## Finance Training Course Pricing Case Study: Valuing Islamic Debt Instruments (Sukuk Bonds) and floating rate instruments

*in*Computational Finance

# Valuation of Floating Rate Debt, Term Finance Certificates, Sukkuks and Corporate Debt

## Introduction to Floating Rate Notes

Three methodologies are presented below for valuation of floating rate notes, term finance certificates, sukkuks and corproates each with a simple numerical example. These are:

- Pricing using the bond pricing model which assumes that the outstanding principal will be redeemed with the coupon payment and scheduled redemption immediately prior to the following reset date.
- Pricing using the forward pricing model which calculates the future floating rate coupons based on a derived forward rate curve.

- Pricing using the Mutual Funds Association of Pakistan’s (MUFAP) published prices.

**Sample instrument**

The examples will be based on the following instrument with the characteristics/ features mentioned below:

TFC of Trust Investment Bank Limit (Code – TRIBL/TFC/040708)

Rating BBB (as on 31 May 2010)

Non Traded

Issue Date = 04^{th} July 2008

Tenor = 5 years

Maturity = 04^{th} July 2013

Coupon Payment Frequency = Semiannual

Floating Rate = 6-month KIBOR + 1.85%

Valuation Date = 31^{st} May 2010

Face Value = PKR 5000

Number of Units = 9000

Day count convention = Actual / Actual

The redemption schedule is as follows:

Period Start (i) | Period End (i+1) | Principal Outstanding | Redemption % of Face Value | Redemption Amount |

04/07/2008 | 04/01/2009 | 45,000,000.00 | 0.020% | 9,000.00 |

04/01/2009 | 04/07/2009 | 44,991,000.00 | 0.020% | 9,000.00 |

04/07/2009 | 04/01/2010 | 44,982,000.00 | 12.495% | 5,622,750.00 |

04/01/2010 | 04/07/2010 | 39,359,250.00 | 12.495% | 5,622,750.00 |

04/07/2010 | 04/01/2011 | 33,736,500.00 | 12.495% | 5,622,750.00 |

04/01/2011 | 04/07/2011 | 28,113,750.00 | 12.495% | 5,622,750.00 |

04/07/2011 | 04/01/2012 | 22,491,000.00 | 12.495% | 5,622,750.00 |

04/01/2012 | 04/07/2012 | 16,868,250.00 | 12.495% | 5,622,750.00 |

04/07/2012 | 04/01/2013 | 11,245,500.00 | 12.495% | 5,622,750.00 |

04/01/2013 | 04/07/2013 | 5,622,750.00 | 12.495% | 5,622,750.00 |

# Valuation Methods

## MUFAP Prices

This method is used by banks to mark to market their TFC, Sukkuk and Corporate Debt portfolio. The methodology employed by MUFAP is in accordance with SECP circular 01 of 2009 dated Jan 6, 2009.

In order to determined the market value on a given date the outstanding principal at the beginning of the current coupon paying period is multiplied with the MUFAP published prices.

For the above example, the MUFAP price published as of 31^{st} May 2010 for the given TFC is 89.8689%. The outstanding principal balance as of the opening date (4/1/2010) of the period that the valuation date falls in is PKR 39,359,250. The market value therefore works out to 39359250×89.8689/100 = PKR 35,371,725.

## Bond Pricing Model

As mentioned earlier, under the bond model the present value of the cash flows due at the end of the current period (including an assumption that the outstanding principal will be redeemed) will be calculated. This involves the following steps:

- Defining the par term structure used to determine the zero curve
- Determination of the zero curve for discounting
- Determination of the floating rate payment for current period based on KIBOR observed at the start of the period and the spread.
- Discount the cash flows, which includes the markup/ coupon payment and the outstanding principal to the valuation date to determine the price.

