Financial Math FM/Bonds

Learning objectives

The Candidate will understand key concepts concerning bonds, and how to perform related calculations.

Learning outcomes

The Candidate will be able to:

• Define and recognize the definitions of the following terms: price, book value, amortization of premium, accumulation of discount, redemption value, par value/face value, yield rate, coupon, coupon rate, term of bond, callable/non-callable.
• Given sufficient partial information about the items listed below, calculate any of the remaining items

• Price, book value, amortization of premium, accumulation of discount. (Note that valuation of bonds between coupon payment dates will not be covered).
• Redemption value, face value.
• Yield rate.
• Coupon, coupon rate.
• Term of bond, point in time that a bond has a given book value, amortization of premium, or accumulation of discount.

Bond

A bond.

A bond is a debt security, in which the issuer, usually a corporation or public institution, owes the holders a debt and is obliged to pay interest (the coupon) and to repay the principal at a later date. A bond is a formal contract to repay borrowed money with interest at fixed intervals. There are two main kinds of bonds: accumulation bonds (zero coupon bonds) and bonds with coupons. An accumulation bond is where the issuer of the bond agrees to pay the face value at a later redemption date, but they are sold at a discount.
Example: a 20 year $1000 face value bond with a 3.5% nominal annual yield would have a price of $502.56.
Bonds with coupons are more common and it's where the issuer of the bond makes period payments (coupons) and a final payment.
Example: a 10 year $1000 par value bond with a 8% coupon convertible semiannually would pay $40 coupons every 6 months and then $1000 at the end of the 10 years.

Terminology and variable naming convention

• $P$ is the price of a bond. The price of a bond P is the amount that the lender, the person buying the bond, pays to the government or corporation issuing the bond.
• $A$ is the price per unit nominal, i.e. $A:=\frac{P}{F}$.
• $F$, is the face amount, face value, par value, or nominal value of a bond which is the amount by which the coupons are calculated, and is printed on the front of the bond.
• $C$ is the redemption value of a bond which is the amount of money paid to the bond holder at the redemption date.
• $R$ is the redemption value per unit nominal, i.e. $R:=\frac{C}{F}$.

• If $R=1$, the bond is redeemable at par
• If $R>1$, the bond is redeemable above par
• If $R<1$, the bond is redeemable below par

• $r$ is coupon rate (or nominal yield) which is the rate per coupon payment period used in determining the amount of coupon.
• $Fr$ is the amount of the coupon
• $g$ is modified coupon rate which is defined by $g:=\frac{Fr}{C}$, i.e. the coupon rate per unit of redemption value instead of per unit of par value (which is the case for $r$).
• $i$ is the yield rate or yield to maturity of a bond, which is the interest rate earned by the investor (i.e. the effective interest rate), assuming the bond is held until it is redeemed.
• $n$ is the number of coupon payment periods from the date of calculation to redemption date.
• $K$ is the present value, that is calculated at the yield rate, of the redemption value of a bond at redemption date, i.e. $K=C{v}^n$ in which $v:=\frac{1}{1+i}$ ($i$ is the yield rate of the bond.
• $G$ is the base amount of a bond which is defined by $Gi:=Fr$, i.e. it is the amount of which an investment at yield rate $i$ produces periodic interest payment that equals the amount of every coupon of the bond.

From now on, unless otherwise specified, the redemption value ($C$) of a bond equals the face amount (par value) ($F$) of the bond. This is also true in the SOA FM Exam. Therefore, we have $g:=\frac{Fr}{C}=\frac{Fr}{F}=r$, i.e. the modified coupon rate is not 'modified' unless otherwise specified.

Four formulas to calculate the price of a bond

Situation in which there are no taxes

In this subsection, we discuss the calculation of price of a bond when there are no taxes. We will discuss the calculation of price of a bond when there are some taxes, namely income tax and capital gains tax.

We will obtain the same price no matter we use which of the following four formulas, because we can use basic formula to derive all other three formulas. The choice of formula is mainly based on what information is given, and we choose the formula for which we can use it most conveniently.

