Fight Finance

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

$$r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.12/12 = 0.01$$

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. The annuity equation can be used to discount equal payments:

$$\begin{aligned} P_\text{0, fully amortising loan} &= \text{PV(annuity of monthly payments)} \\ &= C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ \end{aligned}$$ $$270,000 = C_{\text{monthly}} \times \frac{1}{0.12/12} \left(1 - \frac{1}{(1+0.12/12)^{25 \times 12}} \right)$$ $$\begin{aligned} C_{\text{monthly}} &= 270,000 \div \left(\frac{1}{0.12/12}\left(1 - \frac{1}{(1+0.12/12)^{25 \times 12}} \right) \right) \\ &= 270,000 \div \left(\frac{1}{0.01}\left(1 - \frac{1}{(1+0.01)^{300}} \right) \right) \\ &= 270,000 \div 94.94655125 \\ &= 2,843.705184 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

$$r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.06/12 = 0.005$$

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:

$$\begin{aligned} P_\text{0, fully amortising loan} &= \text{PV(annuity of monthly payments)} \\ &= C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ 450,000 &= C_{\text{monthly}} \times \frac{1}{0.06/12} \left(1 - \frac{1}{(1+0.06/12)^{25 \times 12}} \right) \\ \end{aligned}$$

$$\begin{aligned} C_{\text{monthly}} &= 450,000 \div \left(\frac{1}{0.06/12}\left(1 - \frac{1}{(1+0.06/12)^{25 \times 12}} \right) \right) \\ &= 450,000 \div \left(\frac{1}{0.005}\left(1 - \frac{1}{(1+0.005)^{300}} \right) \right) \\ &= 450,000 \div 155.206864 \\ &= 2,899.356307 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

$$r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.06/12 = 0.005$$

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:

$$\begin{aligned} P_\text{0, fully amortising loan} =& \text{PV(annuity of monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ 320,000 =& C_{\text{monthly}} \times \frac{1}{0.06/12} \left(1 - \frac{1}{(1+0.06/12)^{30 \times 12}} \right) \\ C_{\text{monthly}} =& 320,000 \div \left(\frac{1}{0.06/12}\left(1 - \frac{1}{(1+0.06/12)^{30 \times 12}} \right) \right) \\ =& 320,000 \div \left(\frac{1}{0.005}\left(1 - \frac{1}{(1+0.005)^{360}} \right) \right) \\ =& 320,000 \div 166.7916144 \\ =& 1,918.56168 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

$$r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.06/12 = 0.005$$

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:

$$\begin{aligned} P_\text{0, fully amortising loan} =& \text{PV(annuity of monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ 450,000 =& C_{\text{monthly}} \times \frac{1}{0.06/12} \left(1 - \frac{1}{(1+0.06/12)^{30 \times 12}} \right) \\ C_{\text{monthly}} =& 450,000 \div \left(\frac{1}{0.06/12}\left(1 - \frac{1}{(1+0.06/12)^{30 \times 12}} \right) \right) \\ =& 450,000 \div \left(\frac{1}{0.005}\left(1 - \frac{1}{(1+0.005)^{360}} \right) \right) \\ =& 450,000 \div 166.7916144 \\ =& 2,697.977363 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since this is usually the case by convention and in some countries by law. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

$$r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.09/12 = 0.0075$$

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:

$$\begin{aligned} P_\text{0, fully amortising loan} =& \text{PV(annuity of monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ =& 2,000 \times \frac{1}{0.09/12} \left(1 - \frac{1}{(1+0.09/12)^{30 \times 12}} \right) \\ =& 2,000 \times \frac{1}{0.0075}\left(1 - \frac{1}{(1+0.0075)^{360}} \right) \\ =& 2,000 \times 124.2818657 \\ =& 248,563.7314 \\ \end{aligned}$$

To find the value of the loan in 5 years, remember that the price of any asset or liability is the present value of the future cash flows, and there are 25 years of future monthly payments. The working is nearly identical to that above:

$$\begin{aligned} P_\text{5yrs, fully amortising loan} =& \text{PV(annuity of 25 years of future monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ =& 2,000 \times \frac{1}{0.09/12} \left(1 - \frac{1}{(1+0.09/12)^{\mathbf{25} \times 12}} \right) \\ =& 2,000 \times \frac{1}{0.0075}\left(1 - \frac{1}{(1+0.0075)^{300}} \right) \\ =& 2,000 \times 119.1616222 \\ =& 238,323.2443 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

$$r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.06/12 = 0.005$$

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:

$$\begin{aligned} P_\text{0, fully amortising loan} =& \text{PV(annuity of monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ =& 1,000 \times \frac{1}{0.06/12} \left(1 - \frac{1}{(1+0.06/12)^{30 \times 12}} \right) \\ =& 1,000 \times \frac{1}{0.005}\left(1 - \frac{1}{(1+0.005)^{360}} \right) \\ =& 1,000 \times 166.7916144 \\ =& 166,791.6144 \\ \end{aligned}$$

To find the value of the loan in 20 years, remember that the price of any asset or liability is the present value of the future cash flows, and there are 10 years of future monthly payments. The working is nearly identical to that above, just the number of years remaining has been changed from 30 to 10:

$$\begin{aligned} P_\text{20yrs, fully amortising loan} =& \text{PV(annuity of 10 years of future monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ =& 1,000 \times \frac{1}{0.06/12} \left(1 - \frac{1}{(1+0.06/12)^{\mathbf{10} \times 12}} \right) \\ =& 1,000 \times \frac{1}{0.005}\left(1 - \frac{1}{(1+0.005)^{120}} \right) \\ =& 1,000 \times 90.07345333 \\ =& 90,073.45333 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

$$r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.09/12 = 0.0075$$

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:

$$\begin{aligned} P_\text{0, fully amortising loan} =& \text{PV(annuity of monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ =& 1,500 \times \frac{1}{0.09/12} \left(1 - \frac{1}{(1+0.09/12)^{30 \times 12}} \right) \\ =& 1,500 \times \frac{1}{0.0075}\left(1 - \frac{1}{(1+0.0075)^{360}} \right) \\ =& 1,500 \times 124.2818657 \\ =& 186,422.7985 \\ \end{aligned}$$

