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.

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} ###

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

Quick explanation

###\begin{aligned} P_0 &= \frac{C_\text{1, monthly}}{r_\text{eff monthly}} \\ &= \frac{C_\text{1, monthly}}{ \left( \dfrac{r_\text{APR comp monthly}}{12} \right) } \\ \end{aligned}### ###270,000 = \frac{C_\text{1, monthly}}{\left( \dfrac{0.12}{12} \right) } ### ###C_\text{1, monthly} = 270,000 \times \dfrac{0.12}{12} = 2,700 ###

Longer explanation

To price a loan with equal monthly payments and a principal to pay at the end,

### P_\text{0} = \text{PV(annuity of monthly payments)} + \text{PV(principal)} ###

But since the loan is interest-only, the principal will equal the price since the principal is never paid off, only interest is paid. So:

### P_\text{0} = \text{principal} ###

Since the cash flow is monthly, everything must be measured in months. So the required total return should be an effective monthly rate ##r = r_\text{eff monthly}##, the time should be in months ##T = T_\text{months}##, and the cash flow is monthly with the first occurring in one month ##C_1 = C_\text{1, monthly}##.

Substituting into the above equation,

###P_0 = \frac{C_1}{r} \left(1 - \frac{1}{(1+r)^{T}} \right) + \frac{P_0}{(1 + r)^{T}} ### ###P_0 - \frac{P_0}{(1 + r)^{T}} = \frac{C_1}{r} \left(1 - \frac{1}{(1+r)^{T}} \right) ### ###P_0 \times \left( 1 - \frac{1}{(1 + r)^{T}} \right) = \frac{C_1}{r} \left(1 - \frac{1}{(1+r)^{T}} \right) ### ###P_0 = \frac{C_1}{r} ###

This is the perpetuity without growth equation! This actually makes sense because rather than paying the interest only loan back at maturity, a new interest-only loan can be issued to pay back the old one, and the process can repeat itself forever which makes a perpetuity with no growth.

Remembering that the required return ##r## must be an effective monthly return ##r_\text{eff monthly}##, and that the 12% interest rate given is an APR compounding monthly since the mortgage payments are monthly and by law interest rates are always quoted as APR's compounding at the same frequency as the payments. So the APR must be divided by 12 to get the effective monthly rate: ###\begin{aligned} P_0 &= \frac{C_\text{1, monthly}}{r_\text{eff monthly}} \\ &= \frac{C_\text{1, monthly}}{ \left( \dfrac{r_\text{APR comp monthly}}{12} \right) } \\ \end{aligned}### ###270,000 = \frac{C_\text{1, monthly}}{\left( \dfrac{0.12}{12} \right) } ### ###\begin{aligned} C_\text{1, monthly} &= 270,000 \times \dfrac{0.12}{12} \\ &= 2,700 \\ \end{aligned}###

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

Since the loan is interest-only, the perpetuity without growth formula can be used.

###P_0 = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} ###

This formula gives the price of the mortgage, which is also the principal, at the start of every period just after the interest payment is made. It has no time dimension, so it is the price and principal owing right now (t=0) and also after the 180th payment (t=180 months). This makes sense since the mortgage is interest only, so the principal is never paid down. It is always the same at the start of every period.

Substituting values,

###\begin{aligned} P_t = P_0 = P_{180} &= \frac{C_\text{monthly}}{r_\text{eff monthly}} \\ &= \frac{C_\text{monthly}}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right)} \\ &= \frac{3,000}{\left( \frac{0.06}{12} \right)} \\ &= 600,000 \\ \end{aligned} ###

This is a tricky question since you might assume that the loan would be paid down over time. If it was a fully-amortising loan then yes, it would be paid down over time. But since it's interest-only it is not paid down at all and its price remains constant.

