Hi Brianna!
Absolutely! Let's work through each problem step by step, showing the formulas, substitutions, and calculations.
Question 1: Remaining Balance on a 15-Year Loan
Given:
- Original loan amount = $750,000
- Annual interest rate = 9%
- Loan term = 15 years
- Payments are made monthly
- Payments have been made for 9 years
We want to determine how much is still owed after 9 years.
Step 1: Calculate the monthly interest rate
Since payments are monthly, divide the annual interest rate by 12.
[
r=\frac{0.09}{12}
]
[
\boxed{r=0.0075}
]
Step 2: Determine the total number of payments
[
n=15\times12=180
]
Therefore, the loan requires 180 monthly payments.
Step 3: Calculate the monthly loan payment
We use the loan amortization formula:
[
M=\frac{Pr}{1-(1+r)^{-n}}
]
Where:
- (M) = Monthly payment
- (P) = Original loan amount
- (r) = Monthly interest rate
- (n) = Total number of payments
Substituting:
[
M=\frac{750,000(0.0075)}{1-(1.0075)^{-180}}
]
[
\boxed{M\approx$7,607.00}
]
Bill must make monthly payments of approximately $7,607.
Step 4: Calculate the number of remaining payments
The loan has already been paid for 9 years.
[
9\times12=108
]
Therefore, 108 payments have been made.
The remaining payments are:
[
180-108=72
]
Step 5: Calculate the remaining loan balance
The remaining balance equals the present value of the 72 payments still owed.
We use:
[
B=M\left[\frac{1-(1+r)^{-m}}{r}\right]
]
Where (m) is the number of remaining payments.
Substitute the values:
[
B=7,606.999381\left[\frac{1-(1.0075)^{-72}}{0.0075}\right]
]
[
\boxed{B\approx$422,012.35}
]
Final Answer for Question 1:
[
\boxed{$422,012.35}
]
After making 9 years of payments, the remaining loan balance is approximately $422,012.35.
Notice that the balance is not simply 40% of the original loan amount. This is because early loan payments consist of a relatively large proportion of interest, while later payments repay more principal.
Question 2: Expected Return on a Stock Portfolio
Jason has invested in four stocks.
| StockInvestmentExpected Return | ||
| Nestlé | $48,000 | 14% |
| Apple | $84,000 | 28% |
| Macy's | $36,000 | 10% |
| $72,000 | 12% |
We want to calculate the portfolio's expected return.
Step 1: Calculate the total investment
[
48,000+84,000+36,000+72,000
]
[
\boxed{$240,000}
]
Step 2: Calculate each stock's portfolio weight
The weight of each investment is:
[
w_i=\frac{\text{Investment in Stock}}{\text{Total Portfolio Value}}
]
Nestlé:
[
w_1=\frac{48,000}{240,000}=0.20
]
Apple:
[
w_2=\frac{84,000}{240,000}=0.35
]
Macy's:
[
w_3=\frac{36,000}{240,000}=0.15
]
Twitter:
[
w_4=\frac{72,000}{240,000}=0.30
]
Let's check that the weights total 100%:
[
0.20+0.35+0.15+0.30=1.00
]
Perfect!
Step 3: Apply the weighted-average return formula
The expected portfolio return is:
[
E(R_p)=\sum_{i=1}^{n}w_iE(R_i)
]
Substituting:
[
E(R_p)=(0.20)(0.14)+(0.35)(0.28)+(0.15)(0.10)+(0.30)(0.12)
]
Calculate each component:
[
=0.028+0.098+0.015+0.036
]
[
=0.177
]
Convert to a percentage:
[
\boxed{E(R_p)=17.70%}
]
Final Answer for Question 2:
[
\boxed{17.70%}
]
Jason's portfolio has an expected return of 17.70%.
The important concept here is that we cannot simply average the four return percentages because Jason has invested different amounts in each stock.
Instead, each return must be weighted according to its proportion of the total portfolio.
Question 3: Expected Return Based on Economic Conditions
We're given the following information:
| Economic ConditionProbabilityInvestment Return | ||
| Strong growth | 15% | 28% |
| Medium growth | 45% | 22% |
| Marginal growth | 35% | 14% |
Step 1: Write the expected return formula
For an investment with different possible economic outcomes:
[
E(R)=\sum_{i=1}^{n}P_iR_i
]
Where:
- (P_i) = Probability of each economic condition
- (R_i) = Investment return under that condition
Step 2: Multiply each probability by its corresponding return
Strong growth:
[
0.15\times0.28=0.042
]
Medium growth:
[
0.45\times0.22=0.099
]
Marginal growth:
[
0.35\times0.14=0.049
]
Step 3: Add the weighted returns
[
E(R)=0.042+0.099+0.049
]
[
\boxed{0.190=19.0%}
]
However, there's an important issue with the information provided.
Step 4: Check the probabilities
The probabilities of all possible economic outcomes should total 100%.
Let's add them:
[
15%+45%+35%=95%
]
That means 5% of the probability distribution is missing.
Consequently, 19.0% represents the combined weighted contribution of the three listed outcomes, but it is not necessarily the investment's complete expected return.
If another economic condition has a 5% probability, we would need to know its associated investment return.
The complete formula would be:
[
E(R)=0.190+0.05R_4
]
Where (R_4) represents the return in the missing economic state.
For example, if the missing state has a return of 0%, then the expected return would be 19.0%.
If its return is 10%, then:
[
E(R)=0.190+(0.05)(0.10)
]
[
\boxed{E(R)=19.5%}
]
Final Answer for Question 3:
The three listed outcomes contribute 19.0 percentage points to the expected return.
However, we need the missing 5% probability outcome to calculate the complete expected return accurately.
Final Answers Summary
| QuestionFinal Answer | |
| 1. Remaining loan balance after 9 years | $422,012.35 |
| 2. Expected portfolio return | 17.70% |
| 3. Expected return based on economic conditions | 19.0% from the listed states; additional information needed for the complete expected return |
Key concepts to remember:
- Loan amortization: The remaining balance equals the present value of the payments still owed.
- Portfolio returns: Use a weighted average based on the amount invested in each stock.
- Expected investment returns: Multiply each possible return by its probability, then add the results. Always verify that the probabilities total 100%.
I hope these explanations help you understand not only how to calculate the answers, but also why each formula works!
Michael