Detailed methodology is given below:

### Step 1: Defining the par term structure

#### a) Select an appropriate term structure

Based on the debt instrument being priced an appropriate term structure or structures will be chosen. This is an important process because the zero curve (and the forward curve for the forward pricing model) is derived from it which is in turn used to discount the cash flows (and in the case of the forward curve, calculate the future coupon rates).

For our example we select the KIBOR (offer rates) as our par term structure based on the fact that floating rate payments are based on what the KIBOR will be like in the future. As of the valuation date the KIBOR offer rates are as follows:

KIBOR | Offer Rate |

1 – Week | 12.37% |

2 – Week | 12.31% |

1 – Month | 12.31% |

3 – Month | 12.16% |

6 – Month | 12.27% |

9 – Month | 12.58% |

1 – Year | 12.66% |

2 – Year | 12.77% |

3- Year | 12.86% |

#### b) Extending the term structure

If the outstanding term of the debt instrument is greater than the maximum tenor of the selected par term structure than rates from another available term structure having longer tenors may be used as a proxy to the selected par term structure to supplement it. For example, PKRV rates may be used to extend the term structure to tenors as long as 20 years and 30 years. As of the valuation date the PKRV rates were as follows for tenors greater than 3 years:

Tenor | Treasury Rate |

3yr | 12.47% |

4yr | 12.51% |

5yr | 12.56% |

6yr | 12.60% |

7yr | 12.63% |

8yr | 12.63% |

9yr | 12.63% |

10yr | 12.86% |

15yr | 13.07% |

20yr | 13.21% |

30yr | 12.47% |

#### c) Creating a default term structure

##### i) Adding a spread

The rates quoted for respective term structures are based on the underlying instruments. Treasury term structures are based on treasury instruments that are usually compounded on semiannual basis. Interbank offer rates on the other hand tend to be based on corporates that are compounded on a quarterly basis. These rates, presented on an annualized basis, are therefore usually nominal rates compounded based on the payment frequency of the underlying instruments.

Depending on the debt instrument being priced spreads (positive or negative) may need to be added to the rates to account for the differentials between quoting basis and the payment frequency of the instrument and between the primary and extended par term structures. The resulting rates would then be considered as been quoted on the appropriate basis.

For illustration purposes we assume that no spreads are added to any of the rates in the term structure. The default term structure in our example consists of 0.5 year, 1 year, 2 years, 3 years, 4 years and 5 years tenors. As mentioned above our primary term structure is the KIBOR term structure with rates for tenors exceeding the maximum tenor being taken from the PKRV term structure. The resulting default term structure is as follows:

Tenor (in years) | Default Term Structure |

0.5 | 12.27% |

1 | 12.66% |

2 | 12.77% |

3 | 12.86% |

4 | 12.47% |

5 | 12.51% |

##### ii) Interpolating par rates for intermediate tenors

Depending on the payment frequency of the debt instrument, par rates for intermediate tenors may need to be calculated. For example the sample debt instrument has a semiannual payment frequency. Rates for intermediate tenors 1.5 years, 2.5 years, 3.5 year and 4.5 years would also be required. However this is not readily available in the default term structure. The rate would be interpolated from the rates of the adjacent tenors. For example the rate for tenor 1.5 years will be interpolated from the 1-year and 2-year rates as follows:

T= intermediate tenor =1.5 years

T_{1}= 1 year

T_{2}=2 years

Our default par term structure considering the interpolated rates will be as follows:

Tenor (in years) | Default Term Structure |

0.50 | 12.27% |

1.00 | 12.66% |

1.50 | 12.72% |

2.00 | 12.77% |

2.50 | 12.82% |

3.00 | 12.86% |

3.50 | 12.67% |

4.00 | 12.47% |

4.50 | 12.49% |

5.00 | 12.51% |

### Step 2: Deriving the Zero Curve

We use the bootstrapping method for deriving the zero curve from the par term structure. This is an iterative process that allows us to derive a zero coupon yield curve from the rates/ prices of coupon bearing instruments. The step by step procedure employed is given below:

