# Bond Valuation

Fixed Income Securities and Markets

Question A.1

Given the following bond:

|starting date |30/09/2011 |

|maturity date |30/09/2014 |

|coupon rate |4.00% |

|coupon frequency |annual |

|day count |act/act |

|nominal value |100 |

a) Calculate the price of the security on the 30/09/2011, if the yield to maturity is 5% (NB: Price=PV of future cash flows). b) Given the price and the yield to maturity of the bond, calculate the three components of the (expected) total return of this investment (if you invest 100 Euro). c) What will be the price of this bond after one year (on 30/09/2012) if its yield to maturity is 3%? d) If on 30/09/2012 you decide to sell the bond at the price calculated in the previous question, what will be the return of your investment? How this differs from the expected return calculated in (b)? Comment. Solution

a) Using the pricing formula for bonds:

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b) The three different components are:

a. Coupon: holding the bond until maturity, the investor will receive three coupons of size 4 Euro, therefore the coupon component will be nC=12 b. Capital gain: it is the difference between the price at maturity and the price at purchase, equal to 100-97.26= 2.74 c. The interest on interest is 0.61, from the following formula:

[pic]

The total expected return from this investment is therefore equal to 12+2.74+0.61=15.35. In percentage terms, the return is equal to 15.35 divided by the invested amount (97.26), minus 1, or 15.78%. In annual term the percentage return will be one third of the period return, i.e. 5.12%.

c) Applying again the formula used in (a):

[pic]

d) If the bond is sold on 30.09.2012 at 3% yield to maturity, the realized return of this investment will be: a. Coupon: the bond has been held for one year, therefore the investor will receive one coupon, equal to 4 b. The capital gain will be equal to 101.91-97.26=4.64 c. No interest on interest is gained, because the first coupon is received at the selling date The total realized return of this investment has been 8.64 Euro, which corresponds to an annual return of 8.64%. The annual return is higher then the expected return because the second component (capital gain) has been higher than expected. The decrease in the (required) yield to maturity on this bond has caused an appreciation of 4.64 Euro in one year, while the investor expected a capital gain of 2.74 in three years.

Question A.2

Suppose that a bond is purchased between coupon periods. The days between the settlement date and the next coupon period is 115. There are 183 days in the coupon period. Suppose that the bond purchased has a coupon rate of 7.4% and there are 10 semi-annual coupon payments remaining. a) What is the dirty price for this bond if a 5.6% discount rate is used? b) What is the accrued interest for this bond?

c) What is the clean price?

Solution

The fundamental pricing formula can be used to find the dirty price of the bond. In particular, the dirty price is equal to the present value of all future cash flows due to the bond holder. The present value of each cash flow is calculated in the table below. Both coupon rate and yield to maturity considered in the calculation are semi-annual:

|days (from settlemt) |Event |cash flow |yield to maturity |PV | |-68 |last coupon | | | | |0 |settlement date | | | | |115 |coupon |3.7 |2.80%...

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