Which of the following statements is NOT correct? Borrowers:
Borrowers sell debt. This is because borrowers receive cash at the start for selling the debt contract to the lender. Note that selling debt can also be called being 'shorting' debt.
Which of the following statements is NOT correct? Lenders:
Lenders are owed money by borrowers. Confusingly, lenders' debt is actually an asset, not a liability. Lenders own the asset class debt.
A home loan company advertises an interest rate of 6% pa, payable monthly. Which of the following statements about the interest rate is NOT correct? All rates are given to four decimal places.
In most countries, all interest rates are advertised as annualised percentage rates (APR's). In some countries such as Australia, this is a legal requirement. So the 6% pa interest rate must be an APR. Since home loans are paid monthly, we can assume that it is an APR compounding monthly.
###\begin{aligned} r_\text{apr monthly} &= 0.06 \\ \end{aligned}###The effective monthly rate is easy to find since the 6% APR compounding monthly is defined as 12 times the effective monthly rate.
###\begin{aligned} r_\text{eff monthly} &= \frac{r_\text{apr monthly}}{12} = \frac{0.06}{12} =0.005 \\ \end{aligned}###To find the the effective annual rate, the effective monthly rate needs to be compounded up by twelve periods.
###\begin{aligned} (1+r_\text{eff annual})^1 &= \left(1+r_\text{eff monthly} \right)^{12} \\ &= \left(1+\frac{r_\text{apr monthly}}{12}\right)^{12} \\ &= \left(1+\frac{0.06}{12}\right)^{12} \\ \end{aligned}### ###\begin{aligned} r_\text{eff annual} &= \left(1+\frac{0.06}{12}\right)^{12}-1 = 0.061677812 \\ \end{aligned}###To find the effective semi-annual rate, the effective monthly rate needs to be compounded up by six periods. This is because there's 6 months in half a year.
###\begin{aligned} (1+r_\text{eff semi-annual})^1 &= \left(1+r_\text{eff monthly} \right)^{6} \\ &= \left(1+\frac{r_\text{apr monthly}}{12}\right)^{6} \\ &= \left(1+\frac{0.06}{12}\right)^{6} \\ \end{aligned}### ###\begin{aligned} r_\text{eff semi-annual} &= \left(1+\frac{0.06}{12}\right)^{6}-1 = 0.030377509 \\ \end{aligned}###The APR compounding semi-annually is defined as the effective semi-annual rate times two. Since we've already found the effective semi-annual rate above, we can use that. ###\begin{aligned} r_\text{apr semi-annual} &= r_\text{eff semi-annual} \times 2 = 0.030377509 \times 2 = 0.060755019 \\ \end{aligned}###
A semi-annual coupon bond has a yield of 3% pa. Which of the following statements about the yield is NOT correct? All rates are given to four decimal places.
In most countries, all interest rates are advertised as annualised percentage rates (APR's). In some countries such as Australia, this is a legal requirement. So the 3% pa interest rate must be an APR. Since the bonds pay coupons semi-annually, we can assume that it is an APR compounding every 6 months.
###\begin{aligned} r_\text{apr semi-annual} &= r_\text{apr 6mth} = 0.03 \\ \end{aligned}###The effective monthly rate is easy to find since the 3% APR compounding semi-annually is defined as 2 times the effective semi-annual rate.
###\begin{aligned} r_\text{eff 6mth} &= \frac{r_\text{apr 6mth}}{2} = \frac{0.03}{2} =0.015 \\ \end{aligned}###To find the the effective annual rate, the effective 6 month rate needs to be compounded up by two periods.
###\begin{aligned} (1+r_\text{eff annual})^1 &= \left(1+r_\text{eff 6mth} \right)^{2} \\ &= \left(1+\frac{r_\text{apr 6mth}}{2}\right)^{2} \\ &= \left(1+\frac{0.03}{2}\right)^{2} \\ \end{aligned}### ###\begin{aligned} r_\text{eff annual} &= \left(1+\frac{0.03}{2}\right)^{2}-1 = 0.030225 \\ \end{aligned}###To find the effective monthly rate, the effective 6 month rate needs to be compounded down by six periods.
###\begin{aligned} (1+r_\text{eff monthly})^1 &= \left(1+r_\text{eff 6mth} \right)^{1/6} \\ &= \left(1+\frac{r_\text{apr 6mth}}{2}\right)^{1/6} \\ &= \left(1+\frac{0.03}{2}\right)^{1/6} \\ \end{aligned}### ###\begin{aligned} r_\text{eff monthly} &= \left(1+\frac{0.03}{2}\right)^{1/6}-1 = 0.002484517 \\ \end{aligned}###The APR compounding monthly is defined as the effective monthly rate times 12. Since we've already found the effective monthly rate above, we can use that. ###\begin{aligned} r_\text{apr monthly} &= r_\text{eff monthly} \times 12 = 0.002484517 \times 12 = 0.029814201 \\ \end{aligned}###
Which of the below statements about effective rates and annualised percentage rates (APR's) is NOT correct?
An APR is a discretely compounding annual rate that compounds multiple times per year.
An APR compounding once per year is an effective annual rate.
An effective rate is also a discretely compounding rate but it compounds only once per period. The time period is not necessarily annual, it can be monthly, daily, two years, or any time.