#### a) Develop the cash flows matrix

Given the default par term structure above we calculate the cash flows for coupon bearing instruments for each tenor. The par value for each instrument is assumed to be 100. The instruments are assumed to be at par meaning that the coupon rate is equal to the par rate. An instrument with 0.5 year tenor, means a cash flow at maturity of the face value, 100 plus a coupon of par rate * face value= 12.27%*100/2 = 6.135 or a total of 106.135. An instrument with a tenor of 1 year will have a coupon payment at the end of 0.5 year and a payment of the face value + coupon at the end of year 1, and so on. Further details of the cash flows are given in the matrix below:

Tenor | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |

0.50 | 106.135 | 6.33 | 6.3575 | 6.385 | 6.4075 | 6.43 | 6.3325 | 6.235 | 6.245 | 6.255 |

1.00 | 106.33 | 6.3575 | 6.385 | 6.4075 | 6.43 | 6.3325 | 6.235 | 6.245 | 6.255 | |

1.50 | 106.3575 | 6.385 | 6.4075 | 6.43 | 6.3325 | 6.235 | 6.245 | 6.255 | ||

2.00 | 106.385 | 6.4075 | 6.43 | 6.3325 | 6.235 | 6.245 | 6.255 | |||

2.50 | 106.4075 | 6.43 | 6.3325 | 6.235 | 6.245 | 6.255 | ||||

3.00 | 106.43 | 6.3325 | 6.235 | 6.245 | 6.255 | |||||

3.50 | 106.3325 | 6.235 | 6.245 | 6.255 | ||||||

4.00 | 106.235 | 6.245 | 6.255 | |||||||

4.50 | 106.245 | 6.255 | ||||||||

5.00 | 106.255 |

The columns pertain to a particular tenor; the rows pertain to the duration when a payment is due within the tenor.

#### b) Developing the discounted value of cash flows matrix and the zero curve

As the instruments are assumed to be priced at par the total present value of the future cash flows must equal the face value of the instrument, i.e. 100. In order to derive the discounted cash flows and the zero coupon rates we start with the shortest tenor and work our way to the larger tenors.

For the coupon bearing instrument with tenor of 0.5 year the cash flow due at duration 0.5 year amount to 106.135. We know that the discount value of this total cash flow is 100. The zero coupon rate therefore would be the rate that discounts the cash flows to this value, i.e.

106.135 ÷ (1+ZC_{0.5}/2) = 100

Therefore ZC_{0.5} ={ [106.135 ÷ 100] -1}×2 =12.27%

For a coupon bearing instrument with tenor of 1 year, the cash flow at duration 0.5 year will be discounted using the zero coupon rate determined earlier, i.e. ZC_{0.5}=12.27%. The cash flow at duration 1 will be discounted at ZC_{1} (annually compounded rate) which is an unknown at this point in time. Using the fact the total present value is equal to the face value, we determine ZC_{1} by solving the following equation:

100 = [6.135 ÷ (1+ZC_{0.5}/2)] + [106.135 ÷ (1+ZC_{1}/2)^{2}]

(1+ZC_{1}/2)^{2} = 106.135 ÷ {100 – [6.135 ÷ (1+ZC_{0.5}/2)]}

ZC_{2} = 12.672%

The same iterative and substitution process will be used to determine the zero coupon rates for the other tenors.

