Question

If security currently worth $2000 will be worth $2809.86 three years in the future what is the implied interest rate the investor will earn on the security?

Answer 1

To find the implied interest rate, you can use the formula for compound annual growth rate (CAGR). The formula is:

\[ CAGR = \left( \frac{\text{Ending Value}}{\text{Beginning Value}} \right)^{\frac{1}{\text{Number of Years}}} - 1 \]

In this case:

- The beginning value is the current worth of the security, which is $2000.

- The ending value is the future worth of the security, which is $2809.86.

- The number of years is 3.

\[ CAGR = \left( \frac{2809.86}{2000} \right)^{\frac{1}{3}} - 1 \]

\[ CAGR = \left( 1.40493 \right)^{\frac{1}{3}} - 1 \]

\[ CAGR \approx 0.15 \]

To express this as a percentage, multiply by 100:

\[ CAGR \approx 15\% \]

Therefore, the implied interest rate the investor will earn on the security is approximately 15%.

Answer 2

To find the implied interest rate of a security that grows from $2,000 to $2,809.86 in three years, we use the compound interest formula. After calculations, the implied rate comes to approximately 11.70%.

Step-by-step explanation:

If a security currently worth $2,000 will be worth $2,809.86 three years in the future, to calculate the implied interest rate the investor will earn on the security, we can use the formula for compound interest:

FV = PV (1 + r)^n

Where:

• 
  
• FV is the future value of the security.
• 
  
• PV is the present value of the security (current worth).
• 
  
• r is the interest rate per period.
• 
  
• n is the number of periods (years, in this case).

By plugging in the values:

1. 
   
3. FV = $2,809.86
4. 
   
6. PV = $2,000
7. 
   
9. n = 3 years

And solving for r, the equation becomes:

$2,809.86 = $2,000 (1 + r)^3

Dividing both sides by $2,000 gives us:

1.40493 = (1 + r)^3

Now, by taking the cube root of both sides to solve for r:

(1 + r) = √[3]{1.40493}

Which gives us:

1 + r = 1.11698

Finally, subtracting 1 from both sides:

r = 0.11698

Therefore, the implied interest rate is approximately 11.70%.