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

The payments are monthly so the discount rate must also be a monthly effective rate, not an APR:

### r_\text{eff mthly} = \frac{r_\text{APR comp mthly}}{12} = \frac{0.09}{12} = 0.0075 ###

The net borrowings owing will be the future value of the initial borrowings less the future value of the mortgage loan payments. Loan payments are subtracted since they reduce borrowings. Let T be the time at which the mortgage matures.

###\begin{aligned} V_\text{T, net borrowings} =& V_\text{T, initial borrowings} - V_\text{T, payments} \\ =& V_\text{0, borrowings}(1+r)^{T} - V_\text{0, payments}(1+r)^{T} \\ =& V_\text{0, borrowings}(1+r)^{T} - C_\text{1, 2, 3, ...T}\frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}} \right)(1+r)^{T} \\ =& 400,000 \times (1+0.0075)^{25 \times 12} - 3,300 \times \frac{1}{0.0075}\left(1 - \frac{1}{(1+0.0075)^{25 \times 12}} \right) \times (1+0.0075)^{25 \times 12} \\ =& 400,000 \times 9.40841453 - 3,300 \times 119.1616222 \times 9.40841453 \\ =& 3,763,365.8120 - 3,699,702.3931 \\ =& 63,663.4188 \\ \end{aligned} ###

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

The price of an interest-only loan can be found very quickly using the perpetuity formula without growth.

###P_0 = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} ###

Care has to be taken to match the monthly mortgage payments with a monthly effective required return. The 6% interest rate is assumed to be an annualised percentage rate (APR) by convention and because the law requires rates to be advertised as APR's. Since payments on the mortgage are monthly, we would assume that the 6% APR compounds monthly.

###P_0 = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} ### ###\begin{aligned} C_{\text{monthly}} =& P_0 \times r_\text{eff, monthly} \\ =& P_0 \times \frac{r_\text{apr compounding monthly}}{12} \\ =& 240,000 \times \frac{0.06}{12} \\ =& 1,200 \\ \end{aligned} ###

Commentary

The reason why an interest-only loan's price can be found using the perpetuity equation is quite interesting. In common sense terms, interest-only loans require payment of interest only, until maturity when the principal must also be paid. But if this principal is repaid by re-financing using another interest-only loan, and this goes on forever, then the principal will never be paid off. In this case, interest payments will occur in perpetuity. Hence why the perpetuity equation can be used to value an interest only loan.

A mathematical explanation using the present value of the cash flows is shown below. Note that another name for the principal is the par value or face value which is often represented by ##F_T##.

### P_\text{0} = \text{PV(annuity of monthly payments)} + \text{PV(principal)} ### ###P_0 = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) + \frac{\mathbf{F_T}}{(1 + r_\text{eff monthly})^{T_\text{months}}} ###

Since it's an interest only loan, the principal is never paid off so the principal paid at the end ##(F_T)## will equal the price paid at the start ##(P_0)## .

Substituting ##F_T = P_0## into the above equation,

###P_0 = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) + \frac{\mathbf{P_0}}{(1 + r_\text{eff monthly})^{T_\text{months}}} ### ###P_0 - \frac{P_0}{(1 + r_\text{eff monthly})^{T_\text{months}}} = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) ### ###P_0 \left( 1 - \frac{1}{(1 + r_\text{eff monthly})^{T_\text{months}}} \right) = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) ### ###P_0 = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} ###

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

Since the loan is interest-only, the perpetuity without growth formula can be used.

###P_0 = \frac{C_{\text{monthly}}}{r_\text{eff monthly}} ###

We already know the price ##P_0## and interest rate ##r_\text{eff monthly}##. We're interested in finding the monthly cash flow ##C_{\text{monthly}}##, so make it the subject.

###\begin{aligned} C_{\text{monthly}} &= P_0 \times r_\text{eff monthly} \\ &= P_0 \times \frac{r_\text{apr compounding monthly}}{12} \\ &= 450,000 \times \frac{0.06}{12} \\ &= 2,250 \\ \end{aligned}###

These interest payments are paid monthly in arrears which means they occur at the end of each month.