The matrix of discounted cash flows is given below:

Tenor | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |

0.5 | 100.00 | 5.96 | 5.99 | 6.02 | 6.04 | 6.06 | 5.97 | 5.87 | 5.88 | 5.89 |

1 | – | 94.04 | 5.62 | 5.65 | 5.67 | 5.69 | 5.60 | 5.51 | 5.52 | 5.53 |

1.5 | – | – | 88.39 | 5.31 | 5.32 | 5.34 | 5.26 | 5.18 | 5.19 | 5.20 |

2 | – | – | – | 83.03 | 5.00 | 5.02 | 4.94 | 4.87 | 4.87 | 4.88 |

2.5 | – | – | – | – | 77.97 | 4.71 | 4.64 | 4.57 | 4.58 | 4.58 |

3 | – | – | – | – | – | 73.18 | 4.35 | 4.29 | 4.29 | 4.30 |

3.5 | – | – | – | – | – | – | 69.23 | 4.06 | 4.07 | 4.07 |

4 | – | – | – | – | – | – | – | 65.65 | 3.86 | 3.87 |

4.5 | – | – | – | – | – | – | – | – | 61.73 | 3.63 |

5 | – | – | – | – | – | – | – | – | – | 58.04 |

The resulting zero curve is as follows:

t (in years) | Tenor(in half years) | ZC_{t} |

0.50 | 1 | 12.270% |

1.00 | 2 | 12.672% |

1.50 | 3 | 12.727% |

2.00 | 4 | 12.784% |

2.50 | 5 | 12.833% |

3.00 | 6 | 12.883% |

3.50 | 7 | 12.643% |

4.00 | 8 | 12.403% |

4.50 | 9 | 12.436% |

5.00 | 10 | 12.468% |

### Step 3: Determining the floating coupon rate and coupon amount

The coupon rate is calculated based on the KIBOR rate observed on the starting date of the current coupon paying period. To this is added the pre-specified spread. The current coupon period is from 4^{th} January 2010 to 4^{th} July 2010. The KIBOR 6 month offer rate observed on 4^{th} January 2010 was 12.37%. The pre-specified spread is 1.85%. The coupon payment will be the sum of the KIBOR offer rate plus spread divided by 2 (for the payment frequency). The result is multiplied by the outstanding principal amount at the start of the coupon paying period (adjusted for the remaining time left in the coupon paying period), i.e. ((12.37%+1.85%)/2) × 39,359,250.00 × (Period End Date-Valuation date)/365 = PKR 525,674.31.

### Step 4: Discount the cash flows to determine the price

The price or the value of the debt instrument under this methodology is the discounted value of the markup, the principal redeemed and the outstanding principal balance at the end of the coupon paying period discounted to the valuation date using the appropriate zero coupon rate. The underlying assumption is that the debt holder will sell the bond immediately prior to the reset date.

The number of days remaining in the current coupon paying period is 34 days. The number of days in the entire coupon paying period is 181 days. The tenor in half years is 34/181=0.19 or 0.09 in years. If the tenor is less than 1 period (i.e. 0.5 years), the zero coupon rate used in the calculation is for 1 period. In this case the zero coupon rate for 0.5 years is 12.27%. The cash flows are as follows:

Principal redeemed during the period (as per schedule given above) = PKR 5,622,750

Outstanding principal at the end of the period = PKR 33,736,500

Mark up/ coupon for the period = PKR 525,674.31

Total cash flow due on 4/7/2010 = PKR 39,884,924.31

Discounted Cash flow = Price as at 31/5/2010 = 39884924.31/(1+12.27%/2)^0.19

= PKR 39,441,311.82

## Forward Pricing Model

Under the forward pricing model the cash flows are projected for all future periods in the entire tenor of the contract. In order to determine the future coupon cash flows which are floating rate payments the forward curve needs to be determined. The sum of the discounted values of all future cash flows will be the price of the debt instrument on the valuation date.

The process follows the steps mentioned below:

- Defining the par term structure used to determine the zero curve
- Determination of the zero curve for discounting
- Determine the coupon paying and receiving periods
- Based on the actual tenors computed in step 3 above determine the zero coupon rates applicable for these periods
- Deriving the forward rate curve from the zero curve
- Determination of the floating rate payments using the forward rates computed in step 5 plus the pre-specified spread.
- Discount the cash flows to the valuation date to determine the price.

