Bond Valuation

## Valuation of Fixed Rate Bonds

Valuation of a bond follows the basic concept of value of money. Cash flows in any bonds structure will be discounted in a similar manner discussed in Chapter 1 and each cash flows will be discounted at the same yield or often referred to as yield to maturity (YTM).

### Example 4-1

Lets consider a two-year MGS issuance of MYR1 billion with semiannual coupon. For newly issued MGS, the coupon is only known after the tender result is announced. The coupon will be the average yield of the successful tender. A short list of successful tender is shown in Table 4-11 as an example.

Table 4-1

The interest due on interest payment date for Actual/Actual convention is coupon divided by the coupon payment frequency per year. The MGS will have an interest payment of 1.479% $$\left(\frac{2.958}{2}\right)$$ every six months.

Bondholders will receive the coupon interest every six months. On maturity date of the bond, the bondholders will receive the last coupon interest and the face value of the bond. The value of the MGS is simply the total sum of each cash flow's present value.

PV of Bond = PV of $$C_1$$ + PV of $$C_2$$ +PV of $$C_3$$ + PV of $$C_4$$ + PV of Face Value

Where $$C_1$$, $$C_2$$, $$C_3$$ and $$C_4$$ are the $$1^{st}$$ ,$$2^{nd}$$ ,$$3^{rd}$$ and $$4^{th}$$ coupon interest cash flows, respectively and 'PV' stands for present value.

Diagram 4-1

Diagram 4-1 illustrates the discount factor for the relevant interest period where $$y$$ is the YTM of the bond. The discount factor shown in the diagram, discounted the cash flow only to the start date of the interest period i.e. $$\dfrac{1}{1+\frac{y}{2}}$$ is only applicable to discount:

1. $$1^{st}$$ coupon to ‘Today’
2. $$2^{nd}$$ coupon to $$1^{st}$$ coupon date
3. $$3^{rd}$$ coupon to $$2^{nd}$$ coupon date
4. $$4^{th}$$ coupon to $$3^{rd}$$ coupon date.

Hence, to discount the cash flows to ‘Today’, the discounted value has to be discounted further. Discounting the $$2^{nd}$$ coupon using $$\frac{1}{1+\frac{y}{2}}$$ for instance, only gives a discounted value as at the $$1^{st}$$ coupon date. The value has to be discounted further by $$\frac{1}{1+\frac{y}{2}}$$ to obtain the present value as at ‘Today’.

The same process is repeated for the $$3^{rd}$$ coupon, $$4^{th}$$ coupon and the face value occurring on maturity of the bond. The process is illustrated in the following diagram;

Diagram 4-2

In other words:

$$\textrm{PV of Bond } = \dfrac{C_1}{1+\frac{y}{2}}+ \dfrac{C_2}{\left(1+\frac{y}{2}\right)^2}+ \dfrac{C_3}{\left(1+\frac{y}{2}\right)^3}+ \dfrac{C_4+FV}{\left(1+\frac{y}{2}\right)^4}$$

For practical purposes, the formula can be further generalised using the day count factor:

\begin{align} \textrm{PV of Bond } &= \left(\sum_{i = 1}^{n}{FV.\dfrac{r_i.dc_i}{\prod_{k=1}^{i}{(1+y.dcf_k)}}}\right) + \dfrac{FV}{\prod_{i=1}^{n}{(1+y.dcf_i)}} \\ \textrm{where } FV &= \textrm{ face value of the bond}\\ r_i&= \textrm{ the }i^{th} \textrm{ coupon rate }\\ dc_i&= \textrm { the } i^th \textrm{ coupon period day count factor}\\ dcf_k&= \textrm { the } k^{th} \textrm{ remaining coupon period day count factor}\\ y&= \textrm{ yield of the bond} \end{align}

Equation 4-1

For the non-mathematicians, perhaps the ensuing explanation would help to understand the equation better. $$FV.r_i.dc_i$$ is the coupon interest for the $$i^{th}$$ coupon period. This is being discounted to present value by the term $$\frac{1}{\prod_{k=1}^{i}{(1+y.dcf_k)}}$$, which is the discount factor from value date to the $$i^{th}$$ coupon. If the value date is not on a coupon date, then $$dcf_1$$ will be less than $$dc_1$$ as some days have lapsed and the remaining coupon period become smaller. Each $$dcf$$ will be the same as $$dc$$ for the remainder of the coupon period. If value date is on a coupun date, each $$dcf$$ will be the same as $$dc$$ for all coupon period. The sum of the coupons' present values is represented by $$\left(\sum_{i = 1}^{n}{FV.\frac{r_i.dc_i}{\prod_{k=1}^{i}{(1+y.dcf_k)}}}\right)$$

The last term, $$\frac{FV}{\prod_{i=1}^{n}{(1+y.dcf_i)}}$$, is the present value of the face value whereby $$\frac{1}{\prod_{i=1}^{n}{(1+y.dcf_i)}}$$ is the discount factor from value date to maturity of the bond. The value of the bond is the sum of the present values of the coupons and the face value.

