You are a depositor giving cash at the start, therefore you are lending to the bank. The bank is borrowing from you.
You are a depositor giving cash at the start, therefore you are buying debt from the bank. The bank has sold their debt to you. You've bought the bank's promise to pay you back, which is a contract on a piece of paper.
You are a depositor giving cash at the start, therefore you are investing in the debt asset issued by the bank. In return for you investment, the bank issued you the piece of paper debt contract promising to pay you back the principal plus interest.
The deposit account at the bank is your debt asset since it will give you a future benefit. On the other side of the coin, the deposit account is the bank's debt liability since they owe it to you.
Borrowers owe money to their lenders.
Lenders are owed money by their borrowers who owe them. Strangely, 'owed' is not the past tense of 'owe'. They have completely opposite meanings which doesn't make sense.
Owners of debt assets must be lenders since they are owed money, they expect a positive benefit in the future when they're paid back.
Which of the following statements is NOT correct? Bond investors:
A bond investor buys bonds, which is lending. But a debtor sells bonds, which is borrowing.
A credit card company advertises an interest rate of 18% 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 18% pa interest rate must be an APR. Since credit cards are paid monthly, we can assume that it is an APR compounding monthly.
###\begin{aligned} r_\text{apr monthly} &= 0.18 \\ \end{aligned}###The effective monthly rate is easy to find since the 18% 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{0.18}{12} =0.015 \\ \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{0.18}{12}\right)^{12} \\ \end{aligned}### ###\begin{aligned} r_\text{eff annual} &= \left(1+\frac{0.18}{12}\right)^{12}-1 = 0.195618171 \\ \end{aligned}###To find the effective quarterly rate, the effective monthly rate needs to be compounded up by three periods since there's 3 months in a quarter of a year.
###\begin{aligned} (1+r_\text{eff quarterly})^1 &= \left(1+r_\text{eff monthly} \right)^{3} \\ &= \left(1+\frac{r_\text{apr monthly}}{12}\right)^{3} \\ &= \left(1+\frac{0.18}{12}\right)^{3} \\ \end{aligned}### ###\begin{aligned} r_\text{eff quarterly} &= \left(1+\frac{0.18}{12}\right)^{3}-1 = 0.045678375 \\ \end{aligned}###The APR compounding quarterly is defined as the effective quarterly rate times four. This is because there's 4 quarters in a year (not 3!). Since we've already found the effective quarterly rate above, we can use that. ###\begin{aligned} r_\text{apr quarterly} &= r_\text{eff quarterly} \times 4 = 0.045678375 \times 4 = 0.1827135 \\ \end{aligned}###
Which of the following statements about effective rates and annualised percentage rates (APR's) is NOT correct?
An APR compounding monthly is equal to 12 times the effective monthly rate. There are two steps required to convert an APR compounding monthly to an effective weekly rate:
- Convert the APR into the effective rate that it naturally converts into, an effective monthly rate, by dividing by 12.
- Convert the effective monthly rate into an effective weekly rate by compounding down. Add one and raise it all to the power of the inverse of the number of weeks in a month, all minus one.
The number of weeks per month is about 4, or to be more exact: 52 weeks per year divided by 12 months per year.
Mathematically:
###r_\text{eff monthly} = \dfrac{r_\text{apr comp monthly}}{12}### ###\begin{aligned} r_\text{eff weekly} &= \left( 1 + r_\text{eff monthly} \right)^{1/(\text{number of weeks in a month})} -1 \\ &= \left( 1 + \dfrac{r_\text{apr comp monthly}}{12} \right)^{1/(52/12)} -1 \\ \end{aligned}###A European bond paying annual coupons of 6% offers a yield of 10% pa.
Convert the yield into an effective monthly rate, an effective annual rate and an 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} ###
Since the coupons are paid annually, by convention (and in some countries by law), we assume that the yield is an APR compounding annually. An APR compounding annually is a special case that is also an effective annual rate.
###\begin{aligned} r_\text{eff, monthly} =& (1 + r_\text{eff,annual})^{1/12} - 1 \\ =& (1 + 0.1)^{1/12} - 1 \\ =& 0.00797414 \\ \end{aligned}###
### r_\text{eff, yearly} = 0.1 \text{ as given} ###
###\begin{aligned} r_\text{eff, daily} =& (1 + r_\text{eff, yearly})^{1/365} - 1 \\ =& (1 + 0.1)^{1/365} - 1 \\ =& 0.000261158 \\ \end{aligned}###
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.