Illustration:

 P   Fr  Fr  Fr  Fr  Fr  Fr C
 ↓   ↑   ↑   ↑   ↑   ↑   ↑↗ 
-|---|---|---|---|---|---|----
 0   1   2   3   4   5   6

Proof. It follows from the Template:Colored em that the price is set to equal the present value of future coupons plus present value of the redemption value (i.e. all future payments), so that the price is a 'fair price'. (We treat the time at which the bond price is calculated as Template:Colored em.)

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Template:Colored remark

Proof. By basic formula, $$\begin{array}{cc}P& =Fr{a}_{\bar{n}|i}+C{v}^n\\ & =Fr{a}_{\bar{n}|i}+C(1-i{a}_{\bar{n}|i})\text{by }{a}_{\bar{n}|}=\frac{1-v^n}{i}\iff v^n=1-i{a}_{\bar{n}|}\\ & =(\underset{=Cg}{\underbrace{Fr}}-Ci)a_{\bar{n}|i}+C\text{by }g:=\frac{Fr}{C}\iff Fr=Cg\\ & =C+C(g-i)a_{\bar{n}|i}\\ \end{array}$$

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Proof. $$\begin{array}{cc}P& =\underset{Gi}{\underbrace{Fr}}{a}_{\bar{n}|i}+C{v}^n\\ & =G(1-v^n)+C{v}^n\text{by }{a}_{\bar{n}|i}=\frac{1-v^n}{i}\iff i{a}_{\bar{n}|i}=1-v^n\\ & =G+(C-G)v^n\end{array}$$

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Proof. $$\begin{array}{cc}P& =\underset{Cg}{\underbrace{Fr}}{a}_{\bar{n}|}+\underset{K}{\underbrace{C{v}^n}}\\ & =Cg\left(\frac{1-v^n}{i}\right)+K\\ & =\frac{g}{i}(C-\underset{K}{\underbrace{C{v}^n}})+K\\ & =K+\frac{g}{i}(C-K)\end{array}$$

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In practice, stocks are generally quoted as 'percent'. For example, we buy a quantity of a stock at 80% redeemable at 100% (at par), or at 105% (above par). Sometimes, the nominal value of bond is not specified. In this case, we should express our answer in percent (without the % sign), or equivalently, price per 100 nominal, e.g. price percent is 110 is equivalent to price is 110 per 100 nominal.

Situation in which there are income tax and capital gains tax

When there is income tax, the four formulas to calculate price of a bond ($P$) are slightly modified in a similar way. Suppose the income tax rate is $t_1$. By definition, $Fr$ is counted as income, and $C$ is Template:Colored em counted as income (the gain due to the difference between $P$ and $C$ is taxed by capital gains tax instead). So, under income tax, the bond price is computed with $Fr$ multiplied by $1-t_1$ ($t_{1}Fr$ is the income tax paid per coupon payment). Also, we consider the Template:Colored em of the tax payment to compute the Template:Colored em bond price. Hence, we have the following modified formulas under income tax:

The basic formula becomes $$P=Fr(1-t_1)a_{\bar{n}|i}+\underset{K}{\underbrace{C{v}^n}}.$$ The premium/discount formula becomes $$P=\underset{Fr}{\underbrace{Cg}}(1-t_1)a_{\bar{n}|i}+C{v}^n=C+C(g(1-t_1)-i)a_{\bar{n}|i}.$$

• $P-C$ is premium if $P>C\iff g(1-t_1)>i$
• $P-C$ is discount if $P<C\iff g(1-t_1)<i$ (and Template:Colored em).