Notice that frequency is not anywhere in the equation. Frequency has been taken into account in the calculation of day count factor. When frequency is semi-annual and the day count factor is Actual/Actual (Actual/Actual ICMA, to be specific), Equation 4-1 if applied to the MGS will give the same formula as the one below Diagram 4-2.

Once the tender result is announced, Bank Negara Malaysia (BNM) will publish the price for each successful tender. The published price is the price per MYR100 face value and is referred to as clean price or generally referred to as price. The price is obtained using the following formula;

$$\textrm{Price} = \left(\dfrac{\textrm{PV of MGS}}{\textrm{FV}}\right)\textrm{x}100 - \textrm{ Accrued Interest}$$

Equation 4-2

For bonds, accrued interest is the amount of interest that has been accrued since the last issue or coupon date but is not due for payment until the next coupon payment date. For the newly issued MGS, accrued interest is zero since the value date of the tender is the issue date. The following table shows the calculation for one of the successful tender (Asset Management B) using Equation 4-1.

Table 4-2

Table 4-2 showed a detailed present value calculation of each cash flow of the bond tendered by Asset Management B. Asset Management B submitted a bid for the MGS tender at the yield of 2.960% for a face value amount of MYR10 million. The value of the MGS at the tendered yield is 9,999,614.55 and the price at three decimal places is 99.996 which will be the price published by BNM. The prices for other successful tenders are shown below. Readers can manually check the calculation or use our provided bond calculator in the Calculators section.

Table 4-3

The proceeds for each of them is simply Price x Amount / 100. Equation 4-1 can be applied for valuing the MGS on dates not on coupon date as well.

### Example 4-2

After holding the MGS for a week (seven days), Asset Management B decided to sell the MGS at the yield of 2.9000% for value same day . Interest has accrued for the 7-day holding period and has to be calculated to arrive at the proceeds. Accrued interest, however is irrelevant to the calculation of the present value of the bond. The detailed calculations are shown below in table format.

Table 4-4

The sum of the 'Present Value' column is column is the total value of the MGS an is also the proceed is calculated to be 10,016,667.71. The dirty price (per 100) of the bond is simply proceed / face value x 100. Clean price (per 100), on the other hand is defined as proceed - accrued interest x 100 / face value.

We can get the information to calculate the accrued interest from Table 4-4 by comparing the 'Day Count Factor for Coupon' and 'Day Count Factor for Yield' in the first row. The difference is the accrued period and the accrued interest is the difference between the two divided by 'Day Count Factor for Coupon' and mutiply by the interest amount in the first row. The accrued interest calculated this way is 5,657.38.

The long way of calculating the accrued interest is by using this - 7 days / number of days in the coupon period x face value x coupon rate / coupon frequency. For guidance, it is assumed that the issue date of the bond is June 17, 2020 if readers are comparing the calculation result.

The clean price now can be calculated, $$\frac{10,016,667.71 - 5,657.38}{10,00,0000}\textrm{ x }100 = 100.11$$ (as secondary market trades in two decimal places).

In the original book, the valuation formula was treated separately for Private debt securities (PDS) or now commonly known as corporate bonds (CB). With the use of day count factor introduced in Chapter 1, this is no longer a necessity and all coupon bearing instrument can use Equation 4-1.

#### Par Bonds

A par bond is a bond trading at the price of 100. A bond that is trading at 100 has its coupons discounted at YTM that is the same as the coupon rate. It is one of many characteristics of a bond which will be used to generate discount factor and zero rate curve in later in this chapter and in the following chapters (Chapter 5 and 6).

### Example 4-3

Let’s assumed that the above MGS is purchased for value issue date at the yield of 2.958%. For bond purchased for value issue date or coupon payment date, the dirty price of proceed of the bond is the clean price since accrued interest is zero. The clean price is 100.00

On the other hand, if the MGS was bought at the same yield for value September 17, 2020, there will be accrued interest and the clean price of the bond will not be 100. Using our calculator we obtain a proceed of MYR10,073,811.25, an accrued interest of MYR74,354.10 and a clean price of 99.9946(to four decimal places). The most common reason put forward is that the accrued interest deducted from the calculation of clean price is not discounted value i.e. it takes absolute value of accrued interest without being discounted to present value. In practice, however, dealing prices are quoted in two decimal places and will result in the clean price to be quoted as 100.