Section A $$ Bond Pricing and Yield Curve

All rates should be calculated to $3$ decimal places in $\%($e$.$g$.1\mathrm{.}234\%),$ the discount factors to $5$ decimal places $($e$.$g$.0\mathrm{.}98765)$ and the bond prices to $3$ decimal places $($e$.$g$.100\mathrm{.}234).$

The following are the prices for US Treasury on$-$the$-$run bonds for the closing of $1$st March $2024($the bond prices are the clean prices and they have been converted to decimals from the published $1/32$ format$)$:

Term Maturity Issue Date Coupon Price

$2$y $28/02/202628/02/20244\mathrm{.}625\%100\mathrm{.}191$

$3$y $15/02/202715/02/20244\mathrm{.}125\%99\mathrm{.}469$

$5$y $28/02/202928/02/20244\mathrm{.}250\%100\mathrm{.}441$

$7$y $28/02/203128/02/20244\mathrm{.}250\%100\mathrm{.}381$

$10$y $15/02/203415/02/20244\mathrm{.}000\%98\mathrm{.}575$

You are also given the following information:

$-$ The face value of the bonds is $$100.$

$-$ All bonds are semi$-$annual coupon bonds.

$-$ Ignore weekends for the coupon payment dates $($e$.$g$.,$ for the $2-$year bond, they are $28$ February and $28$ August each year$).$

$-$ Accrued interest $=$ coupon rate $\backslash$times face value $\backslash$times N$1/365,$ where N$1$ is the number of days between the issue date and the quote date $(1$st March $2024),$ exclusive of the day of issue date. $($E$.$g$.$ for the $2-$year bond, N$1=2$ days$).$ For year $3$ N$1=15,$ for year $5$ N$1=2,$ for year $7$ N$1=2$ and for year $10$ N$1=15$

$-$ Both the zero$-$coupon rates and the yield$-$to$-$maturity $($YTM$)$ should be computed as semi$-$annually compounding rates.

$-$ For coupon bonds, the YTM is the rate Y that solves,

$=$

$100$

y

$2$T$-1$ C

N$2$ I y $+$

$1$

y $2$T$-1,$

$(1+2)$N$3$

i$=$O

$(1+2)$

$(1+2)$

where DP is the dirty price, C is the coupon rate, T is the maturity in full numbers $($e$.$g$.$ T $=10$ for the $10-$year bond$),$ N$2$ is number of days between the quote date and the first coupon date, and N$3$ is number of days in the first coupon period $($e$.$g$.$ for the $2-$year bond, N$2=180$ days and N$3=182$ days$)\mathrm{.}1$

$.$ This is the convention for US Treasury bonds, for which the fraction of a year is calculated by Actual$/$Actual $($i$.$e$.,$ the actual number of days divided by the actual total number of days in the period$).$

$-$ The zero$-$coupon rates z$($t$)$ given above for t $\{0\mathrm{.}5,1\}$ are the annualised rates, i$.$e$.$

where D$($t$)$ is the t$-$year discount factor.

$($a$)$ Compute the yield$-$to$-$maturity of all the on$-$the$-$run bonds. You may use the Excel spreadsheet function $$Data $->$ What$-$if$-$Analysis $->$ Goal Seek$$ to find the YTMs$.$

$($b$).$Assuming that your answer to $($a$)$ are the semi$-$annually compounding par yields for the respective maturity T $\{2,3,5,7,10\},$ calculate the $6-$monthly discount factors D$($t$)$ and the semi$-$annual zero$-$coupon rates z$($t$),$ where t $\{0\mathrm{.}5,1,1\mathrm{.}5,...,9\mathrm{.}5,10\}.$ Any required par yields for other maturities should be computed using a linear interpolation method.

C$).$Hence, calculate the price of a $5-$year semi$-$annual coupon bond with an annual coupon rate of $1\mathrm{.}875\%$ and face value of $100.$ Assume that the bond$$s maturity is $1$st March $2029$ and that it has just made its most recent coupon payment. What is its YTM$?$ On the same date, a bond with a maturity of $28$th February $2029$ and a coupon rate of $1\mathrm{.}875\%$ was quoted at $4\mathrm{.}179\%.$ Why do you think the YTMs differ on the two $5-$year bonds?