• Chapter 6 Bonds
  • 6.1 introduction
  • 6.2 bonds alphabet soup and the basic price formula:
  • 6.3 The premuim-discount formula
  • 6.4 Other price formulas
  • 6.5 Bond Amortization Schedule
    • Premium bonds (writing down):
    • Discount bonds (writing up):
    • Bond with level coupons:
  • 6.6 Value of Bond after issuing
  • 6.7 Sell a Bond after issuing
  • 6.8 Yield rate example
  • 6.9 Callable bond
    • European option
    • Bermuda option
    • American option

Chapter 6 Bonds

6.1 introduction

• bonds: government issued,

• last payment occurs at maturity date / redemption date

• purchase date is called issue date,

• term: intervel

• maturity date is fixed, then the bond is noncallable,

• a single payment at a fixed maturity date, it is called zero-coupon bonds / (pure) discount bonds

• a sequence of payment prior to and at the redemption date coupon bonds, as well as an redemption payment additionally.

• length of the interval between coupon payments is coupon period, usually semiannual , half a year;

• indenture is the legal document that specifies the term and conditions of a particular bond;
  
  

6.2 bonds alphabet soup and the basic price formula:

• Face value / Par value $F$, calculate the size of the coupon payments, not the price investor pay, not the redemption value of the bond at maturity.

• $\alpha$, the nominal rate convertible $m$ times per year, $m$ is the number of coupons each year.

• Set $r=\frac {\alpha}{m}$, r is the effective rate for the coupon period.

• both $\alpha$ and r are sometimes called Coupon Rates.

• Amount of each payment is $Fr=\frac{F \alpha}{m}$.

• let $n$ denoted the numbers of coupon periods in the term of the bond. if it is an N- year bond, $n= Nm$.

• $C$ id the redemption amount , the bond holder receives at the bonds maturity date, excluding the regular coupon amount.

• If $F=C$, the bond is said to be a par value bond / redeemable at particular.
  if no redemption amount is specified, assume the bond is a par-value bond.

• Modified Coupon Rate $g=\frac{Fr}{C}$.

• Annual effective yield rate $i$, investor’s effective yield rate per coupon period $j$, and $i=(1+j)^m-1$, usually, the yield rate is specifies as a nominal rate $I$, convertible $m$ times per year, $j=\frac{I}{m}$, and $i = (1+ \frac{I}{m})^m -1$

• Base Amount $G$ is the amount you need to invest now, at interest $j$, to create a perpetuity paying an amount of interest at the end of each coupon period that is equal to the coupon amount. $G$ expresses the coupon amount in terms of the yield rate per coupon period $j$. $G=\frac{Fr}{j}$ .

• The amount of each coupon payment is $Fr=Cg= Gj$.

• Price paid for the bond to yield the investor $j$ per interest period, $v_j = (1+j)^{-1}$.

• Set $K=Cv_j^n$, the value of the redemption at the issue date figured using compound interest and the yield rate received by th e investor.

• Basic price formula $P = (Fr)a$ _{${\overline{n|}}j$} $+Cv_j^n = (Fr)a$ _{${\overline{n|}}j$} $+K$.

6.3 The premuim-discount formula

$P = (Fr)a$ _{${\overline{n|}}j$} $+Cv_j^n$ $= (Cg)a$ _{${\overline{n|}}j$} $+C(1- ja$ _{${\overline{n|}}j$ $)$.}

• Premium-discount pricing formula $P = C+ C(g-j)a$ _{${\overline{n|}}j$}

• Sell at a premuim if the price $P$ is greater than the redemption value $C$, premium / amount of premium $=P-C$, $g>j$.

• Sell at a discount if the price $P$ is less than the redemption value $C$, discount / amount of discount $=C-P$, $j>g$.

• Premium $= P-C = C(g-j)a$ _{${\overline{n|}}j$}

• Discount $= C-P = C(j-g)a$ _{${\overline{n|}}j$}
  
  

6.4 Other price formulas

• From the base amount formula we can get:

  • a_{${\overline{n|}}j$} $=\frac{1-v^n_j}{j}$

  • $Cv_j^n = K$

• $P = (C-G)v^n_j +G$

• $P= Cg \frac{1-v^n_j}{j} +Cv_j^n = \frac{g}{j} C(1-v_j^n) +Cv_j^n$

• Makeham’s Formula: $P = \frac{g}{j} (C-K)+ K$
  
  

6.5 Bond Amortization Schedule

• The applicable interest rate for each interest period is $j$.

• Set book value $B_t$ the balance of the debt at time t, immediately after any time t coupon payment, before redemption payment.

• $B_0 =P$, and $B_n =C$

• interest due at time t,
  $I_t = jB_{t-1}=Cg - C(g-j) v_j^{n-t+1}$

• amount for adjustment of principal
  $P_t = B_{t-1} - B_t = C(g-j)v_j^{n-t+1}$

• $I_ t+ P_t = Cg$

  • premium: $P_t$ is positive;
  • discount: $P_t$ is negative;
  • price equals the redemption value, $P_t=0$.

• $B_t = (1+j)B_{t-1} - Cg$

• $B_t= Fr a$ _{${\overline{n-t|}}j$} $+Cv_j^{n-t}$

• $B_t= C(g-j)a$ _{${\overline{n-t|}}j$} $+C$

Premium bonds (writing down):

Discount bonds (writing up):

Bond with level coupons:

6.6 Value of Bond after issuing

• Value with respect of interest rate $j$(effective yield rate per coupon period) price paid for the bond;

• dirty value/ theoroetical dirty value: $D_T = B_T, 1≤T≤n$.
  discountinue

• clean value continues function.
  
  

6.7 Sell a Bond after issuing

• price of bond purchased cum divident $= D^{j^{-}}_T$

6.8 Yield rate example

6.9 Callable bond

• With agreement / call provision, allow the issuer to hasten the repayment of the debt.
• at designated call date, lockout period,

European option

single call date prior to maturity;
wait until the date;

Bermuda option

Multiple specified date, coincide with coupon dates;
wait until certain day;

American option

Callable at any time following the lockout period.
any day