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} ###
For a price of $100, Vera will sell you a 2 year bond paying semi-annual coupons of 10% pa. The face value of the bond is $100. Other bonds with similar risk, maturity and coupon characteristics trade at a yield of 8% pa.
First price the bond:
###\begin{aligned} p_\text{0, bond, theoretical} =& \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 0.1}{2} \times \frac{1}{0.08/2}\left(1 - \frac{1}{(1+0.08/2)^{2\times2}} \right) + \frac{100}{(1+0.08/2)^{2 \times 2}} \\ =& 5 \times 3.629895224 + 74.62153966 \\ =& 18.14947612 + 85.4804191 \\ =& 103.6298952 \\ \end{aligned} ###
The NPV of the deal is the theoretical price of the bond less the actual asking price:
###\begin{aligned} V_0 =& p_\text{0, bond, theoretical} - p_\text{0, bond, actual} \\ =& 103.6298952 - 100 \\ =& 3.6298952 \\ \end{aligned} ###
For a price of $95, Nicole will sell you a 10 year bond paying semi-annual coupons of 8% pa. The face value of the bond is $100. Other bonds with the same risk, maturity and coupon characteristics trade at a yield of 8% pa.
The coupon rate and yield to maturity are both 8% pa, so the bond is a 'par' bond, which means that its (theoretical) price should equal its face value which is $100. Therefore there's no need to price the bond. But you can anyway, and the price will be $100:
###\begin{aligned} P_\text{0, bond, theoretical} =& \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 0.08}{2} \times \frac{1}{0.08/2}\left(1 - \frac{1}{(1+0.08/2)^{10 \times 2}} \right) + \frac{100}{(1+0.08/2)^{10 \times 2}} \\ =& 4 \times 13.5903263449677 + 45.6386946201292 \\ =& 54.3613053798708 + 45.6386946201292 \\ =& 100 \\ \end{aligned} ###
The NPV of the deal is the theoretical price of the bond less the actual asking price:
###\begin{aligned} V_0 =& P_\text{0, bond, theoretical} - P_\text{0, bond, actual} \\ =& 100 - 95 \\ =& 5 \\ \end{aligned} ###
So Nicole is selling the bond at less than its true value. The bond is under-priced, it's a bargain. By spending $95, Nicole will give you a bond worth $100 which is a great deal since you'll gain $5 which is the NPV.
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.
Bonds A and B are issued by the same company. They have the same face value, maturity, seniority and coupon payment frequency. The only difference is that bond A has a 5% coupon rate, while bond B has a 10% coupon rate. The yield curve is flat, which means that yields are expected to stay the same.
Which bond would have the higher current price?
Since the bonds are the same in every respect except coupon rate, and because the yield curve is flat, the yields on the two bonds must be equal. A bond's price is the present value of future cash flows discounted by the yield, so bond B must have a higher price than bond A because it pays higher coupons (cash flows) and they have the same yield (discount rate).
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} ###
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.
A two year Government bond has a face value of $100, a yield of 2.5% pa and a fixed coupon rate of 0.5% pa, paid semi-annually. What is its price?
To price the bond:
###\begin{aligned} P_\text{0} =& \text{PV(annuity of coupons)} + \text{PV(face value)} \\ =& C_1 \times \frac{1}{r_\text{eff}}\left(1 - \frac{1}{(1+r_\text{eff})^{T}} \right) + \frac{F_T}{(1+r_\text{eff})^{T}} \\ =& \frac{100 \times 0.005}{2} \times \frac{1}{0.025/2}\left(1 - \frac{1}{(1+0.025/2)^{2 \times 2}} \right) + \frac{100}{(1+0.025/2)^{2 \times 2}} \\ =& 0.25 \times 3.878057983 + 95.15242752 \\ =& 0.969514496 + 95.15242752 \\ =& 96.12194202 \\ \end{aligned} ###
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.
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} ###
Bonds A and B are issued by the same Australian company. Both bonds yield 7% pa, and they have the same face value ($100), maturity, seniority, and payment frequency.
The only difference is that bond A pays coupons of 10% pa and bond B pays coupons of 5% pa. Which of the following statements is true about the bonds' prices?
Since bond A's coupon rate (10%) is more than its yield (7%), its price must be more than its face value ($100) so it is a premium bond.
Since bond B's coupon rate (5%) is less than its yield (7%), its price must be less than its face value ($100) so it is a discount bond.
Bonds X and Y are issued by different companies, but they both pay a semi-annual coupon of 10% pa and they have the same face value ($100) and maturity (3 years).
The only difference is that bond X and Y's yields are 8 and 12% pa respectively. Which of the following statements is true?
Bond X is a premium bond because the 10% coupon rate is more than the 8% yield. The yield is what the bond investor deserves, and the coupon is what he receives. Since he receives more than what he deserves, he pays a high price for this bond. He 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.
Bond Y is a discount bond because the 10% coupon rate is less than the 12% yield. The yield is what the bond investor deserves, and the coupon is what he receives. Since he receives less than what he deserves, he pays a low price for this bond. He will have a low income (coupon) return, but a positive capital return since the price will increase to the higher face value, and the sum of the income and capital returns will equal the yield.
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.
Bonds X and Y are issued by different companies, but they both pay a semi-annual coupon of 10% pa and they have the same face value ($100), maturity (3 years) and yield (10%) as each other.
Which of the following statements is true?