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 = 0.0425/12 = 0.003541667###

The loan is interest-only for the first 10 years and the interest rate is expected to remain constant so the monthly payments will all be equal. We can assume that the payments are made in arrears, as is normal. The perpetuity formula can be used to find the monthly payment. Note that the 10 year interest only term is not needed in the perpetuity formula, and the growth rate of the payments is zero since the payments don't grow:

###P_\text{0, interest only loan} = \dfrac{ C_{\text{1, monthly}} }{ r_\text{eff monthly} - g_\text{eff monthly} } ### ###600,000 = \dfrac{ C_\text{1, monthly} }{ 0.0425/12 - 0 } ###

###\begin{aligned} C_{1 \rightarrow 120 \text{, monthly}} &= 600,000 \times 0.0425 / 12 \\ &= 600,000 \times 0.003541667 \\ &= 2,125 \\ \end{aligned} ###

Another way to think about this is to remember that the interest component of the total loan payment is just the effective monthly rate multiplied by the loan price at the start of the month, which is true for fully amortising and interest only loans.

###\begin{aligned} C_\text{1, monthly interest component} &= P_\text{0} \times r_\text{eff monthly} \\ &= P_\text{0} \times r_\text{APR comp monthly} / 12 \\ &= 600,000 \times 0.0425 / 12 \\ &= 600,000 \times 0.003541667 \\ &= 2,125 \\ \end{aligned} ###

To find the monthly payments after year 10 when the loan becomes fully amortising, remember that the original $600,000 amount is still owing since no principal payments were made over the past 10 years while the loan was interest-only. So the $600,000 loan value at month 120 (=10years*12months/year) should equal the present value of the remaining 180 months (=(25-10)years*12months/year) worth of monthly payments. We can present value these 180 monthly payments using the annuity formula to find the unknown monthly loan payment:

###P_\text{120mth, fully amortising} = \text{PV(annuity of 15 years of future monthly payments)} ###

###\begin{aligned} 600,000 &= C_{121 \rightarrow 180 \text{, monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ &= C_{121 \rightarrow 180 \text{, monthly}} \times \frac{1}{0.003541667 } \left(1 - \frac{1}{(1+0.003541667 )^{180}} \right) \\ &= C_{121 \rightarrow 180 \text{, monthly}} \times 132.9295092 \\ \end{aligned} ###

###\begin{aligned} C_{121 \rightarrow 180 \text{, monthly}} &= \dfrac{600,000}{132.9295092} \\ &= 4,513.670468 \\ \end{aligned} ###

It makes sense that the fully amortising loan payments are much bigger than the interest-only loan payments since now the principal must be paid off in addition to the interest.

Just for fun we can calculate the first few interest and principal components of the total loan payments:

The first fully amortising loan payment at month 181 consists of a $2,125 (=600,000*0.0425/12) interest component and a $2,388.670468 (=4,513.670468 - 2,250) principal component.

The second fully amortising loan payment at month 182 consists of a $2,116.540125 (=(600,000-2,388.670468)*0.0425/12) interest component and a $2,397.130343 (=4,513.670468 - 2,116.540125) principal component.

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

Find the value of each interest-only mortgage loan:

###\begin{aligned} V_\text{before} &= \frac{C_\text{1, monthly}}{r_\text{eff mthly} - g_\text{eff mthly}} \\ &= \frac{2,000}{\left( \dfrac{0.0474}{12} - 0 \right)} \\ &= 506,329.1139 \\ \end{aligned}###

###\begin{aligned} V_\text{after} &= \frac{C_\text{1, monthly}}{r_\text{eff mthly} - g_\text{eff mthly}} \\ &= \frac{2,000}{\left( \dfrac{0.0449}{12} - 0 \right)} \\ &= 534,521.1581 \\ \end{aligned}###

###\begin{aligned} \text{Proportional increase} &= \frac{V_\text{after}-V_\text{before}}{V_\text{before}} \\ &= \frac{534,521.1581 - 506,329.1139}{506,329.1139} \\ &= 0.055679287 \\ &\approx 5.6\% \\ \end{aligned}###

Note that the answer is 0.029547 or 2.9547% if the mortgage loans are both fully amortising. Thanks to Shahzada for providing that solution.

