($3000 in this case)
P = the principal amount (amount to deposit initially)
r = annual interest rate (9.5% or 0.095 as a decimal)
n = number of times the interest is compounded per year (semiannually, so 2)
t = number of years (4)
Substituting the given values into the formula, we get:
3000 = P(1 + 0.095/2)^(2*4)
Now we can solve for P:
3000 = P(1 + 0.0475)^8
3000 = P(1.0475)^8
3000 = P(1.434)
P = 3000 / 1.434
P ≈ 2090.78
Therefore, you should deposit approximately $2090.78 now to have $3000 in four years.