Since bond A's coupon rate is equal to its yield, it must be a par bond. Similarly for bond B. So both are par bonds. Also note that they would have a price equal to their par value which is $100.
A four year bond has a face value of $100, a yield of 6% and a fixed coupon rate of 12%, paid semi-annually. What is its price?
###\begin{aligned} P_\text{0, bond, theoretical} =& \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)^{4\times2}} \right) + \frac{100}{(1+0.06/2)^{4 \times 2}} \\ =& 6 \times 7.01969219 + 78.94092343 \\ =& 42.11815314 + 78.94092343 \\ =& 121.0590766 \\ \end{aligned} ###
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.
A firm wishes to raise $20 million now. They will issue 8% pa semi-annual coupon bonds that will mature in 5 years and have a face value of $100 each. Bond yields are 6% pa, given as an APR compounding every 6 months, and the yield curve is flat.
How many bonds should the firm issue?
A common misunderstanding in this question is to divide the amount to be raised by the bond face value. This is wrong because the firm doesn't receive the face value at the start, actually it pays the face value at maturity.
To find the number of bonds that must be issued, divide the amount to be raised by the bond price since that's the cash flow that the issuing firm receives at the start.
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{100 \times 0.08}{2} \times \frac{1}{0.06/2}\left(1 - \frac{1}{(1+0.06/2)^{5\times2}} \right) + \frac{100}{(1+0.06/2)^{5 \times 2}} \\ =& 34.12081135 + 74.40939149 \\ =& 108.5302028 \\ \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{$20m}{$108.5302028} \\ =& 0.1842805m \\ =& 184,280.5 \text{ bonds} \\ \end{aligned} ###
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.
A five year bond has a face value of $100, a yield of 12% and a fixed coupon rate of 6%, paid semi-annually.
What is the bond's 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 0.06}{2} \times \frac{1}{0.12/2}\left(1 - \frac{1}{(1+0.12/2)^{5\times2}} \right) + \frac{100}{(1+0.12/2)^{5 \times 2}} \\ &= 3 \times 7.3600870514147 + 55.8394776915118 \\ &= 22.0802611542441 + 55.8394776915118 \\ &= 77.9197388457559 \\ \end{aligned} ###
Which one of the following bonds is trading at par?
If a bond 'trades at par', it means that it's price is the same as its par value, and par value is a synonym of face value or principal. A par bonds' yield also equals its coupon rate.
Only answer (d) is correct since the bond's face value and price are equal so it is a par bond. The other bonds described in (a), (b) and (c) are all premium bonds.
A firm wishes to raise $8 million now. They will issue 7% pa semi-annual coupon bonds that will mature in 10 years and have a face value of $100 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?
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{100 \times 0.07}{2} \times \frac{1}{0.1/2}\left(1 - \frac{1}{(1+0.1/2)^{10 \times 2}} \right) + \frac{100}{(1+0.1/2)^{10 \times 2}} \\ =& 43.6177362 + 37.68894829 \\ =& 81.30668449 \\ \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{$8m}{$81.30668449 } \\ =& 0.09839289415m \\ =& 98,393 \text{ bonds} \\ \end{aligned} ###
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.
Question 207 income and capital returns, bond pricing, coupon rate, no explanation
For a bond that pays fixed semi-annual coupons, how is the annual coupon rate defined, and how is the bond's annual income yield from time 0 to 1 defined mathematically?
Let: ##P_0## be the bond price now,
##F_T## be the bond's face value,
##T## be the bond's maturity in years,
##r_\text{total}## be the bond's total yield,
##r_\text{income}## be the bond's income yield,
##r_\text{capital}## be the bond's capital yield, and
##C_t## be the bond's coupon at time t in years. So ##C_{0.5}## is the coupon in 6 months, ##C_1## is the coupon in 1 year, and so on.
No explanation provided.
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.
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.
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.
A four year bond has a face value of $100, a yield of 9% and a fixed coupon rate of 6%, paid semi-annually. 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 0.06}{2} \times \frac{1}{0.09/2}\left(1 - \frac{1}{(1+0.09/2)^{4\times2}} \right) + \frac{100}{(1+0.09/2)^{4 \times 2}} \\ &= 3 \times 6.59588606735872 + 70.3185126968858 \\ &= 19.7876582020761 + 70.3185126968858 \\ &= 90.1061708989619 \\ \end{aligned} ###
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} ###
Bonds X and Y are issued by the same 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 pays coupons of 6% pa and bond Y pays coupons of 8% pa. Which of the following statements is true?
Since bond X's coupon rate (6%) is less than its yield (10%), its price must be less than its face value ($100) so it is a discount bond.
Since bond Y's coupon rate (8%) is less than its yield (10%), its price must be less than its face value ($100) so it is also a discount bond.
A 30 year Japanese government bond was just issued at par with a yield of 1.7% pa. The fixed coupon payments are semi-annual. The bond has a face value of $100.
Six months later, just after the first coupon is paid, the yield of the bond increases to 2% 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 2% yield is the discount rate and there are 29.5 years left which is 59 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 1.7% 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{100 \times 0.017}{2} \times \frac{1}{0.02/2}\left(1 - \frac{1}{(1+0.02/2)^{2 \times 29.5}} \right) + \frac{100}{(1+0.02/2)^{2 \times 29.5}} \\ &= 0.85 \times 44.4045887902863 + 55.5954112097137 \\ &= 37.7439004717433 + 55.5954112097137 \\ &= 93.3393116814571 \\ \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} ###
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.
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.