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

The price of an interest-only loan where interest rates are expected to be stable is equal to the price of a perpetuity without growth. See question 29 for why.

When interest rates are 14% pa, then payments of $2,000 per month equate to a borrowing capacity of:

###P_\text{0, r=14%} = \frac{C_\text{1, monthly}}{r_\text{eff monthly}} = \frac{C_\text{1, monthly}}{ \left( \dfrac{r_\text{APR comp monthly}}{12} \right) } = \frac{2,000}{ \left( \dfrac{0.14}{12} \right) } = 171,428.5714 ###

When interest rates are 4.5% pa, then payments of $2,000 per month equate to a borrowing capacity of:

###P_\text{0, r=4.5%} = \frac{C_\text{1, monthly}}{r_\text{eff monthly}} = \frac{C_\text{1, monthly}}{ \left( \dfrac{r_\text{APR comp monthly}}{12} \right) } = \frac{2,000}{ \left( \dfrac{0.045}{12} \right) } = 533,333.3333 ###

The proportional increase in borrowing capacity is:

###\begin{aligned} \text{Proportional increase} &= \dfrac{V_\text{low rates}-V_\text{high rates}}{V_\text{high rates}} \\ &= \dfrac{P_\text{0, r=4.5%} - P_\text{0, r=14%}}{P_\text{0, r=14%}} \\ &= \dfrac{533,333.3333 - 171,428.5714}{171,428.5714} \\ &= 2.11111 \\ &= 211.111\% \\ \end{aligned}###

The increase in borrowing capacity due to lower inflation and lower interest rates was a point discussed by the Reserve Bank of Australia Governor Glenn Stevens in his 1997 speech 'Some Observations on Low Inflation and Household Finances'. Perhaps this is one reason for the high growth in house prices seen in the decades after the late 1990's.

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

Quick method

An interest only loan can be valued as a constant perpetuity ##(C/r)## when interest rates don't change, and a fully amortising loan can be valued as an annuity ##(C/r.(1-(1+r)^{-T}))##. Since the ##C/r## is a common factor, it will cancel out when we find the proportional increase so we don't need to know what the monthly payments ##(C)## are. ###\begin{aligned} & \text{Proportional Increase} = \dfrac{V_\text{0,interest only}}{V_\text{0,fully amortising}} - 1 \\ &= \dfrac{ \left( \dfrac{C_1}{r} \right) }{ \left( \dfrac{C_1}{r} \left( 1 - \dfrac{1}{(1+r)^T} \right) \right) } - 1 \\ &= \dfrac{ 1 }{ \left( 1 - \dfrac{1}{(1+r)^{T}} \right) } - 1 \\ &= \dfrac{ 1 }{ \left( 1 - \dfrac{1}{(1+0.06/12)^{25 \times 12}} \right) } - 1 \\ &= \dfrac{ 1 }{ 0.288602803} - 1 = 0.77603432 = 77.603432\% \end{aligned}###