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 a fully amortising 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) \\ 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} ###
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.
To your surprise, you can actually afford to pay $2,000 per month and your mortgage allows early repayments without fees. If you maintain these higher monthly payments, how long will it take to pay off your mortgage?
The cash flows occur every month, so the discount rate needs to be an effective monthly rate and the time must be measured in months.
First we have to find the amount borrowed when the payments are $1,500 per month for 30 years.
###\begin{aligned} V_0 &= C_{\text{monthly}} \times \dfrac{1}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} \right) \\ &= 1,500 \times \dfrac{1}{\left( \dfrac{0.09}{12} \right) } \left( 1 - \dfrac{1}{\left(1+\dfrac{0.09}{12}\right)^{30 \times 12}} \right) \\ &= 1,500 \times 124.2818657 \\ &= 186,422.7985 \\ \end{aligned}###
When the present value of the 'T' months of $2,000 payments are equal to the amount borrowed, then the loan will be paid off. So the only job left is to solve for T.
###V_0 = \dfrac{C_\text{monthly}}{\left( \dfrac{r_\text{apr comp monthly}}{12} \right) } \left( 1 - \frac{1}{\left(1+\dfrac{r_\text{apr comp monthly}}{12}\right)^{T}} \right) ### ###186,422.7985 = \dfrac{2,000}{\left( \dfrac{0.09}{12} \right) } \left( 1 - \frac{1}{\left(1+\dfrac{0.09}{12}\right)^{T}} \right) ### ###\frac{186,422.7985}{2,000} \left( \frac{0.09}{12} \right) = 1 - \frac{1}{\left(1+\frac{0.09}{12}\right)^{T}} ### ###\frac{1}{\left(1+\frac{0.09}{12}\right)^{T}} = 1 - \frac{186,422.7985}{2,000} \left( \frac{0.09}{12} \right) ### ###\left(1+\frac{0.09}{12}\right)^{-T} = 0.300914506 ### ### \ln \left( \left(1+\frac{0.09}{12}) \right) ^{-T} \right) = \ln \left(0.300914506 \right) ### ### -T \times \ln \left(1+\frac{0.09}{12} \right) = \ln \left(0.300914506 \right) ######\begin{aligned} T &= -\dfrac{\ln \left(0.300914506 \right)} {\ln \left( 1+\dfrac{0.09}{12} \right)} \\ &= 160.7235953 \text{ months}\\\\ &= 13.39363294 \text{ years}\\ \end{aligned}###
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.
Question 56 income and capital returns, bond pricing, premium par and discount bonds
Which of the following statements about risk free government bonds is NOT correct?
Hint: Total return can be broken into income and capital returns as follows:
###\begin{aligned} r_\text{total} &= \frac{c_1}{p_0} + \frac{p_1-p_0}{p_0} \\ &= r_\text{income} + r_\text{capital} \end{aligned} ###
The capital return is the growth rate of the price.
The income return is the periodic cash flow. For a bond this is the coupon payment.
A premium bond's price (##p_0##) is higher than its face value (##p_1##). Therefore: ###p_0 > p_1### ###r_\text{capital} = \frac{p_1-p_0}{p_0} < 0###
So the expected capital return of a premium bond must be negative, not positive.
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} ###
A two year Government bond has a face value of $100, a yield of 0.5% and a fixed coupon rate of 0.5%, paid semi-annually. What is its price?
This is a par bond since the coupon rate is equal to the yield. Therefore the price is equal to the face value, $100.
Using the fixed interest bond pricing formula gives the same answer, but takes a lot longer:
###\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 0.005}{2} \times \frac{1}{0.005/2}\left(1 - \frac{1}{(1+0.005/2)^{2\times2}} \right) + \frac{100}{(1+0.005/2)^{2 \times 2}} \\ =& 0.25 \times 3.975124455 + 74.62153966 \\ =& 0.993781114 + 99.00621889 \\ =& 100 \\ \end{aligned} ###
A firm wishes to raise $10 million now. They will issue 6% pa semi-annual coupon bonds that will mature in 8 years and have a face value of $1,000 each. Bond yields are 10% pa, given as an APR compounding every 6 months, and the yield curve is flat.
How many bonds should the firm issue? All numbers are rounded up.
The key thing to realise is that the firm receives the bond price at the start when it issues the bonds. So to find the number of bonds that must be issued, divide the amount to be raised by the bond price. The firm does not receive the face value at the start, actually it pays the face value at maturity.
