Principles of Finance ACST603
Tutorial 4, Week 4 Valuation of fixed income bonds
Homework questions.
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} ###
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.
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.
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}###
"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.
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.
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} ###
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 $100, Carol will sell you a 5 year bond paying semi-annual coupons of 16% pa. The face value of the bond is $100. Other bonds with similar risk, maturity and coupon characteristics trade at a yield of 12% 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 0.16}{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}} \\ =& 8 \times 7.360087051 + 55.83947769 \\ =& 58.88069641 + 55.83947769 \\ =& 114.7201741 \\ \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} \\ =& 114.7201741 - 100 \\ =& 14.7201741 \\ \end{aligned} ###
For a price of $100, Rad will sell you a 5 year bond paying semi-annual coupons of 16% pa. The face value of the bond is $100. Other bonds with the same risk, maturity and coupon characteristics trade at a yield of 6% pa.
To 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 0.16}{2} \times \frac{1}{0.06/2}\left(1 - \frac{1}{(1+0.06/2)^{5 \times 2}} \right) + \frac{100}{(1+0.06/2)^{5 \times 2}} \\ =& 8 \times 8.53020283677583 + 74.4093914896725 \\ =& 68.2416226942066 + 74.4093914896725 \\ =& 142.651014183879 \\ \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} \\ =& 142.651014183879 - 100 \\ =& 42.6510141838791 \\ \end{aligned} ###
For a price of $100, Andrea will sell you a 2 year bond paying annual coupons of 10% pa. The face value of the bond is $100. Other bonds with the same risk, maturity and coupon characteristics trade at a yield of 6% pa.
To 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}{1} \times \frac{1}{0.06/2}\left(1 - \frac{1}{(1+0.06/2)^{2}} \right) + \frac{100}{(1+0.06/2)^{2}} \\ =& 10 \times 1.83339266642934 + 88.999644001424 \\ =& 18.3339266642934 + 88.999644001424 \\ =& 107.333570665717 \\ \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} \\ =& 107.333570665717 - 100 \\ =& 7.33357066571736 \\ \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.
An investor bought a 10 year 2.5% pa fixed coupon government bond priced at par. The face value is $100. Coupons are paid semi-annually and the next one is in 6 months.
Six months later, just after the coupon at that time was paid, yields suddenly and unexpectedly fell to 2% pa. Note that all yields above are given as APR's compounding semi-annually.
What was the bond investors' historical total return over that first 6 month period, given as an effective semi-annual rate?
Since the bond was issued at par, its initial price ##(P_0)## must equal its face value ##(F_{10} = 100)## and its yield must equal its 2.5% pa coupon rate.
###P_0 = F_{10} = 100###After 6 months the bond pays its first semi-annual coupon of $1.125.
###\begin{aligned} \text{SemiAnnualCoupon} &= \dfrac{\text{AnnualCouponRate} \times \text{FaceValue}}{2} \\ &= \dfrac{0.025\times 100}{2} \\ &= 1.125 \\ \end{aligned}###Therefore the income return on the bond over the first 6 months ##(0<=t<0.5)## given as an effective 6 month rate was:
###r_{\text{income, }0 \rightarrow 0.5, \text{ eff 6mth}} = \dfrac{C_{0.5}}{P_0} = \dfrac{1.125}{100} = 0.0125 ###To find the capital return of the bond, we need to find its new price after the decrease in yields from 2.5% to 2%. Since yields fell, the bond price should have risen so the capital return will be positive. Since one coupon was already paid, there will only be 19 left in the future since the bond's remaining maturity is 9.5 years and it pays coupons twice per year.
