Multiple Choice Question Which of the following has the highest present value? O $$ \$ 1 $$ compounded annually over 3 years at $$ 7 \% $$ O $$ \$ 1 $$ annuity compounded annually over 3 years at $$ 7 \% $$ O $$ \$ 1 $$ compounded annually over 3 years at $$ 5 \% $$ O $$ \$ 1 $$ annuity compounded annually over 3 years at $$ 5 \% $$

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Alright, I need to determine which of the given options has the highest present value. Let's break down each option step by step to understand their present values.

**Option 1: \$1 compounded annually over 3 years at 7%**
This is a simple compound interest scenario. The formula for compound interest is:
$$ A = P \times (1 + r)^n $$
Where:
- $$ A $$ is the amount after n years
- $$ P $$ is the principal amount (\$1)
- $$ r $$ is the annual interest rate (7% or 0.07)
- $$ n $$ is the number of years (3)

Plugging in the numbers:
$$ A = 1 \times (1 + 0.07)^3 $$
$$ A = 1 \times (1.07)^3 $$
$$ A = 1 \times 1.225043 $$
$$ A \approx \$1.225 $$

**Option 2: \$1 annuity compounded annually over 3 years at 7%**
An annuity involves a series of equal payments made at regular intervals. However, the question specifies a single payment of \$1, which implies it's a lump sum rather than an annuity. If it were an annuity, we'd need to know the number of payments. Since it's a single payment, it's essentially the same as Option 1. Therefore, the present value would still be approximately \$1.225.

**Option 3: \$1 compounded annually over 3 years at 5%**
Using the compound interest formula:
$$ A = 1 \times (1 + 0.05)^3 $$
$$ A = 1 \times (1.05)^3 $$
$$ A = 1 \times 1.157625 $$
$$ A \approx \$1.158 $$

**Option 4: \$1 annuity compounded annually over 3 years at 5%**
Again, assuming it's a lump sum, the present value would be the same as Option 3, approximately \$1.158.

**Comparing the Present Values:**
- Option 1: \$1.225
- Option 2: \$1.225
- Option 3: \$1.158
- Option 4: \$1.158

Both Option 1 and Option 2 have the highest present value of approximately \$1.225. Option 3 and Option 4 have a lower present value of approximately \$1.158.

However, it's important to note that if Option 2 were indeed an annuity with multiple payments, the present value would be different. But based on the information provided, both Option 1 and Option 2 are identical in terms of present value.

Option 1 and Option 2 have the highest present value of approximately \$1.225.

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