Therefore answer (c) is incorrect. An effective monthly rate is a monthly rate compounding per month.
A credit card offers an interest rate of 18% pa, compounding monthly.
Find the effective monthly rate, effective annual rate and the effective daily rate. Assume that there are 365 days in a year.
All answers are given in the same order:
### r_\text{eff monthly} , r_\text{eff yearly} , r_\text{eff daily} ###
###\begin{aligned} r_\text{eff monthly} &= \frac{r_\text{apr comp monthly}}{12} \\ &= \frac{0.18}{12} \\ &= 0.015 \\ \end{aligned}###
###\begin{aligned} r_\text{eff yearly} &= \left(1+\frac{r_\text{apr comp monthly}}{12}\right)^{12} - 1 \\ &= \left(1+\frac{0.18}{12}\right)^{12} - 1 \\ &= 0.195618171 \\ \end{aligned}###
###\begin{aligned} r_\text{eff daily} &= \left(1+\frac{r_\text{apr comp monthly}}{12}\right)^{12/365}-1 \\ &= \left(1+\frac{0.18}{12}\right)^{12/365}-1 \\ &= 0.000489608 \\ \end{aligned}###
Calculate the effective annual rates of the following three APR's:
- A credit card offering an interest rate of 18% pa, compounding monthly.
- A bond offering a yield of 6% pa, compounding semi-annually.
- An annual dividend-paying stock offering a return of 10% pa compounding annually.
All answers are given in the same order:
##r_\text{credit card, eff yrly}##, ##r_\text{bond, eff yrly}##, ##r_\text{stock, eff yrly}##
###\begin{aligned} r_\text{credit card, eff yrly} &= \left(1 + \frac{r_\text{credit card, apr comp monthly}}{12} \right)^{12} - 1 \\ &= \left(1 + \frac{0.18}{12} \right)^{12} - 1 \\ &= 0.195618171 \\ \end{aligned}###
###\begin{aligned} r_\text{bond, eff yrly} &= \left(1 + \frac{r_\text{bond, apr comp 6 monthly}}{2} \right)^{2} - 1 \\ &= \left(1 + \frac{0.06}{2} \right)^{2} - 1 \\ &= 0.0609 \\ \end{aligned}###
###\begin{aligned} r_\text{stock, eff yrly} &= \left(1 + \frac{r_\text{stock, apr comp yearly}}{1} \right)^{1} - 1 \\ &= r_\text{stock, apr comp yearly} \\ &= 0.1 \\ \end{aligned}###
On his 20th birthday, a man makes a resolution. He will deposit $30 into a bank account at the end of every month starting from now, which is the start of the month. So the first payment will be in one month. He will write in his will that when he dies the money in the account should be given to charity.
The bank account pays interest at 6% pa compounding monthly, which is not expected to change.
If the man lives for another 60 years, how much money will be in the bank account if he dies just after making his last (720th) payment?
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}###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 a fully amortising 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 loan payments are paid in arrears (at the end of the month).
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} ###
You want to buy an apartment worth $400,000. You have saved a deposit of $80,000. The bank has agreed to lend you the $320,000 as a fully amortising 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?
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} ###
You just signed up for a 30 year fully amortising mortgage loan with monthly payments of $2,000 per month. The interest rate is 9% pa which is not expected to change.
How much did you borrow? After 5 years, how much will be owing on the mortgage? The interest rate is still 9% and is not expected to change.
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} ###
You just signed up for a 30 year fully amortising mortgage loan with monthly payments of $1,500 per month. The interest rate is 9% pa which is not expected to change.
How much did you borrow? After 10 years, how much will be owing on the mortgage? The interest rate is still 9% and is not expected to change.
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.
You just agreed to a 30 year fully amortising mortgage loan with monthly payments of $2,500. The interest rate is 9% pa which is not expected to change.
How much did you borrow? After 10 years, how much will be owing on the mortgage? The interest rate is still 9% and is not expected to change. The below choices are given in the same order.
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.
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 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}} ###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%.
Confusingly, an 'interest payment' is the same thing as a 'coupon payment', but an 'interest rate' is not the same thing as a 'coupon rate'. It makes no sense, but that's the way it is.
Interest rate has the same meaning as yield, required return, cost of debt, total return, discount rate and many others but not a coupon rate.
Confusingly, an 'interest rate' is not the same thing as a 'coupon rate', yet an 'interest payment' is the same thing as a 'coupon payment'. It makes no sense, but that's the way it is.
Interest rate has the same meaning as yield, required return, cost of debt, total return, discount rate and many others but not a coupon rate.
Confusingly, an 'interest rate' is the same thing as a 'yield' or 'return', even though an 'interest payment' is the same thing as a 'coupon payment' but coupon payments are calculated from coupon rates which are not yields. It's baffling.
Interest rate has the same meaning as yield, total required return, cost of debt, discount rate and many others but not a coupon rate.
An 'interest only' loan can also be called a:
The price of an interest only loan (the amount paid by lender to borrower at the start) is not paid off at all until the very end. Only interest payments are made throughout the life of the loan.
Therefore the amount owing at the end of the loan will be equal to the original amount borrowed.
Since the amount at the end is called the principal, face value or par value, we say that the loan is priced at par. It's a par loan.