Intuitive method

Let's find the value of the interest only loan and then the fully amortising loan if the payments are $2,000 per month. ###\begin{aligned} V_\text{0,interest only} &= \dfrac{C_\text{1, monthly}}{r_\text{eff monthly}} \\ &= \dfrac{2,000}{0.06/12} \\ &= 400,000 \\ \end{aligned}### ###\begin{aligned} V_\text{0,fully amortising} &=\dfrac{C_\text{1, monthly}}{r_\text{eff monthly}} \left( 1 - \dfrac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ &= \dfrac{2,000}{0.06/12} \left( 1 - \dfrac{1}{(1+0.06/12)^{25 \times 12}} \right) \\ &= 310,413.728 \\ \end{aligned}### ###\begin{aligned} & \text{Proportional Increase} = \dfrac{V_\text{0,interest only}}{V_\text{0,fully amortising}} - 1 \\ &= \dfrac{400,000 }{310,413.728 } - 1 \\ &= 0.288602803= 28.8602803 \% \end{aligned}###

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

Quick method

An interest only loan can be valued as a constant perpetuity ##(C/r)## when interest rates don't change, and a fully amortising loan can be valued as an annuity ##(C/r.(1-(1+r)^{-T}))##. Since the ##C/r## is a common factor, it will cancel out when we find the proportional increase so we don't need to know what the monthly payments ##(C)## are. ###\begin{aligned} & \text{Proportional Increase} = \dfrac{V_\text{0,interest only}}{V_\text{0,fully amortising}} - 1 \\ &= \dfrac{ \left( \dfrac{C_1}{r} \right) }{ \left( \dfrac{C_1}{r} \left( 1 - \dfrac{1}{(1+r)^T} \right) \right) } - 1 \\ &= \dfrac{ 1 }{ \left( 1 - \dfrac{1}{(1+r)^{T}} \right) } - 1 \\ &= \dfrac{ 1 }{ \left( 1 - \dfrac{1}{(1+0.04/12)^{25 \times 12}} \right) } - 1 \\ &= \dfrac{ 1 }{ 0.631508277 } - 1 \\ &= 0.583510521 = 58.3510521\% \end{aligned}###

Intuitive method

Let's find the value of the interest only loan and then the fully amortising loan if the payments are $2,000 per month. ###\begin{aligned} V_\text{0,interest only} &= \dfrac{C_\text{1, monthly}}{r_\text{eff monthly}} \\ &= \dfrac{2,000}{0.04/12} \\ &= 600,000 \\ \end{aligned}### ###\begin{aligned} V_\text{0,fully amortising} &=\dfrac{C_\text{1, monthly}}{r_\text{eff monthly}} \left( 1 - \dfrac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ &= \dfrac{2,000}{0.04/12} \left( 1 - \dfrac{1}{(1+0.04/12)^{25 \times 12}} \right) \\ &= 378,904.9659 \\ \end{aligned}### ###\begin{aligned} & \text{Proportional Increase} = \dfrac{V_\text{0,interest only}}{V_\text{0,fully amortising}} - 1 \\ &= \dfrac{600,000}{378,904.9659} - 1 \\ &= 0.583510521 = 58.3510521 \% \end{aligned}###

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

The annuity formula with T cash flows sums the present value of each, where the first is at time 1 and the last at time T:

###\dfrac{C}{r}\left(1-\dfrac{1}{(1+r)^T} \right) = \dfrac{C_1}{(1+r)^1} +\dfrac{C_2}{(1+r)^2} + ... + \dfrac{C_T}{(1+r)^T} ###

The annuity formula with 3 cash flows sums the present value of each, where the first is at time 1 and the last at time 3:

###\dfrac{C}{r}\left(1-\dfrac{1}{(1+r)^3} \right) = \dfrac{C_1}{(1+r)^1} +\dfrac{C_2}{(1+r)^2} + \dfrac{C_3}{(1+r)^3} ###

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

The annuity formula is perfectly suited to this problem since the payments are all equal. Since the first cash flow is exactly one month away, the annuity equation will give a present value one month before that which is at time zero. Perfect.

