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 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. $$K=Cv^n$$ 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.

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

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 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 $$1-t_1$$ ($$t_1Fr$$ is the income tax paid per coupon payment). Also, we consider the of the tax payment to compute the  bond price. Hence, we have the following modified formulas under income tax:

The basic formula becomes $$P=Fr(1-{\color{darkgreen}t_1})a_{\overline n|i}+\underbrace{Cv^n}_K.$$ The premium/discount formula becomes $$P=\underbrace{Cg}_{Fr}(1-{\color{darkgreen}t_1})a_{\overline n|i}+Cv^n=C+C(g(1-{\color{darkgreen}t_1})-i)a_{\overline n|i}.$$
 * $$P-C$$ is premium if $$P>C\Leftrightarrow g(1-{\color{darkgreen}t_1})>i$$
 * $$P-C$$ is discount if $$P<C\Leftrightarrow g(1-{\color{darkgreen}t_1})<i$$ (and ).
 * 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 $$P=\underbrace{Gi}_{Fr}(1-{\color{darkgreen}t_1})+Cv^n=G(1-{\color{darkgreen}t_1})+(C-G(1-{\color{darkgreen}t_1}))v^n.$$ The Makeham's formula becomes $$P=\underbrace{Cg}_{Fr}(1-{\color{darkgreen}t_1})a_{\overline n|i}+Cv^n=K+\frac{g(1-{\color{darkgreen}t_1})}{i}(C-K)$$

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 the purchase price by  of $$t_2(C-P)$$ at the redemption date (the  of the capital gain tax paid for the bond).

Serial bond
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, $$C_j=RF_j,\quad C=RF=R\sum_{j}F_j=\sum_{j}C_j,$$ and $$K_j=C_jv^{n_j},\quad K=Cv^{n_j}=\sum_{j}C_jv^{n_j}.$$ In this situation, Makeham's formula is quite useful, and ease calculation. Its usefulness is illustrated in the following.

When there are no taxes, $$P_j=K_j+\frac{g}{i}(C_j-K_j)\Leftrightarrow\sum_{j}P_j=\sum_{j}K_j+\frac{g}{i}\left(\sum_{j}(C_j)-\sum_{j}(K_j)\right)=K+\frac{g}{i}(C-K)=P.$$

Let $$P'$$ be the price of serial bond when there is income tax. When there is income tax, say the rate is $$t_1$$, $$P'_j=K_j+\frac{g(1-t_1)}{i}(C_j-K_j)\Leftrightarrow \sum_{j}P'_j=\sum_{j}K_j+\frac{g(1-t_1)}{i}\left(\sum_{j}(C_j)-\sum_{j}(K_j)\right)=K+\frac{g(1-t_1)}{i}(C-K)=P'.$$

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. 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 $$\sum_{j=1}^{k}\left(t_2\left(\underbrace{RF_j}_{C_j}-\frac{F_j}{F}P''\right)v^{n_j}\right) =t_2\left(1-\frac{P''}{FR}\right)\sum_{j=1}^{k}RF_jv^{n_j} =t_2\left(1-\frac{P''}{C}\right)K =t_2\left(C-P''\right)v^n,$$ which is same as how the capital gain tax for normal bond with single redemption is computed.

{{colored exercise| {Select all correct statement(s). - Without any taxes, a zero-coupon serial bond with several installments must have a lower $$P$$ than a bond with the same $$F$$, with single redemption at a date that is later than all redemption dates of the serial bond. + Suppose a 10-year serial bond that is redeemable by 10 equal installments of amount $$k$$ has the price $$P$$. Then, for ten 1-year bonds that is redeemable by a single installment of amount $$k$$ at the end of period lasted by the bond, purchased one by one such that they last for 10 years in total, the sum of present values of their prices is also $$P$$. - The multiple redemptions in serial bond are the same in amount. - The multiple redemptions in serial bond are at an regular interval. - The multiple redemptions in serial bond are different in present value. }}
 * type="[]"}
 * since the redemption dates are all earlier than the redemption date of another bond, the discounting factor when computing the present values of the redemption payments is closer than one
 * so, the sum of present values is higher than the present value of the redemption payment at a late date
 * therefore, the serial bond must have a higher $$P$$
 * for the serial bond: $$P=kv+kv^2+\cdots+kv^{10}$$
 * for the ten bonds: $$P=\underbrace{kv]_{\text{1st bond}}+\underbrace{kv^2}_{\text{2nd bond}}+\cdots+kv^{10}$$
 * They can be different in amount.
 * They can be at irregular intervals
 * in some special cases, the multiple redemptions are different in amount such that their present values are the same
 * e.g. two redemptions: 1st installment is 1100 at the end of 1st year, 2nd installment is 1210 at the end of 2nd year, then present values of them (at time zero) are both 1000 if the effective annual interest rate is 10%.
 * but this is generally true

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 from the purchasing price to the redemption value of the bond, during the period lasted by the bond.

The reason for this is that there are, 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 , which is by the coupon payment and interest rate, and we call these values.

Since book value measures the 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:

Then, we can have the following recursive formula for computing book value, using this basic formula.

Bond amortization schedule
Since the nature of a bond is quite similar to that of a loan, we can construct a, which is similar to the.

Recall that in the, 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 ($$B_k$$)
 * for the installment, we have mentioned that a corresponding term of bond is ($$Fr$$)
 * for the principal repaid, a similar term is $$\Delta B_k:=B_{k+1}-B_k$$, but since we are constructing schedule, the book value is expected to be decreasing (premium bond), and so $$\Delta B_k<0$$, and we often do  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.

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

Now, we proceed to construct the amortization schedule.

To increase the usefulness of the amortization schedule, we would like to determine a 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:

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

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

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

Callable bond
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 scenario to the. That is, the borrower will choose the option such that the investor has most , as follows:


 * if the redemption value ($$C$$) on all redemption dates are, then if $$iC$$, then that the redemption date will be the  possible date,  (i.e. $$i>g\Leftrightarrow PC$$, the bond was bought at a premium, and thus there will be a at redemption, so the most unfavourable condition to the investor occurs when the  happens at the  time
 * if $$i>g\Leftrightarrow P<C$$, the bond was bought at a discount, and thus there will be a at redemption, so the most unfavourable condition to the investor occurs when the  happens at the  time

In particular, it is common for a callable bond to have redemption values that as the term of the bond, 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:
 * if the redemption value ($$C$$) on all the redemption dates including the maturity date are, then we need to compute the bond price at different possible redemption dates to check which is the , and thus is most  to the investor