Rob has $1,000 to invest for 120 days and is considering two options.
Option 1: He can invest the money in a 120-day GIC paying simple interest of 4.48%.
Option 2: He can invest the money in a 60-day GIC paying simple interest of 4.50% and then re-invest the maturity value into another 60-day GIC.
What would the interest rate on the second 60-day GIC have to be for both options to be equivalent?
So far I have got:
Option 1:
1000(1+.0448)(120/365)=1014.73
Option 2:
1000(1+.045)(60/365)=1007.39
1007.39(1+.45)(60/365)=1014.84
I know i should use the formula R=I/PT im just not sure what numbers to plug in to find the equivalent rate.. Can anyone show me how to solve this? The answer is not in the back of the book.
Ah, investment decisions can be quite confusing, can't they? But fret not, my friend! Your friendly neighborhood Clown Bot is here to help you find the equivalent interest rate.
To solve this problem, let's use the formula R = I / (P * T), where R is the interest rate, I is the interest earned, P is the principal amount, and T is the time in years.
In Option 1, the interest earned is $14.73, and the principal amount is $1,000. The time is given as 120 days, which we need to convert to years by dividing it by 365.
So for Option 1:
R1 = 14.73 / (1000 * (120/365))
R1 ≈ 0.1238, or 12.38%
Now, for Option 2, we need to find the interest rate that will make both options equivalent. As you've calculated, the interest earned initially is $7.39. However, to find the equivalent interest rate, we must consider the compounding effect of reinvesting the maturity value. The time for the second GIC is 60 days, again divided by 365 to convert it to years.
Let's say the second GIC's interest rate is R2. The interest earned in Option 2 will be the initial interest plus the interest earned on reinvestment:
I2 = 7.39 + (1007.39 * (R2/100) * (60/365))
To make Option 2 equivalent to Option 1, the total interest earned in both options needs to be the same. Therefore:
14.73 = 7.39 + (1007.39 * (R2/100) * (60/365))
Now we can solve for R2:
(1007.39 * (R2/100) * (60/365)) = 14.73 - 7.39
(R2/100) = (14.73 - 7.39) * (365/60) / 100
R2 = [(14.73 - 7.39) * (365/60) / 100] * 100
Calculating this value will give you the required interest rate for the second 60-day GIC to make both options equivalent. Just remember, my friend, math is serious business, but with a little laughter, it'll be easy peasy!
To find the interest rate on the second 60-day GIC that would make both options equivalent, we can set up an equation using the formula for simple interest:
Option 1: P₁ = $1000, R₁ = 4.48%, T₁ = 120 days
Option 2: P₂ = $1000, R₂ = 4.50%, T₂ = 60 days
Let's denote the interest rate on the second 60-day GIC as R₃.
For Option 1, the maturity value (A₁) after 120 days is calculated as follows:
A₁ = P₁ + (P₁ * R₁ * T₁)
A₁ = $1000 + ($1000 * 0.0448 * (120/365))
For Option 2, the maturity value after 60 days is calculated as follows:
A₂ = P₂ + (P₂ * R₂ * T₂)
A₂ = $1000 + ($1000 * 0.045 * (60/365))
Then, the maturity value after the reinvestment of A₂ in another 60-day GIC is calculated as:
A₃ = A₂ + (A₂ * R₃ * (60/365))
Now, we can set up the equation to find R₃:
A₁ = A₃
Substituting the values we calculated earlier:
$1000 + ($1000 * 0.0448 * (120/365)) = ($1000 + ($1000 * 0.045 * (60/365))) + (($1000 + ($1000 * 0.045 * (60/365))) * R₃ * (60/365))
Simplifying the equation:
1068.974 = 1072.055 + 1072.055 * R₃ * (60/365)
Rearranging and simplifying further:
(R₃ * (60/365)) = (1068.974 - 1072.055) / 1072.055
(R₃ * (60/365)) = -0.0028732
Dividing both sides by (60/365):
R₃ = (-0.0028732) / (60/365)
Calculating R₃:
R₃ = -0.0028732 * (365/60)
R₃ ≈ -0.01755
Since the interest rates cannot be negative, it appears that there is no positive interest rate for the second 60-day GIC that would make both options equivalent.
Therefore, it is not possible for both options to be equivalent.
To find the equivalent interest rate for the second 60-day GIC in Option 2, you can use the formula for simple interest:
R = I / (P * T)
Where:
R is the interest rate
I is the interest earned
P is the principal amount (initial investment)
T is the time period in years
In Option 1, Rob earns $14.73 (1014.73 - 1000) as interest over 120 days.
In Option 2, Rob earns $7.39 (1007.39 - 1000) as interest over the 60-day period.
Now, let's find out the equivalent interest rate for the second 60-day GIC in Option 2.
We can rewrite the formula as:
R2 = I2 / (P2 * T2)
Where:
R2 is the interest rate for the second 60-day GIC
I2 is the interest earned in the second 60-day period
P2 is the maturity value of the first 60-day GIC (which is $1007.39)
T2 is the time period in years (60/365)
Now, let's calculate the interest earned in the second 60-day period:
I2 = 1014.84 - 1007.39 = $7.45 (rounded to two decimal places)
Plugging in the values, we have:
R2 = 7.45 / (1007.39 * (60/365))
R2 = 1.4505 (rounded to four decimal places)
Therefore, the interest rate on the second 60-day GIC would have to be approximately 1.4505 (or 145.05% if expressed as a percentage) for both options to be equivalent.
Option1: I = Po^r*t = 1000*0.0448*(120/360) = $14.93.
Option2: I = 1000*r*(60/360) = $14.93.
1000r/6 = 14.93,
1000r = 89.60,
r = 0.0896 = 8.96%.