###\begin{aligned} V_0 &= \frac{C_{1}}{r} \left( 1-\frac{1}{(1+r)^T} \right) \\ V_0 &= \frac{C_\text{1 monthly}}{r_\text{eff monthly}} \left( 1-\frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ &= \frac{570}{0.0123} \left( 1-\frac{1}{(1+0.0123)^{10 \times 12}} \right) \\ &= 35,654.3278 \\ \end{aligned}###

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

Use the annuity formula to find the present value of the 10 payments of $80 starting in 3 years (t=3). Keep in mind that the annuity formula gives a value that is one period before the first cash flow at t=3, so the value of the annuity will be at t=2 and needs discounting by 2 periods to get to t=0.

###\begin{aligned} V_\text{0, annuity} &= C_{1} \times \dfrac{1}{r} \left(1 - \dfrac{1}{(1+r)^{10}} \right) \\ V_\text{2, annuity} &= C_{\mathbf{3}} \times \dfrac{1}{r} \left(1 - \dfrac{1}{(1+r)^{10}} \right) \\ &= 80 \times \dfrac{1}{0.1} \left(1 - \dfrac{1}{(1+0.1)^{10}} \right) \\ &= 80 \times 6.144567106 \\ &= 491.56536848 \\ \end{aligned} ###

Now discount this value at year 2 (t=2) to the present (t=0) using the 'present value of the single cash flow' formula: ###\begin{aligned} V_\text{0, annuity} &= \dfrac{V_\text{2, annuity}}{(1+r)^2} \\ &= \dfrac{491.56536848 }{(1+0.1)^{2}} \\ &= 406.252370645 \\ \end{aligned} ###

To discount the $600 payment in 5 years and 6 months (t=5.5) to a value now (a present value), again use the 'present value of the single cash flow' formula:

###\begin{aligned} V_\text{0, single cash flow} &= \dfrac{C_\text{5.5}}{(1+r)^{5.5}} \\ &= \dfrac{600}{(1+0.1)^{5.5}} \\ &= 355.2151514 \\ \end{aligned} ###

Now we just add the present values of the annuity and the single payment in 5.5 years together:

###\begin{aligned} V_\text{0, all} &= V_\text{0, annuity} + V_\text{0, single cash flow} \\ &= 406.252370645 + 355.2151514 \\ &= 761.4675221 \\ \end{aligned} ###

Here's all the steps in one big formula:

###\begin{aligned} V_{0} &= \dfrac{C_{3} \times \dfrac{1}{r_\text{eff annual}} \left(1 - \dfrac{1}{(1+r_\text{eff annual})^{10}} \right)}{(1+r_\text{eff annual})^2} + \dfrac{C_{5.5}}{(1+r_\text{eff annual})^{5.5}} \\ &= \dfrac{80 \times \dfrac{1}{0.1} \left(1 - \dfrac{1}{(1+0.1)^{10}} \right)}{(1+0.1)^2} + \dfrac{600}{(1+0.1)^{5.5}} \\ &= \dfrac{80 \times 6.144567106}{(1+0.1)^2} + \dfrac{600}{(1+0.1)^{5.5}} \\ &= 406.2523706 + 355.2151514 \\ &= 761.4675221 \\ \end{aligned} ###

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

The effective monthly interest rate can be calculated by dividing the annualised percentage rate compounding per month by 12.

###\begin{aligned} r_\text{eff mthly} &= r_\text{APR comp monthly} / 12 \\ &= 0.06/12 \\ &= 0.005 \\ \end{aligned}###

The present value of the annuity of end-of-month payments can be calculated using the ordinary annuity equation. Let the current time at which the man is 20 years old be time zero (t=0).

###\begin{aligned} V_0 &= \frac{C_\text{1, monthly}}{r_\text{eff monthly}}\left(1-\dfrac{1}{(1+r_\text{eff monthly})^{T_\text{months}}}\right) \\ &= \frac{30}{0.005}\left(1-\dfrac{1}{(1+0.005)^{720}}\right) \\ &= 5,834.580469 \\ \end{aligned}###

To find the value in 720 months ##(=60\text{ years}\times12\text{ months/year})## we can just future value the present value.

