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COUPON BONDS AND DERIVING YIELD CURVES. This question gives some

information for the valuation of bonds and wants from you to do some mathematical

operations in $R.$ Do not fear the longness of the question, the answer is easy to find.

The spot rate is the yield at present prevailing for zero coupon bonds of a given

maturity. The t$-$year yield yot of a zero coupon bond is determined as follows:

$y_{0t}={?}^t\frac{F}{n_0}-1$

where $F$ is the face value or principal of the bond, $P_0$ is the purchase price of the

bond, and $t$ is the zero coupon bond's term to maturity. The t$-$year spot rate is

denoted here by yot, which represents an implied spot rate on a loan to be made at

time zero and repaid in its entirety at time $t.$ Spot rates may be estimated from

bonds with known future cash flows and their current prices. We are able to obtain

spot rates from yields implied from series of bonds when we assume that the Law

of One Price holds. Recall that the Law of One Price maintains that securities

generating identical cash flows must sell for the same price.

The yield curve represents yields or spot rates of bonds with varying terms to

maturity. For example, at a given moment in time, the yield or spot rate for one$-$

year bonds may be $4\%(y=0\mathrm{.}04),$ while the yield for five$-$year bonds may be

$0\mathrm{.}06).$ This section is concerned with how interest rates or yields vary with

maturities of bonds. The simplest bonds to work with from an arithmetic

perspective are pure discount notes, also known as zero coupon notes, which make

no interest payments. Such notes make only one payment at one point in time $-$

on the maturity date of the note. Determining the relationship between yield and

term to maturity for these bonds is quite trivial. The return that one obtains from

a pure discount note is strictly a function of capital gains; that is$,$ the difference

between the face value $F$ of the note and its purchase price $P_0.$ Short$-$term U$.$S$.$

Treasury bills are an example of pure discount $($or zero coupon$)$ notes. Coupon

bonds are somewhat more difficult to work with from an arithmetic perspective

because they make payments to bondholders at a variety of different periods. Since

they make multiple payments, coupon bonds are analogous to portfolios of pure

discount bonds.

A coupon bond may be treated as a portfolio of pure discount notes, with each

coupon being treated as a separate note maturing on the date the coupon is paid.

Using coupon bonds slightly complicates the process for determining yields, but

Table $1$: Coupon bonds A$,$ B$,$ and C

is necessary when there aren't pure discount notes maturing in key time periods.

Consider an example involving three bonds whose terms and prices are given in

table $1.$ The three bonds are trading at known prices with a total of eight annual

coupon payments among them $($two for bond A and three each for bonds $B$ and $C).$

Bond yields or spot rates must be determined simultaneously to avoid associating

contradictory rates for the annual coupons on each of the three bills. Let $Dt$ be

the discount function for time $t$; that is$,Dt=\frac{1}{(1+y0t{)}^t}.$ Since yot spot rate or

discount rate that equates the present value of a bond with its current price, the

following equations may be solved for discount functions then spot rates:

$947\mathrm{.}376=50D_1+1050D_2$

$904\mathrm{.}438=60D_1+60D_2+1,060D_3$

$980\mathrm{.}999=90D_1+90D_2+1,090D_3$

This system of equations may be represented by the following system of matrices:

To solve this system we first invert matrix $CF$ to obtain $C{F}^{-1}.$ We then use this

inverse matrix to premultiply vector $P_0$ to obtain vector $d$ : Accordingly, the matrix

multiplication is as the following

$C{F}^{-1},P_0=d$

$($a$)$ Find the value of $d=(D_1,D_2,D_3)$ by using the inverse matrix operation

$($solve$($X$))$ and matrix multiplication operations in R$.$

$($b$)$ After finding the $D$ values from solving this system for vector $d,$ the spot rates

yot can be obtained via

$Dt=\frac{1}{(1+y0t{)}^t}$

$\frac{1}{D}t=(1+y0t{)}^t$

$y0t=\frac{1}{D_t^{\frac{1}{t}}}-1$

Then, find the three spot rates $($yot$1,$ yot$2,$ yotz$)$ numerically by solving the last

equation in the equation series above