FRN Pricing and Valuation

## Floating Rate Bonds

Floating rate bond is a bond that does not pay fixed rate coupon as per bonds in previous discussion. Instead, the coupon of the bond is tied to a reference rate either with or without spread. For MYR denominated FRBs, 6-month KLIBOR is often used as reference rate for the coupon.

Coupon fixing for the FRB is typically set on the start of the coupon period and is referred to as fixing in advance. Coupon fixing that is set at the end of the interest period is referred to as fixing in arrears. In addition, the fixing can also be set to be the average of reference rate over a period e.g. average of the reference rate over five (5) working days. The variation in the fixing for FRB cannot be handled by generalised formula and has to be covered separately. Unless otherwise stated, all discussion on FRB will refer to FRBs with coupon fixing in advance based on the reference rate on the fixing date.

Let’s consider a 2-year floating rate bond paying coupon semiannually with its coupons being fixed by a MYR 6M KLIBOR. The coupons are fixed in advanced and the first coupon is fixed at 2.80%. Diagram 4-3 illustrates the timing of the fixing and cash flow of the coupon interests. Diagram 4-3

### Valuation

Valuing FRB still uses the same present value of cash flows concept applied in various products covered in previous chapters. Unlike other products discussed in Chapter 1, Chapter 2 and the earlier part of this chapter, part of FRB's cash flows are unknown since some of the coupons (2nd to 4th coupons in Diagram 4-3) are yet to be fixed.

The unknown coupon rate can be forecasted based on the no arbitrage condition stated in Equation 1-10. The equation is restated below for easy reference.

\begin{align} \left( 1 + r_{l}dcf_{l} \right) &= \left( 1 + r_{s}dcf_{s} \right) \left( 1 + r_{f}dcf_{f} \right)\\ \textrm {where } dcf_{l} &= \textrm{day count factor for the full period}\\ dcf_{s} &= \textrm{day count factor for the deposit period}\\ dcf_{l} &= \textrm{day count factor for the remaining period}\\ \end{align}

The equation states that an investment for a period is equivalent to an investment of a shorter period with the maturity amount being reinvested (or compounded) at a forward rate for another investment period. In other words, a 1-year investment is equivalent to a 6-month investment being rolled into another 6-month investment on its maturity. If the relationship doesn't hold, there will be an arbitrage opportunity, at least theoretically.

For illustration, Table 2-3 in Chapter 2 will be used as our KLIBOR curve for valuation of the FRB. The curve however is insufficient as the rate is only up to 1 year while the FRB has two years to its maturity. A longer curve is required and is a subject at a later chapter. Our illustration, however is mathematical in nature and will not require numerical example.

The 2nd fixing can actually be presumed to be the forward rate as it is an arbitrage free forward rate. Equation 2-3 can be used to calculate the forward rate. For easy reference, Equation 2-3 is restated below:

\begin{align} r_f= \left(\dfrac{DF_s}{DF_l}-1\right)\dfrac{365}{t_f}\\ \textrm{where }DF_s &= \textrm{discount factor applicable for } t_s\\ DF_l &= \textrm{discount factor applicable for } t_l \end{align}

As a result, the cash flow can be calculated and discounted to the current date. For now, lets assume that the FRB is 6-month in nature. On issue date, the value of the bonds will be:

\begin{align} \textrm{ FRB Value } &= \dfrac{\textrm{Face Value} (1 + r_{6}dfc_{6})}{(1 + r_{6}dfc_{6})} \\ &= \textrm{Face Value} \end{align}

Equation 4-3

If the FRB has one year to maturity, the first coupon will be known on issue date while the second coupon rate can be the arbitrage free forward rate calculated using Equation 2-3. The coupon cash flows and the face value due on maturity can then be discounted at the relevant rate for the tenor of the cash flows. The sum of these discounted value will be the FRB value and can be represented the following equation:

\begin{align} \textrm{ FRB Value } &= \dfrac{\textrm{Face Value} ( r_{6}dfc_{6})}{(1 + r_{6}dfc_{6})} + \dfrac{\textrm{Face Value} (1 + r_{6x12}dfc_{6x12})}{(1 + r_{12}dfc_{12})} \\ \end{align}

On issue date, however, the FRB value will simply be the face value, FV, of the FRB as shown below:

\begin{align} \textrm{ FRB Value } &= \dfrac{\textrm{Face Value} ( r_{6}dfc_{6})}{(1 + r_{6}dfc_{6})} + \dfrac{\textrm{Face Value} (1 + r_{6x12}dfc_{6x12})}{(1 + r_{6}dfc_{6})(1 + r_{6x12}dfc_{6x12})} \\ &= \dfrac{\textrm{Face Value} (1 + r_{6}dfc_{6})}{(1 + r_{6}dfc_{6})} \\ &= \textrm{Face Value} \end{align}

In conclusion, on issue date or coupon date, FRB value is its face value under three conditions:

