Pricing Bonds

Section 4.3 Pricing Bonds

Theorem 4.3.1. Setting price of bond with periodic coupons..

Given a bond with stated rate of

$r_{bond}\text{,}$ nominal rate of

$r_{nom}$ maturity (face) value of

$MV\text{,}$ and which matures in t years with coupons provided m times a year, the purchase price is given by

$\begin{equation*} MV (1+\frac{r_{nom}}{m})^{-n} + MV \cdot \frac{r_{bond}}{m} \cdot \frac{1-(1+\frac{r_{nom}}{m})^{-n}}{\frac{r_{nom}}{m}} \end{equation*}$

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Proof.

The yearly coupon value equals $MV \cdot r_{bond}$ broken up into m payments of size $R = MV \frac{r_{bond}}{m}\text{.}$ These coupons can be thought of as a regular annuity with $n = m \cdot t$ payments and finding the present value of these yields the second part of this formula.

The maturity value also is factored into the price by using the compound interest formulas with the nominal rate and moving back n periods to the time when the bond is sold. This is the first part of this formula.

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Corollary 4.3.2.

For Bonds where bond rate = nominal rate, purchase price equals maturity value

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Proof.

In this case, $r_{nom} = r_{bond} = r$ gives

\begin{align*} P & = MV (1+\frac{r}{m})^{-n} + MV \cdot \frac{r}{m} \cdot \frac{1-(1+\frac{r}{m})^{-n}}{\frac{r}{m}}\\ & = MV \left ( (1+\frac{r}{m})^{-n} + \frac{r}{m} \cdot \frac{1-(1+\frac{r}{m})^{-n}}{\frac{r}{m}}[ \right ] \\ & = MV \left ( (1+\frac{r}{m})^{-n} + 1-(1+\frac{r}{m})^{-n}[ \right ] \\ & = MV \cdot 1 = F \end{align*}

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Example 4.3.3.

A 14% bond with semiannual coupons, face value of $MV = \$ 40,000,000$ for 25 years and with nominal annual rate of 10%.

$\begin{align*} n & = 25 \cdot 2 = 50\\ s & = \frac{1-(1+\frac{0.10}{2})^{-50}}{\frac{0.10}{2}}\\ \text{Price On Issue Date} & = \$ 40000000(1+\frac{0.10}{2})^{-50} + \$40000000 \cdot \frac{0.14}{2} \cdot s\\ & = 3488149.08 + 2800000 \cdot 18.2559254605524\\ & = 3488149.08 + 51116591.29 = \end{align*}$

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@interact(layout=dict(top=[['MV'],['m','t'],['r1','r2']]))
def _(r1 = input_box(0.10,label="nominal rate",width=6),
      r2 = input_box(0.14,label="stated bond rate",width=6),
      m = input_box(2,label="coupons per year",width=4),
      t = input_box(25,label="years",width=4),
      MV = input_box(40000,label="Bond Face Value",width=10)):
    n = t*m                       # number of periods = years?
​
    term = (1-(1+r1/m)^(-n))/(r1/m)
    print "Coupon Multiplier = ",term.n()
    valueCoupons = MV*r2/2 * term   # Annual coupons are the biannual payments 
    valueBondEnd = MV*(1+r1/m)^(-n) # Present value of F in 25 years
    print "Coupons Value = ",valueCoupons
    print "Redemption PV = ",valueBondEnd
    print "Purchase Price of Bond = ",valueCoupons+valueBondEnd

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An issuing company may choose to repurchase a bond before the bond matures and may agree to purchase it back at par value. However, the bond is said to be purchased at a discount if its price is less than its redemption value. On the other hand, it is purchased at a premium if its purchase price is greater than its redemption value.

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To determine the fair value of the bond immediately following a coupon payment, adjust the formula above for determining purchase price at the onset of the bond so that it focuses value at one of the time periods in the bonds life span.

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Corollary 4.3.4. Book Value of a Bond.

For a bond convertible m times a year and with a yield rate convertible m times a year, if the Face Value is FV and Book Value is BV and Redemption Value is RV, when redeeming immediately after the kth coupon payment yields

$\begin{equation*} BV = FV \cdot \frac{r_{nom}}{m} \cdot \frac{(1 - (1+\frac{r_{bond}}{m})^{-t \cdot m + k})}{\frac{r_{bond}}{m}} + RV \cdot (1+\frac{r_{bond}}{m})^{-t \cdot m + k} \end{equation*}$

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Example 4.3.5.

Suppose a 10 year $\$ 2000$ bond earns interest at 10.2% convertible semiannually and that the yield rate is 7.1% convertible semiannually. If the redemption value immediately after the 13th coupon payment is $\$ 2300.00\text{,}$ the book value BV is given by

$\begin{gather*} BV = \$ 2212.70 = \$ 2000 \cdot \frac{0.102}{2} \cdot \frac{1 - (1+ \frac{0.071}{2})^{-20 + 13}}{\frac{0.071}{2}} + RV \cdot (1+\frac{0.071}{2})^{-20 + 13} \end{gather*}$

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@interact(layout=dict(top=[['MV','k'],['m','t'],['r1','r2'],['RV']]))
def _(r1 = input_box(0.071,label="nominal rate",width=6),
      r2 = input_box(0.102,label="stated bond rate",width=6),
      m = input_box(2,label="coupons per year",width=4),
      t = input_box(10,label="years",width=4),
      k = input_box(13,label="Coupon Number",width=4),
      MV = input_box(2000,label="Bond Face Value",width=10),
      RV = input_box(2030,label="Redemption Value at coupon number",width=10)):
    n = t*m                       
    term = (1-(1+r1/m)^(-n+k))/(r1/m)
    valueCouponsRemain = MV*r2/m * term   # Annual coupons are the m payments 
    valueMV = RV*(1+r1/m)^(-n+k) # Present value of MV
    BV = round(100*(valueCouponsRemain+valueMV))/100
    print "Book Value of Bond after the ",k,"th coupon = ",BV.n()

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Corollary 4.3.6. Redemption Value of a Bond.

$\begin{equation*} RV = \left [ BV - FV \cdot \frac{r_{bond}}{m} \cdot \frac{(1 - (1+\frac{r_{nom}}{m})^{-t \cdot m + k})}{\frac{r_{nom}}{m}} \right ] / (1+\frac{r_{nom}}{m})^{-t \cdot m + k} \end{equation*}$

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@interact(layout=dict(top=[['MV','k'],['m','t'],['r1','r2'],['BV']]))
def _(r1 = input_box(0.071,label="nominal rate",width=6),
      r2 = input_box(0.102,label="stated bond rate",width=6),
      m = input_box(2,label="coupons per year",width=4),
      t = input_box(10,label="years",width=4),
      k = input_box(13,label="Coupon Number",width=4),
      MV = input_box(2000,label="Bond Face Value",width=10),
      BV = input_box(2212.70,label="Book Value at coupon number",width=10)):
    n = t*m                  
    
    term = (1-(1+r1/m)^(-n+k))/(r1/m)
    valueCouponsRemain = MV*r2/m * term   # Annual coupons are the m payments 
    RV = (BV - valueCouponsRemain)/(1+r1/m)^(-n+k) # Present value of MV
    RV = round(100*RV)/100
    print "Redemption Value of Bond after the ",k,"th coupon = ",RV.n()