Calculate the price of a newly issued ten year bond with a face value of $100, a yield of 8% pa and a fixed coupon rate of 6% pa, paid annually. So there's only one coupon per year, paid in arrears every year.
Since the bond pays coupons annually, we can assume that the yield is given as an annualised percentage rate (APR) compounding every year. Therefore there's no need to do anything because an APR compounding annually is an effective annual rate and since the coupons are annual, we can simply use the 8% rate in our equations:
###\begin{aligned} p_\text{0, bond} &= \text{PV(Annuity of coupons)} + \text{PV(Face value)} \\ &= \text{Coupon} \times \frac{1}{r_\text{eff}}\left(1 - \frac{1}{(1+r_\text{eff})^{T}} \right) + \frac{\text{Face}}{(1+r_\text{eff})^{T}} \\ &= (100 \times 0.06) \times \frac{1}{0.08}\left(1 - \frac{1}{(1+0.08)^{10}} \right) + \frac{100}{(1+0.08)^{10}} \\ &= 6 \times 6.710081399 + 46.31934881 \\ &= 40.26048839 + 46.31934881 \\ &= 86.5798372 \\ \end{aligned} ###
At the risk of making the annual-coupon paying bond pricing formula look more confusing, here is the version with specific names for the types of returns being used in the working above:
###\begin{aligned} p_\text{0, bond} &= \text{AnnualCoupon} \times \frac{1}{r_\text{eff yearly}}\left(1 - \frac{1}{(1+r_\text{eff yearly})^{T_\text{yearly periods}}} \right) + \frac{\text{Face}}{(1+r_\text{eff yearly})^{T_\text{yearly periods}}} \\ &= \text{AnnualCoupon} \times \frac{1}{r_\text{APR comp yearly}/1}\left(1 - \frac{1}{(1+r_\text{APR comp yearly}/1)^{T_\text{yearly periods}}} \right) + \frac{\text{Face}}{(1+r_\text{APR comp yearly}/1)^{T_\text{yearly periods}}} \\ &= \left( \frac{100 \times 0.06}{1} \right) \times \frac{1}{0.08/1}\left(1 - \frac{1}{(1+0.08/1)^{10}} \right) + \frac{100}{(1+0.08/1)^{10}} \\ &= 86.5798372 \\ \end{aligned} ###
Calculate the price of a newly issued ten year bond with a face value of $100, a yield of 8% pa and a fixed coupon rate of 6% pa, paid semi-annually. So there are two coupons per year, paid in arrears every six months.
Since the bond pays coupons semi-annually, we can assume that the yield is given as an annualised percentage rate (APR) compounding every 6 months. Therefore we'll divide the 8% APR compounding semi-annually by 2 to get the yield as an effective 6 month rate:
###\begin{aligned} p_\text{0, bond} &= \text{PV(Annuity of coupons)} + \text{PV(Face value)} \\ &= \text{Coupon} \times \frac{1}{r_\text{eff}}\left(1 - \frac{1}{(1+r_\text{eff})^{T}} \right) + \frac{\text{Face}}{(1+r_\text{eff})^{T}} \\ &= \left( \frac{100 \times 0.06}{2} \right) \times \frac{1}{0.08/2}\left(1 - \frac{1}{(1+0.08/2)^{10\times2}} \right) + \frac{100}{(1+0.08/2)^{10 \times 2}} \\ &= 3 \times 13.59032634 + 45.63869462 \\ &= 40.77097903 + 45.63869462 \\ &= 86.40967366 \\ \end{aligned} ###
At the risk of making the bond pricing formula look more confusing, here is the version with specific names for the types of returns being used in the working above:
###\begin{aligned} p_\text{0, bond} &= \text{SixMonthCoupon} \times \frac{1}{r_\text{eff 6mth}}\left(1 - \frac{1}{(1+r_\text{eff 6mth})^{T_\text{6mth periods}}} \right) + \frac{\text{Face}}{(1+r_\text{eff 6mth})^{T_\text{6mth periods}}} \\ &= \text{SixMonthCoupon} \times \frac{1}{r_\text{APR comp 6mth}/2}\left(1 - \frac{1}{(1+r_\text{APR comp 6mth}/2)^{T_\text{6mth periods}}} \right) + \frac{\text{Face}}{(1+r_\text{APR comp 6mth}/2)^{T_\text{6mth periods}}} \\ &= \left( \frac{100 \times 0.06}{2} \right) \times \frac{1}{0.08/2}\left(1 - \frac{1}{(1+0.08/2)^{10\times2}} \right) + \frac{100}{(1+0.08/2)^{10 \times 2}} \\ &= 86.40967366 \\ \end{aligned} ###
"Buy low, sell high" is a phrase commonly heard in financial markets. It states that traders should try to buy assets at low prices and sell at high prices.
Traders in the fixed-coupon bond markets often quote promised bond yields rather than prices. Fixed-coupon bond traders should try to:
Since buying debt is lending and selling debt is borrowing, then 'buy at low yields, sell at high yields' is the same as 'lend at low yields, borrow at high yields'. Clearly this is a bad idea!
Lenders want to receive high yields and borrowers want to pay low yields. Therefore it's wiser to 'lend at high yields, borrow at low yields', which is equivalent to 'buy at high yields, sell at low yields'.
