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.
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}###
Question 339 bond pricing, inflation, market efficiency, income and capital returns
Economic statistics released this morning were a surprise: they show a strong chance of consumer price inflation (CPI) reaching 5% pa over the next 2 years.
This is much higher than the previous forecast of 3% pa.
A vanilla fixed-coupon 2-year risk-free government bond was issued at par this morning, just before the economic news was released.
What is the expected change in bond price after the economic news this morning, and in the next 2 years? Assume that:
- Inflation remains at 5% over the next 2 years.
- Investors demand a constant real bond yield.
- The bond price falls by the (after-tax) value of the coupon the night before the ex-coupon date, as in real life.
Higher inflation reflects higher prices of goods and services, but since bonds are not consumption assets, bond prices will not increase with inflation.
However, higher inflation will reduce the real bond yield. Because we assume that investors demand a constant real bond yield, then the nominal bond yield must increase by approximately the same amount as inflation, which is 2% pa since inflation grew from 3 to 5% pa.
The Fisher equation shows the relationship between nominal and real returns. There is an exact and an approximate formula:
###1+r_\text{real} = \dfrac{1+r_\text{nominal}}{1+r_\text{inflation}}### ###r_\text{real} \approx r_\text{nominal} - r_\text{inflation}###For the nominal bond yield to increase, the bond price must fall. This would have happened today as soon as the news of higher inflation was released. The bond price is likely to have fallen by about 4% because if the coupon rate is low then the bond price is mostly affected by the change in the present value of the face value which is received in 2 years. ###\text{change in present value of face} = 1-\dfrac{1}{(1+0.02)^2} = 0.03883 \approx 4\%###
The bond was originally issued at par which means that the price originally equaled the par (also called face) value. Then the price dropped due to the higher inflation news. But over the next 2 years until the bond matures, the bond price will slowly appreciate back up to its face value. Note that the bond price will fall by the (after-tax) value of the coupon the night before each ex-coupon date, but on each ex-coupon date, the price will be a little higher than the last time, until it reaches the face value. On the day before the final ex-coupon date, the bond price will equal the face value plus the value of the coupon, and then it will fall to zero. This can be best seen in the familiar saw-tooth graph of bond prices.
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 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.