Bootstrapping Earnings
Bootstrapping earnings (or bootstrap effect) occurs when a company’s earnings increase because of... Read More
A coupon-paying bond’s pricing and valuation are the same as those of a dividend-paying stock. The difference is that the cash flows are coupons and not dividends. Fixed income forward and futures have several problems that are related to the carry arbitrage model.
$$ \text{Accrued interest} = \text{Accrual period} × \text{Periodic coupon amount} $$
The formula for a bond where the quoted price includes the accrued interest is:
$$ F_0=FV(S_0+CC_0-CB_0) $$
Where the quoted price of a bond does not include accrued interest, the spot bond price will be:
$$ S_0=\text{Quoted bond price}+\text{Accrued interest}= B_0+AI_0 $$
The future price is:
$$ F_0=FV(B_0+AI_0-PVCI) $$
Where:
The quoted futures price is:
$$ Q_0=[\frac{1}{CF}]{FV[B_0+AI_0 ]-AI_T-FVCI} $$
Where:
The following information relates to a six-month Euro bond futures contract with a value of €50,000. The underlying is a 5% bond quoted at €110 with an accrued interest of €1.00. Suppose there are no coupon payments due until after the futures contract expires.
$$ \begin{array}{lc|lc} \textbf{Futures Contract} & & \textbf{Underlying Bond} & \\ \hline \text{Euro Bond Contract Value} & €50,000 & \text{Quoted Bond price} & €110 \\ \hline \text{Conversion factor} & 0.65 & \text{Accrued interest since } & €1.00 \\ {} & & \text{last coupon payment} & \\ \hline \text{Time to contract expiration} & 0.5 & \text{Accrued interest at} & €3.00 \\ {} & & \text{futures contract} & \\ {} & & \text{expiration} & \\ \hline \text{Accrued interest over the} & 0 & & \\ \text{life of futures contract} & & & \\ \hline \text{Risk-free rate} & 4.00\% & & \\ \end{array} $$
The equilibrium bond futures price based on the carry arbitrage model is closest to:
$$ QF_0\left(T\right)= \left[\frac{1}{CF\left(T\right)}\right]\times FV_{0,T}\left[B_0(T + Y)+ AI_0\right]– AI_T – FVCI_{0,T} $$
Where:
Therefore,
$$ QF_0\left(T\right)=\left[\frac{1}{0.65}\right]\times 110+1 \times 1.040.5– 3 – 0= €169.54 $$
The value of a bond future is the change in price since the previous day’s settlement. This is because bond futures are marked to market. The futures value is captured at the end of the day during the bond settlement, at which time the contract value is zero. The value of a bond is the present value of the difference in forward prices. Carry benefits reduce forward prices and carry costs increase forward prices.
Question
Consider a $100 par, 4% semiannual coupon bond with a spot price of $100 that matures in 200 days. The bond has just made a coupon payment and the next coupon payment will be made after 60 days. What will be the value of the bond after 120 days?
Given that the risk-free rate of interest is 8%, the value of the forward contract on the bond to the long position is closest to:
- $112.72.
- $14.18.
- $127.
Solution
The correct answer is B.
$$ \text{Coupon payment} = 4\%\times 0.5\times 100 = $2 $$
Present value of coupon payment \(= \frac{2}{\left(1.08\right)^\frac{160}{365}}=$1.93\)
$$ \begin{align*} F_0(T) & =\left(110-1.93\right)\times\left(1.08\right)^\frac{200}{365}=$112.72 \\ V_t\left(\text{Long} \right)&=\left(S_t-PVC_t\right)-\frac{F_0\left(T\right)}{\left(1+r_f\right)^{T-t}} \end{align*} $$
After 120 days, only one coupon payment is due in 40 days (160-120) before the contract maturity in 80 days (200-120).
$$ \begin{align*} PVC_t & =\frac{2}{\left(1.08\right)^\frac{40}{365}} =$1.9832 \\ V_{120} \left(\text{Long} \right) &=$127-1.9832-\frac{$112.72}{\left(1.08\right)^\frac{80}{365}}=$14.18 \end{align*} $$
Reading 33: Pricing and Valuation of Contingent Claims
LOS 33 (d) Describe how fixed-income forwards and futures are priced, and calculate and interpret their no-arbitrage value.