You saved $25,000.00 and want to diversify your monies. You invest 40% in a Treasury bond for 3 years at 4.25% APR compounded annually You place 35% in a CD at 3.15% APR for 3 years compounded annually 25% you invest in a stock plan that increases 8% the first year, decreases in value by 4% the second year, and increases by 6% the third year. You must show all of your work for each question in order to receive any credit. 1.) How much money will you start with in each account? Show your work below. (Hint: be sure your amounts add to $25,000.00) Bond: CD: Stock: 2.) What will be the balance for each type of investment at the end of the third year? Be sure to show your work for each investment below. Bond: CD: Stock: 3.) What is your total gain from all the investments combined? Show your work below. 4.) If you had invested 25% in treasury bonds and 40% in stock, would you have more or less of a gain after the three years? Show all of your work below.

[tex]\boxed{\begin{array}{l}\underline{\textsf{Compound Interest Formula}}\\\\A=P\left(1+\dfrac{r}{n}\right)^{nt}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$A$ is the final amount.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}[/tex]

Treasury bond:

P = $10,000
r = 4.25% = 0.0425
n = 1 (annually)
t = 3 years

Therefore:

[tex]\textsf{Treasury bond balance}=10000\left(1+\dfrac{0.0425}{1}\right)^{1 \cdot 3}\\\\\\\textsf{Treasury bond balance}=10000\left(1.0425\right)^{3}\\\\\\\textsf{Treasury bond balance}=11329.96[/tex]

So, there will be $11,329.96 in the Treasury bond account at the end of the third year.

CD account:

P = $8,750
r = 3.15% = 0.0315
n = 1 (annually)
t = 3 years

Therefore:

[tex]\textsf{CD balance}=8750\left(1+\dfrac{0.0315}{1}\right)^{1 \cdot 3}\\\\\\\textsf{CD balance}=8750\left(1.0315\right)^{3}\\\\\\\textsf{CD balance}=9603.20[/tex]

So, there will be $9,603.20 in the CD account at the end of the third year.

The stock plan is more complicated as it increases by 8% the first year, decreases in value by 4% the second year, and increases by 6% the third year. Therefore, we need to calculate the value at the end of each year and compound accordingly.

[tex]\textsf{Stock balance (end of year 1)}=6250\left(1+0.08\right)\\\\\textsf{Stock balance (end of year 1)}=6250\left(1.08\right)\\\\\textsf{Stock balance (end of year 1)}=6750[/tex]

[tex]\textsf{Stock balance (end of year 2)}=6750\left(1-0.04\right)\\\\\textsf{Stock balance (end of year 2)}=6750\left(0.96\right)\\\\\textsf{Stock balance (end of year 2)}=6480[/tex]

[tex]\textsf{Stock balance (end of year 3)}=6480\left(1+0.06\right)\\\\\textsf{Stock balance (end of year 3)}=6480\left(1.06\right)\\\\\textsf{Stock balance (end of year 3)}=6868.80[/tex]

So, there will be $6,868.80 in the stock account at the end of the third year.

[tex]\dotfill[/tex]

Question 3

To find the total gain, we sum up the balances of all three accounts at the end of the third year and then subtract the initial investment of $25,000.

[tex]\textsf{Total gain}=11329.96+9603.20+6868.80-25000\\\\\textsf{Total gain}=2801.96[/tex]

Therefore, the total gain from all the investments combined is $2,801.96.

[tex]\dotfill[/tex]

Question 4

To determine if we would have more or less of a gain after the three years if we had invested 25% in treasury bonds and 40% in stock, we need to repeat the calculations we performed in question 2, but with the new allocations.

Treasury bond:

P = 25% of $25,000 = $6,250
r = 4.25% = 0.0425
n = 1 (annually)
t = 3 years

Therefore:

[tex]\textsf{Treasury bond balance}=6250\left(1+\dfrac{0.0425}{1}\right)^{1 \cdot 3}\\\\\\\textsf{Treasury bond balance}=6250\left(1.0425\right)^{3}\\\\\\\textsf{Treasury bond balance}=7081.22[/tex]

So, there will be $7,081.22 in the Treasury bond account at the end of the third year.

Stock:

P = 40% of $25,000 = $10,000
r = 8%, -4%, 6%

Therefore:

[tex]\textsf{Stock balance (end of year 1)}=10000\left(1+0.08\right)\\\\\textsf{Stock balance (end of year 1)}=10000\left(1.08\right)\\\\\textsf{Stock balance (end of year 1)}=10800[/tex]

[tex]\textsf{Stock balance (end of year 2)}=10800\left(1-0.04\right)\\\\\textsf{Stock balance (end of year 2)}=10800\left(0.96\right)\\\\\textsf{Stock balance (end of year 2)}=10368[/tex]

[tex]\textsf{Stock balance (end of year 3)}=10368\left(1+0.06\right)\\\\\textsf{Stock balance (end of year 3)}=10368\left(1.06\right)\\\\\textsf{Stock balance (end of year 3)}=10990.08[/tex]

So, there will be $10,990.08 in the stock account at the end of the third year.

Sum up the two new account balances with the previously calculated CD balance:

[tex]7081.22+10990.08+9603.20=27674.50[/tex]

Now, subtract the initial investment of $25,000 to calculate the new gain:

[tex]\textsf{New gain}=27674.50-25000\\\\\textsf{New gain}=2674.50[/tex]

As $2,674,50 is less than $2,801.96, we would have less of a gain after the three years if 25% was invested in treasury bonds and 40% was invested in stock.