To calculate the bond 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{1,000 \times 0.06}{2} \times \frac{1}{0.1/2}\left(1 - \frac{1}{(1+0.1/2)^{8 \times 2}} \right) + \frac{1,000}{(1+0.1/2)^{8 \times 2}} \\ =& 325.1330868 + 458.111522\\ =& 783.2446088 \\ \end{aligned} ###
To find the number of bonds to issue right now:
###D_\text{0, new bonds} = P_\text{0,bond} . n_\text{bonds}######\begin{aligned} n_\text{bonds} =& \frac{D_\text{0, new bonds}}{P_\text{0,bond}} \\ =& \frac{$10m}{$783.2446088 } \\ =& 0.012767404m \\ =& 12,767.4 \text{ bonds} \\ \end{aligned} ###
Fractions of a bond can't be issued, so round up to the nearest whole bond which is 12,768 bonds.
Note that issuing bonds is the same thing as selling bonds or lending. At the start the firm sells the bond contract in exchange for the bond price cash payment. At maturity, the firm will pay the bond face value to the lender. The lender can also be called the bond holder, investor or financier.
Bonds X and Y are issued by the same US company. Both bonds yield 6% pa, and they have the same face value ($100), maturity, seniority, and payment frequency.
The only difference is that bond X pays coupons of 8% pa and bond Y pays coupons of 12% pa. Which of the following statements is true?
Bonds X and Y are both premiums bond because their 8 and 12% coupon rates are more than their 6% yields. The yield is what the bond investors deserve, and the coupon rate is what they receive. Since they receive more than what they deserve, they pay a high price for these bonds. They will have a high income (coupon) return, but a negative capital return since the price will fall to the lower face value, and the sum of the income and capital returns will equal the yield.
Question 25 bond pricing, zero coupon bond, term structure of interest rates, forward interest rate
A European company just issued two bonds, a
- 2 year zero coupon bond at a yield of 8% pa, and a
- 3 year zero coupon bond at a yield of 10% pa.
What is the company's forward rate over the third year (from t=2 to t=3)? 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_{0-3})^3 = (1+r_{0-2})^2(1+r_{2-3}) ### ###(1+0.1)^3 = (1+0.08)^2(1+r_{2-3}) ### ###1+r_{2-3} = \frac{(1+0.1)^3}{(1+0.08)^2} ### ###\begin{aligned} r_{2-3} =& \frac{(1+0.1)^3}{(1+0.08)^2} - 1 \\ =& 0.14111797 \\ \end{aligned} ###A 10 year Australian government bond was just issued at par with a yield of 3.9% pa. The fixed coupon payments are semi-annual. The bond has a face value of $1,000.
Six months later, just after the first coupon is paid, the yield of the bond decreases to 3.65% pa. What is the bond's new price?
When pricing bonds or stocks or any asset, the future cash flows and discount rates are the only things that are important. So the 3.65% yield is the discount rate and there are 9.5 years left which is 19 six-month periods. Note that since the bond was issued at par, its initial yield and coupon rate must have been equal. Since it's a fixed coupon bond, the coupon rate will never change so it will still be 3.9% into the future.
Using the bond price equation:
###\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{1000 \times 0.039}{2} \times \frac{1}{0.0365/2}\left(1 - \frac{1}{(1+0.0365/2)^{2 \times 9.5}} \right) + \frac{1000}{(1+0.0365/2)^{2 \times 9.5}} \\ &= 19.5 \times 15.9344866766596 + 709.195618150963 \\ &= 310.722490194862 + 709.195618150963 \\ &= 1,019.91810834582 \\ \end{aligned} ###
Question 213 income and capital returns, bond pricing, premium par and discount bonds
The coupon rate of a fixed annual-coupon bond is constant (always the same).
What can you say about the income return (##r_\text{income}##) of a fixed annual coupon bond? Remember that:
###r_\text{total} = r_\text{income} + r_\text{capital}###
###r_\text{total, 0 to 1} = \frac{c_1}{p_0} + \frac{p_1-p_0}{p_0}###
Assume that there is no change in the bond's total annual yield to maturity from when it is issued to when it matures.
Select the most correct statement.
From its date of issue until maturity, the income return of a fixed annual coupon:
A premium bond's price is more than its face value. But as time goes by, the bond price, measured just after a coupon payment, will fall down to its face value. While the price falls, the dollar coupon is constant since it's equal to the fixed coupon rate multiplied by the face value of the bond. However, the income return is the dollar coupon divided by the bond price. Therefore the income return will rise due to the falling bond prices and constant dollar coupon.