• this gives us a convenient way to check that whether there is capital gain, and we should be careful that the time period for which $g$ and $i$ is computed must be the same, so that the comparison is fair, and valid (the time period is not necessarily one year)

The base amount formula becomes $$P=\underset{Fr}{\underbrace{Gi}}(1-t_1)+C{v}^n=G(1-t_1)+(C-G(1-t_1))v^n.$$ The Makeham's formula becomes $$P=\underset{Fr}{\underbrace{Cg}}(1-t_1)a_{\bar{n}|i}+C{v}^n=K+\frac{g(1-t_1)}{i}(C-K)$$

On the other hand, capital gains tax is a tax levied on difference between the redemption value of a stock (or other asset) and purchase price if and only if it is strictly lower than redemption value. When there is capital gains tax, say the rate is $t_2$, we need to Template:Colored em the purchase price by Template:Colored em of $t_2(C-P)$ at the redemption date (the Template:Colored em of the capital gain tax paid for the bond). Template:Colored example Template:Colored example Template:Colored exercise

Serial bond

Template:Colored definition Then, for serial bond, we have the following equation. $$F=F_1+F_2+\cdots +F_k$$ in which $F_i$ is the nominal amount that will be redeemed after $n_i$ years, and other notations with subscript $i$ corresponds to this nominal amount. Also, by definition, $$C_j=R{F}_j,C=RF=R\sum _j{F}_j=\sum _j{C}_j,$$ and $$K_j=C_j{v}^{n_j},K=C{v}^{n_j}=\sum _j{C}_j{v}^{n_j}.$$ In this situation, Makeham's formula is quite useful, and ease calculation. Its usefulness is illustrated in the following.

When there are no taxes, $$P_j=K_j+\frac{g}{i}(C_j-K_j)\iff \sum _j{P}_j=\sum _j{K}_j+\frac{g}{i}(\sum _j(C_j)-\sum _j(K_j))=K+\frac{g}{i}(C-K)=P.$$ Template:Colored example Let $P^{\prime }$ be the price of serial bond when there is income tax. When there is income tax, say the rate is $t_1$, $$P{\text{'}}_j=K_j+\frac{g(1-t_1)}{i}(C_j-K_j)\iff \sum _{j}P{\text{'}}_j=\sum _j{K}_j+\frac{g(1-t_1)}{i}(\sum _j(C_j)-\sum _j(K_j))=K+\frac{g(1-t_1)}{i}(C-K)=P^{\prime }.$$ Template:Colored example Let $P^{″}$ be the price of serial bond when there are both income and capital gains tax. If the bond is sold at discount (and there is income tax), i.e. $g(1-t_1)<i$, there is capital gain. Template:Colored em that the capital gain at time $n_j$ is $$C_j-\left(\frac{F_j}{F}\right)P^{″},$$ ($F_j/F$ is the portion of bond, in terms of nominal value, redeemed, corresponding to the redemption value $C_j$) and thus the total present value of the capital gains tax (say at rate $t_2$) payable is $$\sum _{j=1}^k\left(t_2(\underset{C_j}{\underbrace{R{F}_j}}-\frac{F_j}{F}{P}^{″})v^{n_j}\right)=t_2(1-\frac{P^{″}}{FR})\sum _{j=1}^{k}R{F}_j{v}^{n_j}=t_2(1-\frac{P^{″}}{C})K=t_2(C-P^{″})v^n,$$ which is same as how the capital gain tax for normal bond with single redemption is computed. Template:Colored example Template:Colored exercise

Book value

From the previous section, we can see that $P$ of a bond is generally different from $C$. This implies that the value of the bond is Template:Colored em from the purchasing price to the redemption value of the bond, during the period lasted by the bond.

The reason for this Template:Colored em is that there are Template:Colored em, and also there are change in value caused by the interest rate. In the previous section, we have determined that the initial value ($P$) of the bond and also the ending value ($C$) of the bond. In this section, we will also determine the value Template:Colored em, which is Template:Colored em by the coupon payment and interest rate, and we call these Template:Colored em values Template:Colored em. Template:Colored definition Template:Colored remark

Since book value measures the Template:Colored em of a bond, we use a formula for computing it which is similar to the basic formula (which measures the value of the bond, to determine a fair price), as follows:

Proof. It follows from the fact that book value at time $k$ is measuring the adjusted value at time $k$.