1. The current coupon has not been fixed
2. If the current coupon has been fixed but it is the same as the current reference rate.
3. Valuation is for value interest payment date
When the current coupon is known;

\begin{align} \textrm{ FRB Value } &= \dfrac{\textrm{Face Value} ( r_{c}dcf_{c})}{(1 + r_{r}dcf_{r})} + \dfrac{\textrm{Face Value} (1 + r_{rx12}dcf_{rx12})}{(1 + r_{r}dcf_{r})(1 + r_{rx12}dcf_{rx12})} \\ &= \dfrac{\textrm{Face Value} (1 + r_{c}dcf_{c})}{(1 + r_{r}dcf_{r})} \\ \textrm{ where } r_c &= \textrm{ coupon rate} \\ dcf_c &= \textrm{ discount factor for coupon period } \\ r_r &= \textrm {yield for the remaining coupon period } \\ dcf_r &= \textrm { day count factor for the remaining coupon period }\\ r_{rx12} &= \textrm{ forward yield from end of } t_r \textrm { end of } t_{12}\\ dcf_{rx12} &= \textrm{ forward period day count factor for } t_{rx12} \end{align}

Equation 4-3

The above equation is nothing more the the NID formula described in Chapter 1. FRB coupon however, is rarely quoted flat (without margin) against the reference rate. Instead, the coupon is frequently quoted as reference rate plus (or minus) margin e.g 6-month KLIBOR + 150 bps (basis points). The above equation need to be generalised to include two items;

1. coupon margin above the reference rate. This is the 150 bps mentioned above.
2. investor required spread. An FRB traded at 100 implies that the required spread is the same as the coupon margin. The required spread determined the discount factors to discount the projected cash flows.
Hence;

\begin{align} \textrm{ FRB Value } &= \dfrac{\textrm{Face Value} ( r_{c}dcf_{c})}{(1 + (r_{r}+s)dcf_{r})} + \dfrac{\textrm{Face Value} (1 + (r_{rx12} + m)dfc_{rx12})}{(1 + (r_{r}+s)dcf_{r})(1 + (r_{rx12}+s)dcf_{rx12})} \\ \textrm{ where } r_c &= \textrm{ coupon rate} \\ dcf_c &= \textrm{ day count factor for coupon period } \\ r_r &= \textrm {rate (from reference curve) for the remaining coupon period } \\ dcf_r &= \textrm { day count factor for the remaining coupon period }\\ r_{rx12} &= \textrm{ forward rate from end of } t_r \textrm { end of } t_{12}\\ dcf_{rx12} &= \textrm{ forward period day count factor for } t_{rx12}\\ m &= \textrm{coupon margin above reference rate}\\ s &= \textrm{investor's spread requirement}\\ \end{align}

Equation 4-4

Which can now be generalised into:

\begin {align} \textrm{FRB Value} &= \left( FV\sum_{i=1}^n \left((c_i+m_i).\prod_{k=1}^{i} d_{k-1\textrm{x}k} \right)\right) + FV.\prod_{i=1}^{n} d_{i-1\textrm{x}i}\\ \textrm{ where } c_i &= \textrm{ forward reference rate for the } i^{th} \textrm{ coupon period } \\ m_i &= \textrm{ spread for the } i^{th} \textrm{ coupon period }\\ d_{k-1\textrm{x}k} &= \textrm{ forward discount factor for the period between } (k-1)^{th} \textrm{ and }k^{th} \\ &=\dfrac{1}{1 + (r_{k-1xk}+ s)dcf_{k-1xk}}\\ \prod_{k=1}^{i} d_{k-1\textrm{x}k} &= \textrm{ discount factor to the } k^{th} \textrm{ period }\\ d_{i-1\textrm{x}i} &= \textrm{ forward discount factor for the period between } (i-1)^{th} \textrm{ and }i^{th} \\ \prod_{i=1}^{n} d_{i-1\textrm{x}i} &= \textrm{ discount factor to the } n^{th} \textrm{ period } \end{align}

Equation 4-5

The above equation uses the discount factors for each coupon period and can be confusing. It can be simplified into:

\begin {align} \textrm{FRB Value} &= \left( FV\sum_{i=1}^n (c_i+m_i). d_i \right) + FV.d_n\\ \textrm{ where } d_i &= \textrm{ discount factor to the } i^{th} \textrm{ period }\\ &=\prod_{k=1}^{i} d_{k-1\textrm{x}k}\\ d_n &= \textrm{ discount factor to the } n^{th} \textrm{ period }\\ &= \prod_{i=1}^{n} d_{i-1\textrm{x}i} \end{align}

Equation 4-6

Table 4-5

For the next discussion, the following Table 4-5 will be used as the reference rate curve.

#### Example 4-4

Consider an FRB bearing quarterly coupon of 3M KLIBOR + 150 bps with one year remaining to maturity. Today is June 17, 2020 and the current fixing will be 4.25% (2.75% + 1.50%). There is a buyer willing to buy the bond at KLIBOR + 100 bps. What is the buyer's price of the bond?

From all our previous discussion, there basically two items which we need to have for valuation: cash flow and discount factor.

Unknown cash flows will be projected by using the forward rate from the underlying curve in Table 4-5. The margin of 1.50% will be added to the forward rate and the sum is treated as the forward coupon rate in which the cash flow is generated. To discount the cash flow, we will use Equation 4-5 to calculate the discount factor. Table 4-6 displays all our calculation in the process.

We also included buttons to recalculate the values for two other spreads: 1.00% and 2.00%. Users can also change the rates in Table 4-5, to simulate other rate scenarios for valuing the FRB.

Table 4-6

Alternatively, if the FRB can also be revalued using an appropriate curve for the credit. This would avoid the need to calculate forward discount factor to get to the discount factor and may prove to be a faster exercise. We have created a calculator for FRN based on the methodology described above.