A bond's yield is a discount rate, so higher yields lead to lower prices. This is obvious when considering the bond pricing formula where every instance of the yield '##r##' is in the denominator of a fraction, so dividing by a bigger number (the higher yield) leads to a lower bond price, and vice versa.
###\begin{aligned} P_\text{0, bond} =& \text{Coupon} \times \frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}} \right) + \frac{\text{Face}}{(1+r)^{T}} \\ \end{aligned} ###This inverse relationship between yields and prices is the reason why these phrases are all equivalent:
- Buy at low bond prices, sell at high bond prices.
- Lend at high yields, borrow at low yields.
- Buy at high yields, sell at low yields.
Bonds X and Y are issued by the same US company. Both bonds yield 10% pa, and they have the same face value ($100), maturity, seniority, and payment frequency.
The only difference is that bond X and Y's coupon rates are 8 and 12% pa respectively. Which of the following statements is true?
Bond X has a coupon rate that is only 8%, less than its 10% yield. Therefore bond X's price will be less than its face value, so it is a discount bond.
Bond Y has a coupon rate that is 12%, more than its 10% yield. Therefore bond Y's price will be more than its face value, so it is a premium bond.
Question 48 IRR, NPV, bond pricing, premium par and discount bonds, market efficiency
The theory of fixed interest bond pricing is an application of the theory of Net Present Value (NPV). Also, a 'fairly priced' asset is not over- or under-priced. Buying or selling a fairly priced asset has an NPV of zero.
Considering this, which of the following statements is NOT correct?
The NPV of buying any fairly priced asset is zero. Therefore the NPV of buying a fairly priced bond is also zero. Whether the bond is a premium or discount bond is irrelevant, it's unrelated to the NPV of buying it.
The fair price of a bond is the present value (PV) of its expected future cash flows, which is the present value of coupons and face value:
###\begin{aligned} P_\text{0, bond} &= PV(\text{coupons}) + PV(\text{face value}) \\ &= \frac{C_1}{r} \left(1 - \frac{1}{(1+r)^T} \right) + \frac{F_T}{(1+r)^T} \\ \end{aligned}###The net present value (NPV) of buying an asset is the present value of costs less gains.
###\begin{aligned} NPV &= -PV(\text{costs}) + PV(\text{gains}) \\ \end{aligned}###The cost of a bond is its price, and the gains from a bond are the coupons and face value. Since the price of a fairly priced bond equals the present value of the coupons and face value, then the net present value of buying a fairly priced bond must be zero.
Mathematically, we can re-arrange the bond price formula to be in the same form as the NPV formula, which shows that the NPV must be zero:
###P_\text{0, bond} = PV(\text{coupons}) + PV(\text{face value}) ### ###\underbrace{0}_{\text{NPV}} = -\underbrace{P_\text{0, bond}}_{PV(\text{costs})} + \underbrace{PV(\text{coupons}) + PV(\text{face value})}_{PV(\text{gains})} ###Note that premium bonds can also be fairly priced. The NPV of buying a fairly priced premium bond is zero. The term 'premium' does not indicate that the bond's price is above (or below) the fair price, it indicates that the bond's price is above its face value which is usually the $100 or $1,000 that's paid at maturity. Premium bonds have a higher price than their face value because the coupon rate is more than the total required return (the yield). Therefore investors are willing to pay a high price for the bond, higher than the face value, making the bond a premium bond. The highest price investors will pay for the bond will be the price that makes the NPV zero.
The theory of fixed interest bond pricing is an application of the theory of Net Present Value (NPV). Also, a 'fairly priced' asset is not over- or under-priced. Buying or selling a fairly priced asset has an NPV of zero.
Considering this, which of the following statements is NOT correct?
From the bond pricing formula, the required return r is in the denominator of each fraction so any increase in r causes a decrease in the price P and vice versa:
###P_\text{0, bond} = C_\text{1,2,3,...,T} \times \frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}} \right) + \frac{F_\text{T}}{(1+r)^{T}} ###When the required return rises, the bond price falls.
When the required return falls, the bond price rises.
This is not only true for bonds but for any asset including shares and land.
The required return of a fairly priced bond is also its IRR. Remember that the IRR is the discount rate that makes the NPV zero.
###\begin{aligned} NPV &= C_0 + \frac{C_1}{(1+r)^1} + \frac{C_2}{(1+r)^2} + ... + \frac{C_T}{(1+r)^T} \\ 0 &= C_0 + \frac{C_1}{(1+r_{irr})^1} + \frac{C_2}{(1+r_{irr})^2} + ... + \frac{C_T}{(1+r_{irr})^T} \\ \end{aligned} ###Re-arranging the bond-pricing equation:
###P_\text{0, bond} = C_\text{1,2,3,...,T} \times \frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}} \right) + \frac{F_\text{T}}{(1+r)^{T}} ### ###\underbrace{0}_{\text{NPV}} = -\underbrace{P_\text{0, bond}}_{PV(\text{cost})} + \underbrace{C_\text{1,2,3,...,T} \times \frac{1}{r_\text{IRR}}\left(1 - \frac{1}{(1+r_\text{IRR})^{T}} \right) + \frac{F_\text{T}}{(1+r_\text{IRR})^{T}}}_{PV(\text{gains})} ###Because the NPV of buying a fairly priced bond is zero, the bond's yield is equivalent to the IRR of buying it too.
