You want to buy an apartment priced at $300,000. You have saved a deposit of $30,000. The bank has agreed to lend you the $270,000 as an interest only loan with a term of 25 years. The interest rate is 12% pa and is not expected to change.
What will be your monthly payments? Remember that mortgage payments are paid in arrears (at the end of the month).
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}###
You just signed up for a 30 year interest-only mortgage with monthly payments of $3,000 per month. The interest rate is 6% pa which is not expected to change.
How much did you borrow? After 15 years, just after the 180th payment at that time, how much will be owing on the mortgage? The interest rate is still 6% and is not expected to change. Remember that the mortgage is interest-only and that mortgage payments are paid in arrears (at the end of the month).
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.
You just borrowed $400,000 in the form of a 25 year interest-only mortgage with monthly payments of $3,000 per month. The interest rate is 9% pa which is not expected to change.
You actually plan to pay more than the required interest payment. You plan to pay $3,300 in mortgage payments every month, which your mortgage lender allows. These extra payments will reduce the principal and the minimum interest payment required each month.
At the maturity of the mortgage, what will be the principal? That is, after the last (300th) interest payment of $3,300 in 25 years, how much will be owing on the mortgage?
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} ###
You want to buy an apartment worth $300,000. You have saved a deposit of $60,000.
The bank has agreed to lend you $240,000 as an interest only mortgage loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments?
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}} ###You want to buy an apartment priced at $500,000. You have saved a deposit of $50,000. The bank has agreed to lend you the $450,000 as an interest only loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments?
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.
Question 239 income and capital returns, inflation, real and nominal returns and cash flows, interest only loan
A bank grants a borrower an interest-only residential mortgage loan with a very large 50% deposit and a nominal interest rate of 6% that is not expected to change. Assume that inflation is expected to be a constant 2% pa over the life of the loan. Ignore credit risk.
From the bank's point of view, what is the long term expected nominal capital return of the loan asset?
The inflation rate in this question is a red herring, it's not needed to answer the question. Since the loan is interest-only, the amount borrowed at the start will be the same as the amount to be paid at maturity. Therefore the nominal capital return must be zero, since there is no increase in the nominal value of the loan, measured just after each interest payment. Therefore the nominal capital return on the loan is zero.
To summarise:
- The nominal capital return will be 0%. The real capital return will be about -2%.
- The nominal income return will be 6% which is the interest rate. The real income return will be also be about 6%, or just a touch less.
- The nominal total return will be 6%. The real total return will be about 4%.
A prospective home buyer can afford to pay $2,000 per month in mortgage loan repayments. The central bank recently lowered its policy rate by 0.25%, and residential home lenders cut their mortgage loan rates from 4.74% to 4.49%.
How much more can the prospective home buyer borrow now that interest rates are 4.49% rather than 4.74%? Give your answer as a proportional increase over the original amount he could borrow (##V_\text{before}##), so:
###\text{Proportional increase} = \frac{V_\text{after}-V_\text{before}}{V_\text{before}} ###Assume that:
- Interest rates are expected to be constant over the life of the loan.
- Loans are interest-only and have a life of 30 years.
- Mortgage loan payments are made every month in arrears and all interest rates are given as annualised percentage rates compounding per month.
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.
In Australia in the 1980's, inflation was around 8% pa, and residential mortgage loan interest rates were around 14%.
In 2013, inflation was around 2.5% pa, and residential mortgage loan interest rates were around 4.5%.
If a person can afford constant mortgage loan payments of $2,000 per month, how much more can they borrow when interest rates are 4.5% pa compared with 14.0% pa?
Give your answer as a proportional increase over the amount you could borrow when interest rates were high ##(V_\text{high rates})##, so:
###\text{Proportional increase} = \dfrac{V_\text{low rates}-V_\text{high rates}}{V_\text{high rates}} ###
Assume that:
- Interest rates are expected to be constant over the life of the loan.
- Loans are interest-only and have a life of 30 years.
- Mortgage loan payments are made every month in arrears and all interest rates are given as annualised percentage rates (APR's) compounding per month.
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.