To find the value of the loan in 10 years, remember that the price of any asset or liability is the present value of the future cash flows, and there are 20 years of future monthly payments. The working is nearly identical to that above:

$$\begin{aligned} P_\text{20, fully amortising loan} =& \text{PV(annuity of 20 years of future monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ =& 1,500 \times \frac{1}{0.09/12} \left(1 - \frac{1}{(1+0.09/12)^{\mathbf{20} \times 12}} \right) \\ =& 1,500 \times \frac{1}{0.0075}\left(1 - \frac{1}{(1+0.0075)^{240}} \right) \\ =& 1,500 \times 111.144954 \\ =& 166,717.431\\ \end{aligned}$$

The above method is sometimes called the 'prospective' method of finding the loan amount owing since it present values future payments owing. Another method is the retrospective method which looks into the past and subtracts principal payments from the original amount borrowed to find the amount owing. Both methods give the same answer.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The cash flows occur every month, so the discount rate needs to be an effective monthly rate and the time must be measured in months.

First we have to find the amount borrowed when the payments are $1,500 per month for 30 years.

$$\begin{aligned} V_0 &= C_{\text{monthly}} \times \dfrac{1}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} \right) \\ &= 1,500 \times \dfrac{1}{\left( \dfrac{0.09}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\dfrac{0.09}{12}\right)^{30 \times 12}} \right) \\ &= 1,500 \times 124.2818657 \\ &= 186,422.7985 \\ \end{aligned}$$

When the present value of the 'T' months of $2,000 payments are equal to the amount borrowed, then the loan will be paid off. So the only job left is to solve for T.

$$V_0 = \dfrac{C_\text{monthly}}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( 1 - \frac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} \right)$$ $$186,422.7985 = \dfrac{2,000}{\left( \dfrac{0.09}{12} \right) } \left( 1 - \frac{1}{\left(1+\dfrac{0.09}{12}\right)^{T}} \right)$$ $$\frac{186,422.7985}{2,000} \left( \frac{0.09}{12} \right) = 1 - \frac{1}{\left(1+\frac{0.09}{12}\right)^{T}}$$ $$\frac{1}{\left(1+\frac{0.09}{12}\right)^{T}} = 1 - \frac{186,422.7985}{2,000} \left( \frac{0.09}{12} \right)$$ $$\left(1+\frac{0.09}{12}\right)^{-T} = 0.300914506$$ $$\ln \left( \left(1+\frac{0.09}{12}) \right) ^{-T} \right) = \ln \left(0.300914506 \right)$$ $$-T \times \ln \left(1+\frac{0.09}{12} \right) = \ln \left(0.300914506 \right)$$

$$\begin{aligned} T &= -\dfrac{\ln \left(0.300914506 \right)} {\ln \left( 1+\dfrac{0.09}{12} \right)} \\ &= 160.7235953 \text{ months}\\\\ &= 13.39363294 \text{ years}\\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

$$r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.09/12 = 0.0075$$

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:

$$\begin{aligned} P_\text{0, fully amortising loan} =& \text{PV(annuity of monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ =& 2,500 \times \frac{1}{0.09/12} \left(1 - \frac{1}{(1+0.09/12)^{30 \times 12}} \right) \\ =& 2,500 \times \frac{1}{0.0075}\left(1 - \frac{1}{(1+0.0075)^{360}} \right) \\ =& 2,500 \times 124.2818657 \\ =& 310,704.6642 \\ \end{aligned}$$

To find the value of the loan in 10 years, remember that the price of any asset or liability is the present value of the future cash flows, and there are 20 years of future monthly payments. The working is nearly identical to that above:

$$\begin{aligned} P_\text{10yrs, fully amortising loan} =& \text{PV(annuity of 20 years of future monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ =& 2,500 \times \frac{1}{0.09/12} \left(1 - \frac{1}{(1+0.09/12)^{\mathbf{20} \times 12}} \right) \\ =& 2,500 \times \frac{1}{0.0075}\left(1 - \frac{1}{(1+0.0075)^{240}} \right) \\ =& 2,500 \times 111.144954 \\ =& 277,862.3851 \\ \end{aligned}$$

The above method is sometimes called the 'prospective' method of finding the loan amount owing since it present values future payments owing. Another method is the retrospective method which looks into the past and subtracts principal payments from the original amount borrowed to find the amount owing. Both methods give the same answer.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.

$$r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.08/12 = 0.00666667$$

The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:

$$\begin{aligned} P_\text{0, fully amortising loan} =& \text{PV(annuity of monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ 360,000 =& C_{\text{monthly}} \times \frac{1}{0.08/12} \left(1 - \frac{1}{(1+0.08/12)^{30 \times 12}} \right) \\ C_{\text{monthly}} =& 360,000 \div \left(\frac{1}{0.08/12}\left(1 - \frac{1}{(1+0.08/12)^{30 \times 12}} \right) \right) \\ =& 360,000 \div \left(\frac{1}{0.0066667}\left(1 - \frac{1}{(1+0.0066667)^{360}} \right) \right) \\ =& 360,000 \div 136.2834941 \\ =& 2,641.552466 \\ \end{aligned}$$