A bond maturing in 10 years has a coupon rate of 4% pa, paid semi-annually. The bond's yield is currently 6% pa. The face value of the bond is $100. What is its price?
###\begin{aligned} p_\text{0, bond} &= \text{PV(annuity of coupons)} + \text{PV(face value)} \\ &= \text{coupon} \times \frac{1}{r_\text{eff}}\left(1 - \frac{1}{(1+r_\text{eff})^{T}} \right) + \frac{\text{face}}{(1+r_\text{eff})^{T}} \\ &= \frac{100 \times 0.04}{2} \times \frac{1}{0.06/2}\left(1 - \frac{1}{(1+0.06/2)^{10\times2}} \right) + \frac{100}{(1+0.06/2)^{10 \times 2}} \\ &= 2 \times 14.8774748604555 + 55.3675754186335 \\ &= 29.754949720911 + 55.3675754186335 \\ &= 85.1225251395445 \\ \end{aligned} ###
A three year bond has a fixed coupon rate of 12% pa, paid semi-annually. The bond's yield is currently 6% pa. The face value is $100. What is its price?
###\begin{aligned} P_\text{0, bond} &= \text{PV(annuity of coupons)} + \text{PV(face value)} \\ &= \text{coupon} \times \frac{1}{r_\text{eff}}\left(1 - \frac{1}{(1+r_\text{eff})^{T}} \right) + \frac{\text{face}}{(1+r_\text{eff})^{T}} \\ &= \frac{100 \times 0.12}{2} \times \frac{1}{0.06/2}\left(1 - \frac{1}{(1+0.06/2)^{3\times2}} \right) + \frac{100}{(1+0.06/2)^{3 \times 2}} \\ &= 6 \times 5.41719144387819 + 83.7484256683654 \\ &= 32.5031486632691 + 83.7484256683654 \\ &= 116.251574331635 \\ \end{aligned} ###
Note that the coupon rate is more than the yield, so the price must be more than the face value. In other words, this is a premium bond. Since there is only one multiple choice answer choice more than the face value, that must be the correct price.
Which one of the following bonds is trading at a discount?
A discount bond's price will be less than its face value (hence the bond trades at a 'discount' to its face value), and its coupon rate will be less than its yield. The first 3 bonds are actually premium bonds and the fourth is a par bond, so none are discount bonds.
Let the 'income return' of a bond be the coupon at the end of the period divided by the market price now at the start of the period ##(C_1/P_0)##. The expected income return of a premium fixed coupon bond is:
A premium bond sells at a premium to its face value. Therefore its price ##(P_0)## is higher than its face value ##(F_T)##: ###P_\text{0, premium bond} > F_{T}###
Note that the face value is also called the principal or par value. A bond investor receives the face value it at maturity which is at the end ##(t=T)##, while the price is paid at the start ##(t=0)##.
Since the bond price is currently more than face value, we know that the bond price will fall down to equal the face value over time as more coupons are paid. Therefore the expected capital return of the bond is negative since its price will fall over time. Because the total return is the sum of the capital and income returns, the total return will be less than the income return. This is the answer to this question.
In relation to the coupon rate, a premium bond's income return will be less than its coupon rate.
The income return ##(r_\text{income})## is defined as the coupon cash flow at the end of the period ##(C_1)## divided by the price now ##(P_0)##:
###r_\text{income} = C_1 / P_0 ###The coupon rate is defined as the coupon cash flow at the end of the period ##(C_1)## divided by the face value at maturity ##(F_T)##:
###\text{CouponRate} = C_1 / F_T ###Since the price of a premium bond is greater than its face value, then the expected income return must be less than the coupon rate:
###r_\text{income, premium bond} < \text{CouponRateOfAPremumBond}###Which one of the following bonds is trading at a premium?
A premium fixed-coupon bond's price is greater than its face value, and its coupon rate is greater than its yield. The only bond for which this is true is the five-year bond with a $2,000 face value whose yield to maturity is 7.0% and coupon rate is 7.2% paid semi-annually.
An investor bought two fixed-coupon bonds issued by the same company, a zero-coupon bond and a 7% pa semi-annual coupon bond. Both bonds have a face value of $1,000, mature in 10 years, and had a yield at the time of purchase of 8% pa.
A few years later, yields fell to 6% pa. Which of the following statements is correct? Note that a capital gain is an increase in price.
The yield on a bond is equivalent to its required return or discount rate. If yields fall, then the future payments are being discounted by less, so the price of the bonds will increase. This can be seen clearly in the below bond price equation since all amounts are divided by the yield ##r##, so clearly if ##r## falls then we're dividing by less so the price must increase.
###\begin{aligned} p_\text{0, bond} &= \text{PV(annuity of coupons)} + \text{PV(face value)} \\ &= \text{coupon} \times \frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}} \right) + \frac{\text{face}}{(1+r)^{T}} \\ \end{aligned} ###
The fall in yields and rise in bond prices corresponds to a positive capital return. This increase in price should happen straight away as soon as the news of the lower 6% pa yield arrives.