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Template:Colored remark Then, we can have the following recursive formula for computing book value, using this basic formula. Template:Colored proposition

Proof. First, we claim that $$a_{\overline{n-k}|}=v+v{a}_{\overline{n-k-1}|},$$ which is true since $$a_{\overline{n-k}|}=\frac{1-v^{n-k}}{i}=\frac{1-v+v-v^{n-k}}{i}=\frac{(1\cancel{+i}-1)/(1+i)}{\cancel{i}}+v\cdot \frac{1-v^{n-k-1}}{i}=v+v{a}_{\overline{n-k-1}|}.$$ Then, $$\begin{array}{cccc}& & B_k& =Fr{a}_{\overline{n-k}}+C{v}^{n-k}\\ & & & =Fr(v+v{a}_{\overline{n-k-1}|})+C{v}^{n-k}\\ & & & =v(Fr+\underset{=B_{k+1}}{\underbrace{Fr{a}_{\overline{n-(k+1}|}+C{v}^{n-(k+1)}}})\\ & \iff & B_k(1+i)& =Fr+B_{k+1}\\ \end{array}$$

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Bond amortization schedule

Since the nature of a bond is quite similar to that of a loan, we can construct a Template:Colored em, which is similar to the Template:Colored em.

Recall that in the Template:Colored em, the columns correspond to payment (or installment), interest paid, principal repaid, and outstanding balance. So, to construct a similar amortization schedule, we need to determine which term of bonds correspond to these terms.

• for the outstanding balance, we have mentioned that a corresponding term of bond is Template:Colored em ($B_k$)
• for the installment, we have mentioned that a corresponding term of bond is Template:Colored em ($Fr$)
• for the principal repaid, a similar term is $\mathrm{\Delta }{B}_k:=B_{k+1}-B_k$, but since we are constructing Template:Colored em schedule, the book value is expected to be decreasing (premium bond), and so $\mathrm{\Delta }{B}_k<0$, and we often do Template:Colored em want to deal with the negative sign, so we define an alternative term as follows:

Then, for the interest paid, we can determine it in a similar way compared with that in loan (multiplying the outstanding balance from the previous end of period by interest rate), namely multiplying the book value from the previous end of period by interest rate, i.e. Template:Colored proposition

Proof. It follows from the definition of interest.

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Then, we have an similar formula which links $P_k$ and $I_k$, compared to the situation of loan. Template:Colored proposition

Proof. Since $P_k=-\mathrm{\Delta }{B}_{k-1}=Fr-i{B}_{k-1}$ by recursive formula of book value, and $I_k=i{B}_{k-1}$, $$P_k+I_k=Fr\cancel{-i{B}_{k-1}+i{B}_{k-1}}=Fr=k\text{th coupon},$$ since each coupon is of the same amount $Fr$.

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Now, we proceed to construct the amortization schedule.

To increase the usefulness of the amortization schedule, we would like to determine a Template:Colored em for the book value, principal adjustment, etc. at different period.

To do this, we start from $B_0$. Recall the basic formula of book value. Since it is in the same form compared to the basic formula of bond price, we have an analogous premium/discount formula for book value, as follows: Template:Colored proposition

Proof. The proof is identical to the proof for premium/discount formula of bond price, except that $n$ is replaced by $n-k$.

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By definition, the coupon payment is $Cg=Fr$.