Steps 1 and 2 are the same as for the bond pricing model. Detailed methodology for steps 3 -7 are given below:

### Step 3: Determine the coupon paying/ receiving periods

For each future payment we determine the length of the period and the duration between the due date and the valuation or pricing date:

Period Start | Period End | Days to Coupon/ redemption | Days in period | Tenor (in half years)(Basis=Actual/Actual) | Tenor in years |

04/01/2010 | 04/07/2010 | 34 | 181 | 0.19 | 0.09 |

04/07/2010 | 04/01/2011 | 218 | 184 | 1.18 | 0.59 |

04/01/2011 | 04/07/2011 | 399 | 181 | 2.20 | 1.10 |

04/07/2011 | 04/01/2012 | 583 | 184 | 3.17 | 1.58 |

04/01/2012 | 04/07/2012 | 765 | 182 | 4.20 | 2.10 |

04/07/2012 | 04/01/2013 | 949 | 184 | 5.16 | 2.58 |

04/01/2013 | 04/07/2013 | 1130 | 181 | 6.24 | 3.12 |

Period Start = start of the period for which coupon is assessed

Period End = date on which the payments are to be made

Days to coupon/ redemption = number of days between the date when coupon/ redeemed principal will be paid and the valuation date = Period End-Valuation Date

Days in period = number of days in the coupon paying period = Period End –Period Start

Tenor (in half years) = Duration between when the coupon will be paid and the valuation date in years (depending on the day count convention) = Days to Coupon/Days in period

### Step 4: Determine the zero coupon rates applicable

Based on the duration between the valuation date and the coupon payment date the zero coupon rates are determined by interpolating between the zero-coupon rates derived earlier. For tenors less than 1 period (0.5 years) we will use the zero coupon rates for tenor = 0.5 year. These rates will be used to discount the cash flows as well as to determine forward rates which will used be to calculate the future coupon payments under the debt instrument.

For example, for tenor 3.12 years the zero coupon rate is determined by interpolating the rates for 3 years and 3.5 years respectively as follows:

The interpolated zero coupon rates applicable for the pricing of this sample debt instrument are as follows:

Period End | Tenor in years | Interpolated Zero Coupon Rates |

04/07/2010 | 0.09 | 12.270% |

04/01/2011 | 0.59 | 12.344% |

04/07/2011 | 1.10 | 12.684% |

04/01/2012 | 1.58 | 12.737% |

04/07/2012 | 2.10 | 12.794% |

04/01/2013 | 2.58 | 12.841% |

04/07/2013 | 3.12 | 12.824% |

### Step 5: Derive the forward rates

In order to derive forward rates from the zero coupon rates for successive interest rate periods the bootstrapping methodology has been employed. In particular the following formula has been used:

Where t is the tenor in years, ZC_{t} is the zero coupon rate for a tenor of t years and FC_{t-0.5,t} is the forward rate for the period (t-0.5,t).

For example, the forward rate for the interest rate period 0.5 years to 1 year using zero coupon rates is

The forward rates are as follows is our example:

T | FC_{t-0.5,t} |

0.50 | 12.270% |

1.00 | 13.075% |

1.50 | 12.836% |

2.00 | 12.957% |

2.50 | 13.026% |

3.00 | 13.133% |

3.50 | 11.210% |

4.00 | 10.731% |

4.50 | 12.701% |

5.00 | 12.756% |

Applying this methodology to the zero curves interpolated using the actual payment periods, the forward rates to be used are as follows:

Period Start | Period End | Interpolated Forward Rates |

04/01/2010 | 04/07/2010 | 12.270% |

04/07/2010 | 04/01/2011 | 12.319% |

04/01/2011 | 04/07/2011 | 13.343% |

04/07/2011 | 04/01/2012 | 12.382% |

04/01/2012 | 04/07/2012 | 13.438% |

04/07/2012 | 04/01/2013 | 12.431% |

04/01/2013 | 04/07/2013 | 13.874% |

For example, the forward rate for the period 4/1/2011 and 4/7/2012 will be calculated by used the zero coupon rate for tenors 0.59 and 1.10:

