An investor has $100,000 to invest in three types of
bonds: short-term, intermediate-term, and
long-term.How much should she invest in each type
to satisfy the given conditions?
Short-term bonds pay 4% annually,
intermediate-term bonds pay 5%, and long-term
bonds pay 6%. The investor wishes to realize a total
annual income of 5.1%, with equal amounts
invested in short- and intermediate-term bonds.
I don't understand the question at all, this is just an example problem for the chapter. Please help!
Important info at the end:
"with equal amounts
invested in short- and intermediate-term bonds. "
So let each of those amounts be x
then the amount invested at long-term be
100,000 - 2x
He wants to end up with a 5.1% profit on the total 100,000 or $5100
Now it is easy....
.04x + .05x + .06(100000-2x) = 5100
.09x + 6000 - .12x = 5100
-.03x = -900
x = 30,000
so .....
I will the substitutions up to you.
Also check the answers to see if they satisfy the original conditions.
No problem! Let's break down the question and try to understand it step by step.
The investor has $100,000 to invest in three types of bonds: short-term, intermediate-term, and long-term. The interest rates for each type of bond are provided: short-term bonds pay 4% annually, intermediate-term bonds pay 5%, and long-term bonds pay 6%.
The investor wants to achieve a total annual income of 5.1% from their investments. Additionally, they want to invest equal amounts in short-term and intermediate-term bonds.
To solve this problem, we can use a system of equations.
Let's say the amount invested in short-term bonds is x, the amount invested in intermediate-term bonds is also x, and the amount invested in long-term bonds is y.
We can set up the following equations based on the given information:
Equation 1: 0.04x + 0.05x + 0.06y = 0.051 * $100,000
This equation represents the total annual income the investor desires. It is calculated by multiplying the total investment ($100,000) by the desired interest rate (5.1%).
Equation 2: x + x + y = $100,000
This equation represents the requirement to invest equal amounts in short-term and intermediate-term bonds, totaling $100,000.
Now we can solve these equations simultaneously to find the values of x and y.
By simplifying Equation 2, we get:
2x + y = $100,000
Now we have a system of two equations with two variables:
0.09x + 0.06y = $5,100
2x + y = $100,000
To solve this system, you can use substitution or elimination method. This will give you the values of x and y, which represent the amount the investor should invest in each type of bond to satisfy the given conditions.
It's important to note that this is just an example problem, as you mentioned. The process explained here is applicable to similar problems involving investing in different bonds or assets to achieve specific goals.
No worries! I'm here to help you understand the problem and solve it step-by-step.
From the problem, we know that the investor wants to invest in three types of bonds: short-term, intermediate-term, and long-term. The interest rates for each type of bond are:
- Short-term bonds: 4% annual income
- Intermediate-term bonds: 5% annual income
- Long-term bonds: 6% annual income
The investor wants a total annual income of 5.1% with equal amounts invested in short-term and intermediate-term bonds.
To solve this problem, we can use a system of equations.
Let's denote the amount invested in short-term bonds as x, and the amount invested in intermediate-term bonds as y.
According to the condition that equal amounts are invested in short-term and intermediate-term bonds, we can say that x = y.
Now, let's calculate the annual income from each bond:
- Short-term bond annual income: x * 4%
- Intermediate-term bond annual income: y * 5%
- Long-term bond annual income: (100,000 - x - y) * 6%
The total annual income should be 5.1%, which is equal to the sum of the annual incomes from each bond type. Therefore, the equation is:
x * 4% + y * 5% + (100,000 - x - y) * 6% = 5.1%
Now, let's solve this equation step-by-step:
1. Convert the percentages to decimals:
0.04x + 0.05y + 0.06(100,000 - x - y) = 0.051
2. Distribute 0.06:
0.04x + 0.05y + 6000 - 0.06x - 0.06y = 0.051
3. Combine like terms:
-0.02x - 0.01y = -5940
4. Multiply both sides by -100 (to eliminate the decimals):
2x + y = 594,000
5. Use the condition that x = y:
2x + x = 594,000
6. Combine like terms:
3x = 594,000
7. Divide both sides by 3:
x = 198,000
Now, substitute the value of x back into the equation x = y:
198,000 = y
Therefore, the investor should invest $198,000 in short-term bonds, $198,000 in intermediate-term bonds, and the remaining amount ($100,000 - x - y) in long-term bonds.