Both bonds would have been discount bonds when first issued, since their coupon rates (0% and 7%) were less than their yields (8%), and therefore their prices would have been less than their face values. After yields fell to 6% and the bond prices rose, the zero coupon bond would have still been a discount bond, but the 7% coupon bond would have been a premium bond.
In these tough economic times, central banks around the world have cut interest rates so low that they are practically zero. In some countries, government bond yields are also very close to zero.
A three year government bond with a face value of $100 and a coupon rate of 2% pa paid semi-annually was just issued at a yield of 0%. What is the price of the bond?
When the yield is zero, there is no time value of money. Therefore we can just sum cash flows like an accountant. Over the 3 year bond's maturity there will be 6 semi-annual coupon payments of $1 each, and the face value paid at maturity.
###\begin{aligned} P_\text{0, bond} &= 6 \times C + F \\ &= 6 \times 1 + 100 = 106 \\ \end{aligned}###
Interestingly, the normal way to value a fixed-coupon bond using the annuity equation will not work since there will be a divide by zero problem which is mathematically impossible:
###\begin{aligned} P_0 &= C_\text{1} \times \frac{1}{r_\text{eff 6mth}} \left( 1 - \frac{1}{(1+r_\text{eff 6mth})^{T}} \right) + \frac{F_T}{(1+r_\text{eff 6mth})^T} \\ &= 1 \times \color{red}{\frac{1}{0}} \left( 1 - \frac{1}{(1+0)^{6}} \right) + \frac{100}{(1+0)^6} \\ \end{aligned}###
Which is mathematically undefined, so that is a dead-end.
But present-valuing the individual payments separately will still work.
###\begin{aligned} P_0 &= \frac{C_\text{0.5 yr}}{(1+r_\text{eff 6mth})^1} + \frac{C_\text{1 yr}}{(1+r_\text{eff 6mth})^2} + \frac{C_\text{1.5 yr}}{(1+r_\text{eff 6mth})^3} + \frac{C_\text{2 yr}}{(1+r_\text{eff 6mth})^4} +\frac{C_\text{2.5 yr}}{(1+r_\text{eff 6mth})^5} + \frac{C_\text{3 yr}}{(1+r_\text{eff 6mth})^6} + \frac{F_\text{3 yr}}{(1+r_\text{eff 6mth})^6} \\ &= \frac{1}{(1+0)^1} + \frac{1}{(1+0)^2} + \frac{1}{(1+0)^3} + \frac{1}{(1+0)^4} +\frac{1}{(1+0)^5} + \frac{1}{(1+0)^6} + \frac{100}{(1+0)^6} \\ &= 1+1+1+1+1+1+100 \\ &= 6 \times 1 + 100 \\ &= 106 \\ \end{aligned}###
A 10 year bond has a face value of $100, a yield of 6% pa and a fixed coupon rate of 8% pa, paid semi-annually. What is its price?
###\begin{aligned} P_\text{0} =& \text{PV(annuity of semi-annual coupons)} + \text{PV(face value)} \\ =& C_\text{1,2..T} \times \frac{1}{r_\text{eff 6mth}}\left(1 - \frac{1}{(1+r_\text{eff 6mth})^{T}} \right) + \frac{F_\text{T}}{(1+r_\text{eff 6mth})^{T}} \\ =& \frac{100 \times 0.08}{2} \times \frac{1}{0.06/2}\left(1 - \frac{1}{(1+0.06/2)^{10\times2}} \right) + \frac{100}{(1+0.06/2)^{10 \times 2}} \\ =& 59.50989944 + 55.36757542 \\ =& 114.8774749 \\ \end{aligned} ###
Below are some statements about loans and bonds. The first descriptive sentence is correct. But one of the second sentences about the loans' or bonds' prices is not correct. Which statement is NOT correct? Assume that interest rates are positive.
Note that coupons or interest payments are the periodic payments made throughout a bond or loan's life. The face or par value of a bond or loan is the amount paid at the end when the debt matures.
Answer d is a false statement. A zero coupon bond's coupon rate is obviously zero and assuming that yields are positive, then the coupon rate is less than the yield which means that zero coupon bonds are discount bonds. Their price should be less than their face value.
Question 35 bond pricing, zero coupon bond, term structure of interest rates, forward interest rate
A European company just issued two bonds, a
- 1 year zero coupon bond at a yield of 8% pa, and a
- 2 year zero coupon bond at a yield of 10% pa.
What is the company's forward rate over the second year (from t=1 to t=2)? Give your answer as an effective annual rate, which is how the above bond yields are quoted.
Using the term structure of interest rates equation, also known as the expectations hypothesis theory of interest rates,
###(1+r_{0-2})^2 = (1+r_{0-1})(1+r_{1-2}) ### ###(1+0.1)^2 = (1+0.08)(1+r_{1-2}) ### ###1+r_{1-2} = \frac{(1+0.1)^2}{1+0.08} ### ###\begin{aligned} r_{1-2} =& \frac{(1+0.1)^2}{1+0.08} - 1 \\ =& 0.12037037 \\ \end{aligned} ###Question 143 bond pricing, zero coupon bond, term structure of interest rates, forward interest rate
An Australian company just issued two bonds:
- A 6-month zero coupon bond at a yield of 6% pa, and
- A 12 month zero coupon bond at a yield of 7% pa.