**Step 6: Determine the cash flows**

The cash flows for the debt instrument are as follows:

Period End (i+1) | Forward Rate | Coupon Rate(Forward Rate + Spread) | Markup | Principal Redeemed | Total Cash flow |

04/07/2010 | 12.370% | 14.22%* | 525,674.31 | 5,622,750.00 | 6,148,424.31 |

04/01/2011 | 12.319% | 14.17% | 2,390,124.81 | 5,622,750.00 | 8,012,874.81 |

04/07/2011 | 13.343% | 15.19% | 2,135,728.61 | 5,622,750.00 | 7,758,478.61 |

04/01/2012 | 12.382% | 14.23% | 1,600,434.47 | 5,622,750.00 | 7,223,184.47 |

04/07/2012 | 13.438% | 15.29% | 1,289,368.79 | 5,622,750.00 | 6,912,118.79 |

04/01/2013 | 12.431% | 14.28% | 802,978.40 | 5,622,750.00 | 6,425,728.40 |

04/07/2013 | 13.874% | 15.72% | 442,071.86 | 5,622,750.00 | 6,064,821.86 |

* this rate is based on the six month KIBOR rate observed on 4/1/2010, i.e.12.37% plus spread 1.85%

The coupon payments are based on KIBOR + spread where spread is given as 185 basis points. The KIBOR rate will be the forward rate derived. Therefore the floating Coupon Rate above is the Forward Rate + Spread and the Markup is (Forward Rate+ Spread)*Outstanding Principal at the start of the period.

For the period ended 4/7/2011 the markup for the fractional period is (12.37%+1.85%)/2 × 39359250 × 34/181 = 525,674.31

For the period ended 4/1/2011, the markup for the entire period is charged. This works out to (12.319% + 1.85%)/2× 33736500 = 2,390,124.81

### Step 7: Discount the cash flows and calculated the price

The next step is to discount the cash flows using the interpolated zero coupon rates. The resulting present values are given below:

Period End (i+1) | Tenor in years | Interpolated Zero Coupon Rate | Total Cash flow | PV of Cash Flow |

04/01/2009 | -1.39 | 0.000% | – | – |

04/07/2009 | -0.91 | 0.000% | – | – |

04/01/2010 | -0.40 | 0.000% | – | – |

04/07/2010 | 0.09 | 12.270% | 6,148,424.31 | 6,080,039.63 |

04/01/2011 | 0.59 | 12.344% | 8,012,874.81 | 7,463,994.57 |

04/07/2011 | 1.10 | 12.684% | 7,758,478.61 | 6,775,013.16 |

04/01/2012 | 1.58 | 12.737% | 7,223,184.47 | 5,939,843.85 |

04/07/2012 | 2.10 | 12.794% | 6,912,118.79 | 5,326,191.03 |

04/01/2013 | 2.58 | 12.841% | 6,425,728.40 | 4,661,656.30 |

04/07/2013 | 3.12 | 12.824% | 6,064,821.86 | 4,114,405.81 |

Price | 40,361,144.34 |

For example for the period ended 04/07/2010, the discount cash flows are calculated as follows:

PV of Fixed Leg = 6064821.86 ÷ (1+12.824%/2)^{3.12×2 }= PKR 4,114,405.81

The price of the debt instrument is the sum of the present values of the individual cash flows which works out to PKR 40,361,144.34.

## Floating rate note comparative prices

The methodologies all give separate results for the price which are summarized below:

Price (PKR) as at 31/5/2010 | |

MUFAP | 35,371,725.02 |

Bond Pricing Model | 39,441,311.82 |

Forward Pricing Model | 40,361,144.34 |

The methodology based on MUFAP prices gave the most conservative result whereas that based on the forward pricing model gave the highest valuation.