A. Find the period of deferral in each of the following deferred annuity problem.

1. Monthly payments of P1,000 for 9 years that will start 9 months from now

2. Monthly payments of P200 for 6 years that will start at the end of 10 years

3. Semi-annual payments of P12,700 for 5 years that will start 2 years from now

B. Solve the following problems completely.

1. A loan is to be repaid quarterly for 5 years that will start at the end of 2 years. If

interest rate is 6% converted quarterly, how much is the loan if the quarterly

payment is P10,000?

2. A cash loan is to be repaid by paying P13,500 quarterly for 3 years starting at the

end of 4 years. If interest rate is 12% convertible quarterly, how much is the cash

loan?

3. A car is to be purchased in monthly payments of P19,500 for 5 years starting at

the end of 3 months. How much is the cash value of the car if the interest rate

used is 10% converted monthly?

Pa help naman po!

$\mathbf{For \space the \space first \space part}$

#1.

We believe that the period of deferral would be $9$ months because the payment would be delayed for $9$ months.

#2.

We believe that the period of deferral would be $10$ years because we gotta wait for $10$ years before we can finally pay my dues.

#3.

We believe that the period of deferral would be $2$ years because we can finally pay our dues after waiting $2$ years.

$\mathbf{For \space the \space second \space part}$

#1.

We recognize that the problem above is a deferred annuity problem. The payment will start at the end of two years (The payment is deferred by two years) and the payment will last five years.

The term "$6$% converted quarterly", I believe, would mean that the interest rate is $6$ percent per year divided by 4, giving $\frac{0.06}{4}$ or $0.015$. In short, the interest rate $6$% is compounded quarterly.

The amount of the loan to be paid for five years would be the present value of the loan at the end of five years. Using the formula

$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$ $$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$

where....

$A$ is the amount of each payment of an ordinary annuity, $i$ is the interest rate, $n$ is the number of payment periods, $k$ is the number of deferred periods.

In this problem, we see that the number of payment periods if we pay quarterly for a year would be $4$. We will pay the amount for five years, so the number of payment periods is now $\left(\frac{4}{year}\right)(5 \space years) = 20 $. The number of deferred periods is $\left( \frac{4}{year} \right)(2 \space years) = 8$ because the interest rate already took effect even if there is no payment within the deferred period.

Now, we have...

$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$ $$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$ $$k|P = A \left( \frac{\left(1+\left(\frac{0.06}{4}\right)\right)^20-1}{\left(\frac{0.06}{4}\right)\left(1+\left(\frac{0.06}{4}\right)\right)^20}\right) \left(\frac{1}{(1+\left(\frac{0.06}{4}\right))^8} \right)$$ $$k|P = 152407.91$$

Therefore, the present value of the loan after five years would be $\color{green}{152407.91}$

#2.

I recognize that the problem above is a deferred annuity problem. The payment will start at the end of four years (The payment is deferred by four years) and the payment will last three years.

The term "$12$% converted quarterly", I believe, would mean that the interest rate is $12$ percent per year divided by 4, giving $\frac{0.12}{4}$ or $0.03$. In short, the interest rate $12$% is compounded quarterly.

The amount of the loan to be paid for five years would be the present value of the loan at the end of five years. Using the formula

$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$ $$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$

where....

$A$ is the amount of each payment of an ordinary annuity, $i$ is the interest rate, $n$ is the number of payment periods, $k$ is the number of deferred periods.

In this problem, we see that the number of payment periods if we pay quarterly for a year would be $4$. We will pay the amount for three years, so the number of payment periods is now $\left(\frac{4}{year}\right)(3 \space years) = 12 $. The number of deferred periods is $\left( \frac{4}{year} \right)(4 \space years) = 16$ because the interest rate already took effect even if there is no payment within the deferred period.

Now, we have...

$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$ $$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$ $$k|P = A \left( \frac{\left(1+\left(\frac{0.12}{4}\right)\right)^12-1}{\left(\frac{0.12}{4}\right)\left(1+\left(\frac{0.12}{4}\right)\right)^12}\right) \left(\frac{1}{(1+\left(\frac{0.12}{4}\right))^16} \right)$$ $$k|P = 83740.58$$

Therefore, the present value of the loan after three years would be $\color{green}{83740.58}$

#3.

We recognize that the problem above is a deferred annuity problem. The payment will start at the end of three months (The payment is deferred by three months) and the payment will last five years.

The term "$10$% converted monthly", I believe, would mean that the interest rate is $10$ percent per year divided by 12, giving $\frac{0.10}{12}$ or $\frac{1}{120}$. In short, the interest rate $10$% is compounded monthly.

The amount of the car to be paid for five years would be the present value of the car at the end of five years. Using the formula

$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$ $$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$

where....

$A$ is the amount of each payment of an ordinary annuity, $i$ is the interest rate, $n$ is the number of payment periods, $k$ is the number of deferred periods

In this problem, we see that the number of payment periods if we pay monthly for a year would be $12$. We will pay the amount for five years, so the number of payment periods is now $\left(\frac{12}{year}\right)(5 \space years) = 60 $. The number of deferred periods is $3$ because the interest rate already took effect even if there is no payment within the deferred period.

Now, we have...

$$k|P = A(P/A,i\%,n)(P/F,i\%,k)$$ $$k|P = A \left( \frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$$ $$k|P = A \left( \frac{\left(1+\left(\frac{0.10}{12}\right)\right)^60-1}{\left(\frac{0.10}{12}\right)\left(1+\left(\frac{0.10}{12}\right)\right)^60}\right) \left(\frac{1}{(1+\left(\frac{0.10}{12}\right))^3} \right)$$ $$k|P = 895207.49$$

Therefore, the cash value of the car if the interest rate

used is 10% converted monthly would be $\color{green}{895207.49}$

hope my answers would be correct. Alternate solutions are encouraged....

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