What is the company's forward rate from 6 to 12 months? Give your answer as an APR compounding every 6 months, which is how the above bond yields are quoted.
Using the term structure of interest rates equation, also known as the expectations hypothesis theory of interest rates,
###\left(1+\frac{r_{\text{0}\rightarrow\text{12mth, apr 6mth}}}{2}\right)^2 = \left(1+\frac{r_{\text{0}\rightarrow\text{6mth, apr 6mth}}}{2}\right)^1 \left(1+\frac{r_{\text{6}\rightarrow\text{12mth apr 6mth}}}{2}\right)^1 ### ###\left(1+\frac{0.07}{2}\right)^2 = \left(1+\frac{0.06}{2}\right)^1 \left(1+\frac{r_{\text{6}\rightarrow\text{12mth apr 6mth}}}{2}\right)^1 ### ###\left(1+\frac{r_{\text{6}\rightarrow\text{12mth, apr 6mth}}}{2}\right)^1 = \frac{\left(1+\frac{0.07}{2}\right)^2}{\left(1+\frac{0.06}{2}\right)^1} ### ###\begin{aligned} r_{\text{6}\rightarrow\text{12mth, apr 6mth}} &= \left( \frac{\left(1+\frac{0.07}{2}\right)^2}{\left(1+\frac{0.06}{2}\right)^1} - 1 \right) \times 2 \\ &= 0.0800 \\ \end{aligned} ###Question 96 bond pricing, zero coupon bond, term structure of interest rates, forward interest rate
An Australian company just issued two bonds paying semi-annual coupons:
- 1 year zero coupon bond at a yield of 8% pa, and a
- 2 year zero coupon bond at a yield of 10% pa.
What is the forward rate on the company's debt from years 1 to 2? Give your answer as an APR compounding every 6 months, which is how the above bond yields are quoted.
Using the term structure of interest rates equation, also known as the expectations hypothesis theory of interest rates,
###\left(1+r_\text{0-2yrs, eff 6mth}\right)^4 = \left(1+r_\text{0-1yrs, eff 6mth}\right)^2\left(1+r_\text{1-2yrs, eff 6mth}\right)^2 ### ###\left(1+\frac{r_\text{0-2yrs, apr 6mth}}{2}\right)^4 = \left(1+\frac{r_\text{0-1yrs, apr 6mth}}{2}\right)^2\left(1+\frac{r_\text{1-2yrs, apr 6mth}}{2}\right)^2 ### ###\left(1+\frac{0.1}{2}\right)^4 = \left(1+\frac{0.08}{2}\right)^2\left(1+\frac{r_\text{1-2yrs, apr 6mth}}{2}\right)^2 ### ###\left(1+\frac{r_\text{1-2yrs, apr 6mth}}{2}\right)^2 = \frac{\left(1+\frac{0.1}{2}\right)^4}{\left(1+\frac{0.08}{2}\right)^2} ### ###\begin{aligned} r_\text{1-2yrs, apr 6mth} &= \left( \left( \frac{\left(1+\frac{0.1}{2}\right)^4}{\left(1+\frac{0.08}{2}\right)^2} \right)^{1/2} - 1 \right) \times 2\\ &= 0.1202 \\ \end{aligned} ###You're trying to save enough money to buy your first car which costs $2,500. You can save $100 at the end of each month starting from now. You currently have no money at all. You just opened a bank account with an interest rate of 6% pa payable monthly.
How many months will it take to save enough money to buy the car? Assume that the price of the car will stay the same over time.
The cash flows occur every month, so the discount rate needs to be an effective monthly rate. This also means that time (T), the unknown variable, will be measured in months.
Since the money needed to buy the car will be ready in the future, we need the future value of the annuity, not the present value. An easy way to do this is to find the present value of the annuity of $100 monthly payments, and then grow that amount out to the future. This is a simple step which doesn't require memorisation of the 'future value of an annuity equation'. Then we can solve for the only unknown variable which is 'T'.
###V_0 = \dfrac{C_\text{monthly}}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( \vcenter{ 1 - \dfrac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} } \right) ### ###V_T = \dfrac{C_\text{monthly}}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( \vcenter{ 1 - \dfrac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} } \right) \left(1+\dfrac{r_\text{apr comp monthly}}{12} \right) ^{T}### ###2,500 = \dfrac{100}{\left( \dfrac{0.06}{12} \right) } \left( \vcenter{ 1 - \dfrac{1}{\left(1+\dfrac{0.06}{12}\right)^{T}} } \right) \left(1+\dfrac{0.06}{12} \right) ^{T} ### ###2,500 = \dfrac{100}{\left( \dfrac{0.06}{12} \right) } \left( \vcenter{ \left(1+\dfrac{0.06}{12} \right) ^{T} - \dfrac{\left(1+\dfrac{0.06}{12} \right) ^{T}}{\left(1+\dfrac{0.06}{12}\right)^{T}} } \right) ### ###2,500 = \dfrac{100}{\left( \dfrac{0.06}{12} \right) } \left( \left(1+\dfrac{0.06}{12}\right)^{T} -1 \right) ### ###\dfrac{2,500}{100} \left( \dfrac{0.06}{12} \right) +1 = \left(1+\dfrac{0.06}{12}\right)^{T} ### ###\ln \left( \left( 1+\frac{0.06}{12}) \right) ^{T} \right) = \ln \left( \dfrac{2,500}{100} \left( \dfrac{0.06}{12} \right) +1 \right) ### ###T \times \ln \left( 1+\frac{0.06}{12} \right) = \ln \left( \vcenter{ \dfrac{2,500}{100} \left( \dfrac{0.06}{12} \right) +1} \right) ######\begin{aligned} T &= \dfrac{\ln \left( \dfrac{2,500}{100} \times \dfrac{0.06}{12} +1 \right)} {\ln \left( 1+\dfrac{0.06}{12} \right)} \\ &= \dfrac{\ln \left( 1.125 \right) } {\ln \left( 1.005 \right)} \\ &= 23.6154497 \text{ months}\\ \end{aligned}###
Your main expense is fuel for your car which costs $100 per month. You just refueled, so you won't need any more fuel for another month (first payment at t=1 month).