Then, using these, we can determine $I_k$ and $P_k$ as follows: Template:Colored corollary

Proof. For the formula for $I_k$, by the proposition about formula of $I_k$ and premium/discount formula of book value, we have $$I_k=i{B}_{k-1}=i(C+C(g-i)a_{\overline{n-k+1}|i}).$$

For the formula of $P_k$, by the proposition about relationship between $P_k$ and $I_k$, we have $$\begin{array}{cc}{P}_k& =\text{coupon}-I_k=Cg-i(C+C(g-i)a_{\overline{n-k+1}|i})\\ & =Cg-Ci-Ci(g-i)a_{\overline{n-k+1}|i}\\ & =(g-i)(C-Ci{a}_{\overline{n-k+1}|i})\\ & =(g-i)(C-C\cancel{i}\cdot \frac{1-v^{n-k+1}}{\cancel{i}})\\ & =C(g-i)(1-(1-v^{n-k+1}))\\ & =C(g-i)v^{n-k+1}.\end{array}$$

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After that, we can construct the amortization schedule as follows:

Bond amortization schedule

$k$	Coupon	$I_k=i{B}_{k-1}$	$P_k=\text{coupon}-I_k$	$B_k$
$0$	N/A	N/A	N/A	$C+C(g-i)a_{\bar{n}|i}$
$1$	$Cg$	$i{B}_0=i(C+C(g-i))a_{\bar{n}|i}$	$Cg-I_1=C(g-i)v^n$	$C+C(g-i)a_{\overline{n-1}|i}$
$2$	$Cg$	$i{B}_1=i(C+C(g-i))a_{\overline{n-1}|i}$	$Cg-I_2=C(g-i)v^{n-1}$	$C+C(g-i)a_{\overline{n-2}|i}$
$⋮$
$m$	$Cg$	$i{B}_{m-1}=i(C+C(g-i))a_{\overline{n-m+1}|i}$	$Cg-I_m=C(g-i)v^{n-m+1}$	$C+C(g-i)a_{\overline{n-m}|i}$
$⋮$
$n-1$	$Cg$	$i{B}_{n-1}=i(C+C(g-i))a_{\bar{2}|i}$	$Cg-I_{n-1}=C(g-i)v^2$	$C+C(g-i)a_{\bar{1}|i}$
$n$	$Cg$	$i{B}_n=i(C+C(g-i))a_{\bar{1}|i}$	$Cg-I_n=C(g-i)v$	$C+\underset{0}{\underbrace{C(g-i)a_{\bar{0}|i}}}$
Total	$nCg$	$\text{total coupon}-\text{total principal adjustment}=nCg-C(g-i)a_{\bar{n}|i}$	$C(g-i)a_{\bar{n}|i}$

Callable bond

Template:Colored definition Template:Colored remark Illustration of callable bond:

     possible redemption dates
           |-----^----|
-|---------|----------|----
 0         t          n

Because of the callable nature of this kind of bond, the term of the bond is uncertain. So, there is problem in computing prices, yield rates, etc.

To solve this, we assume the Template:Colored em scenario to the Template:Colored em . That is, the borrower will choose the option such that the investor has most Template:Colored em, as follows:

• if the redemption value ($C$) on all redemption dates are Template:Colored em, then if $i<g\iff P>C$, then Template:Colored em that the redemption date will be the Template:Colored em possible date, Template:Colored em (i.e. $i>g\iff P<C$), Template:Colored em that the redemption date will be the Template:Colored em possible date

• this is because if $i<g\iff P>C$, the bond was bought at a premium, and thus there will be a Template:Colored em at redemption, so the most unfavourable condition to the investor occurs when the Template:Colored em happens at the Template:Colored em time
• if $i>g\iff P<C$, the bond was bought at a discount, and thus there will be a Template:Colored em at redemption, so the most unfavourable condition to the investor occurs when the Template:Colored em happens at the Template:Colored em time

• if the redemption value ($C$) on all the redemption dates including the maturity date are Template:Colored em, then we need to compute the bond price at different possible redemption dates to check which is the Template:Colored em , and thus is most Template:Colored em to the investor

In particular, it is common for a callable bond to have redemption values that Template:Colored em as the term of the bond Template:Colored em, i.e. the later the redemption, the lower the redemption value, and if the bond is not called, the bond is redeemed at the redemption value. We have a special name for the difference between the redemption value (through call) and the par value: Template:Colored definition Template:Colored example Template:Colored remark Template:Colored example