###\begin{aligned} P_\text{0.5, bond} =& \text{PV(AnnuityOfCoupons)} + \text{PV(FaceValue)} \\ =& \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 0.025}{2} \times \frac{1}{0.02/2}\left(1 - \frac{1}{(1+0.02/2)^{9.5 \times 2}} \right) + \frac{100}{(1+0.02/2)^{9.5 \times 2}} \\ =& 1.125 \times 17.2260085 + 82.7739915 \\ =& 21.53251062 + 82.7739915 \\ =& 104.3065021 \\ \end{aligned} ###Now to find the capital return over the past 6 months:
###\begin{aligned} r_{\text{total, }0 \rightarrow 0.5, \text{ eff 6mth}} &= \dfrac{P_{0.5} - P_0}{P_0} \\ &= \dfrac{104.3065021 - 100}{100} \\ &= 0.043065021 \\ \end{aligned}###Therefore the total return is:
###\begin{aligned} r_{\text{capital, }0 \rightarrow 0.5, \text{ eff 6mth}} &= r_{\text{capital, }0 \rightarrow 0.5, \text{ eff 6mth}} + r_{\text{income, }0 \rightarrow 0.5, \text{ eff 6mth}} \\ &= \dfrac{P_{0.5} - P_0}{P_0} + \dfrac{C_{0.5}}{P_0} \\ &= \dfrac{104.3065021 - 100}{100} + \dfrac{1.25}{100} \\ &= 0.043065021 + 0.0125 \\ &= 0.055565021 \\ \end{aligned}###An investor bought a 20 year 5% pa fixed coupon government bond priced at par. The face value is $100. Coupons are paid semi-annually and the next one is in 6 months.
Six months later, just after the coupon at that time was paid, yields suddenly and unexpectedly rose to 5.5% pa. Note that all yields above are given as APR's compounding semi-annually.
What was the bond investors' historical total return over that first 6 month period, given as an effective semi-annual rate?
Since the bond was issued at par, its initial price ##(P_0)## must equal its face value ##(F_{20} = 100)## and its yield must equal its 5% pa coupon rate.
###P_0 = F_{20} = 100###After 6 months the bond pays its first semi-annual coupon of $2.50.
###\begin{aligned} \text{SemiAnnualCoupon} &= \dfrac{\text{AnnualCouponRate} \times \text{FaceValue}}{2} \\ &= \dfrac{0.05 \times 100}{2} \\ &= 2.5 \\ \end{aligned}###Therefore the income return on the bond over the first 6 months ##(0<=t<0.5)## given as an effective 6 month rate was:
###r_{\text{income, }0 \rightarrow 0.5, \text{ eff 6mth}} = \dfrac{C_{0.5}}{P_0} = \dfrac{2.5}{100} = 0.025 ###To find the capital return of the bond, we need to find its new price after the increase in yields from 5% to 5.5%. Since yields rose, the bond price should have fallen so the capital return will be negative. Since one coupon was already paid, there will only be 39 left in the future since the bond's remaining maturity is 19.5 years and it pays coupons twice per year.
###\begin{aligned} P_\text{0.5, bond} =& \text{PV(AnnuityOfCoupons)} + \text{PV(FaceValue)} \\ =& \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 0.05}{2} \times \frac{1}{0.055/2}\left(1 - \frac{1}{(1+0.055/2)^{19.5 \times 2}} \right) + \frac{100}{(1+0.055/2)^{19.5 \times 2}} \\ =& 2.5 \times 23.74024884 + 34.71431569 \\ =& 59.3506221 + 34.71431569 \\ =& 94.06493779 \\ \end{aligned} ###Now to find the capital return over the past 6 months:
###\begin{aligned} r_{\text{total, }0 \rightarrow 0.5, \text{ eff 6mth}} &= \dfrac{P_{0.5} - P_0}{P_0} \\ &= \dfrac{94.06493779 - 100}{100} \\ &= -0.059350622 \\ \end{aligned}###Therefore the total return is:
###\begin{aligned} r_{\text{capital, }0 \rightarrow 0.5, \text{ eff 6mth}} &= r_{\text{capital, }0 \rightarrow 0.5, \text{ eff 6mth}} + r_{\text{income, }0 \rightarrow 0.5, \text{ eff 6mth}} \\ &= \dfrac{P_{0.5} - P_0}{P_0} + \dfrac{C_{0.5}}{P_0} \\ &= \dfrac{94.06493779 - 100}{100} + \dfrac{2.5}{100} \\ &= -0.059350622 + 0.025 \\ &= -0.034350622 \\ \end{aligned}###