You have $2,500 in a bank account which pays interest at a rate of 6% pa, payable monthly. Interest rates are not expected to change.
Assuming that you have no income, in how many months time will you not have enough money to fully refuel your car?
The payments are monthly so let's first convert the interest rate, which must be an APR compounding per month, to an effective monthly rate.
###r_\text{eff monthly} = r_\text{apr comp monthly}/12 = 0.06/12 = 0.005###
The annuity equation can be used to find the time when the $2,500 runs out after repeated $100 withdrawals at the end of the month.
###V_0 = C_\text{1, monthly} \times \frac{1}{r_\text{eff mthly}} \left( 1 - \frac{1}{(1+r_\text{eff mthly})^{T}} \right) ### ###2,500 = 100 \times \frac{1}{0.005} \left( 1 - \frac{1}{(1+0.005)^{T}} \right) ### ###\frac{2,500 \times 0.005}{100} = 1 - \frac{1}{(1+0.005)^{T}} ### ###(1+0.005)^{-T} = 1 - \frac{2,500 \times 0.005}{100} ### ###\ln\left((1+0.005)^{-T}\right) = \ln\left(1 - \frac{2,500 \times 0.005}{100}\right) ### ###-T . \ln\left(1+0.005\right) = \ln\left(1 - \frac{2,500 \times 0.005}{100}\right) ######\begin{aligned} T &= -\frac{\ln\left(1 - \frac{2,500 \times 0.005}{100}\right)}{\ln\left(1+0.005\right)} \\ &= 26.77298872 \text{ months} \\ &= 27 \text{ months, rounded up to get the feasible solution.} \\ \end{aligned}###
The decimal answer is not a feasible solution since we only refuel at the end of each month, not at a fractional time period 0.77298872 way through the month. Therefore we round this number up to get the first time that we go to the petrol station and can not afford to fully refuel. This will be at t=27, the end of the 27th month.
You really want to go on a back packing trip to Europe when you finish university. Currently you have $1,500 in the bank. Bank interest rates are 8% pa, given as an APR compounding per month. If the holiday will cost $2,000, how long will it take for your bank account to reach that amount?
There are only two cash flows here, the start amount ##(V_0 = 1,500)## and the end amount ##(V_T = 2,000)##. Therefore the 'present value of a single future cash flow' formula will work well: ##V_0 = V_T/(1+r)^T##. We just need to solve for T, the payment time.
The only complicating factor is the 8% annualised percentage rate (APR) compounding per month. It must be converted to an effective rate. The easiest way is to convert it to an effective monthly rate since the APR compounding monthly is just the effective monthly rate multiplied by 12. So let's divide it by 12 to get the effective monthly rate:
###r_\text{apr, comp monthly} = 12 \times r_\text{eff monthly}### ###0.08 = 12 \times r_\text{eff monthly}### ###\begin{aligned} r_\text{eff monthly} &= \dfrac{0.08}{12} \\ &= 0.006666\dot{6} \\ \end{aligned}###Remember that since our effective monthly rate is measured in months, our time T will also be in months, not years! Let's now apply the formula:
###V_0 = \frac{V_t}{\left(1+r_\text{eff monthly} \right) ^{t}} ### ###V_0 = \frac{V_t}{\left(1+\frac{r_\text{apr, comp monthly}}{12} \right) ^{t}} ### ###1,500 = \frac{2,000}{\left(1+\frac{0.08}{12} \right) ^{t}} ### ###\left(1+\frac{0.08}{12} \right) ^{t} = \frac{2,000}{1,500} ### ###\ln \left( \left( 1+\frac{0.08}{12}) \right) ^{t} \right) = \ln \left( \frac{2,000}{1,500} \right) ### ###t \times \ln \left( 1+\frac{0.08}{12} \right) = \ln \left( \frac{2,000}{1,500} \right) ######\begin{aligned} t =& \frac{\ln \left( \frac{2,000}{1,500} \right)}{\ln \left( 1+\frac{0.08}{12} \right)} \\ =& \frac{\ln \left(1.333333333 \right)}{\ln \left(1.006666667 \right)} \\ =& 43.29599261 \text{ months}\\ =& \frac{43.29599261}{12} \text{ years}\\ =& 3.607999384 \text{ years}\\ \end{aligned}###