###\begin{aligned} V_\text{T months} &= V_0(1+r_\text{eff monthly})^{T_\text{months}} \\ V_{720} &= 5,834.580469\times(1+0.005)^{720} \\ &= 211,628.4731 \\ \end{aligned}###

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

All answers are mathematically equivalent except (e).

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

The annuity formula can be applied to find the present value of the 6 equal payments from t=2 to 7. But care must be taken since the present value of an annuity is one period before the first cash flow (at t=2), so the whole annuity value will be at t=1 so it needs to be discounted back one extra period to get a present value.

###\begin{aligned} V_0 &= C_0 - \dfrac{ C_2.\dfrac{1}{r}\left( 1-\dfrac{1}{(1+r)^6} \right) }{(1+r)^1} \\ &= 5,000 - \dfrac{ 1,000 \times \dfrac{1}{0.1}\left( 1-\dfrac{1}{(1+0.1)^6} \right) }{(1+0.1)^1} \\ &= 5,000 - 3,959.3279 \\ &= 1,040.6721 \\ \end{aligned}###

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

When the yield is zero, there is no time value of money. Therefore we can just sum cash flows like an accountant.

###\begin{aligned} V_0 &= T \times C \\ &= 5 \times 10 = 50 \\ \end{aligned}###

Interestingly, the normal way to value an annuity with the annuity equation will not work since there will be a divide by zero problem which is mathematically impossible:

###\begin{aligned} V_0 &= C_\text{1} \times \frac{1}{r_\text{eff yrly}} \left( 1 - \frac{1}{(1+r_\text{eff yrly})^{T}} \right) \\ &= 1 \times \color{red}{\frac{1}{0}} \left( 1 - \frac{1}{(1+0)^{5}} \right) \\ \end{aligned}###

Since 1/0 is mathematically undefined, that is a dead-end.

But present-valuing the individual payments separately will still work.

###\begin{aligned} P_0 &= \frac{C_\text{1 yr}}{(1+r_\text{eff yrly})^1} + \frac{C_\text{2 yr}}{(1+r_\text{eff yrly})^2} + \frac{C_\text{3 yr}}{(1+r_\text{eff yrly})^3} + \frac{C_\text{4 yr}}{(1+r_\text{eff yrly})^4} +\frac{C_\text{5 yr}}{(1+r_\text{eff yrly})^5} \\ &= \frac{10}{(1+0)^1} + \frac{10}{(1+0)^2} + \frac{10}{(1+0)^3} + \frac{10}{(1+0)^4} +\frac{10}{(1+0)^5} \\ &= 10+10+10+10+10 \\ &= 5 \times 10 \\ &= 50 \\ \end{aligned}###

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

We will use the annuity equation and the present value of a single cash flow equation. Keep in mind that the annuity equation gives a value that is one period before the first cash flow at t=6.5, so the value of the annuity will be at t=5.5 and needs discounting by 5.5 periods to get to t=0.

###\begin{aligned} V_{0} &= \dfrac{C_\text{6.5, annual} \times \dfrac{1}{r_\text{eff annual}} \left(1 - \dfrac{1}{(1+r_\text{eff annual})^{10}} \right)}{(1+r_\text{eff annual})^{5.5}} + \dfrac{C_{4.25}}{(1+r_\text{eff annual})^{4.25}} \\ &= \dfrac{80 \times \dfrac{1}{0.1} \left(1 - \dfrac{1}{(1+0.1)^{10}} \right)}{(1+0.1)^{5.5}} + \dfrac{500}{(1+0.1)^{4.25}} \\ &= \dfrac{80 \times 6.144567106}{(1+0.1)^{5.5}} + \dfrac{500}{(1+0.1)^{4.25}} \\ &= 291.0191113 + 236.810101 \\ &= 624.4847522 \\ \end{aligned} ###

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

No explanation provided.