Interest Rate Sensitivity
The financial markets both impact and are controlled by the interest rate sensitivity. Therefore, understanding the term structure of the interest rates and the intrinsic price of the fixed income security is one of the core aspects of the financial markets. In this paper, we will focus on the valuation of bonds with embedded options such as call and put through binomial interest rate tree. Additionally, we will also comment on the concept of effective duration and how the same affect the bonds. Important to note, while an option free bond can be easily valued using the spot rates, however, the same methodology cannot be applied to bonds with embedded options because any change in the future interest rates will affect the underlying cash flows associated with these bonds and also the probability of exercising call or put options. Therefore, in order to value such bonds, binomial interest rate tree is one suitable method and the same has been discussed below.
Binomial Interest Rate Valuation
The binomial interest rate tree framework assumes that the interest rates have an equal probability of taking one of the two possible values in the next period and the set of possible interest rate paths to be used in order to value the bonds with this binomial model is called binomial interest rate tree.
Valuation of bonds with embedded option
Unlike straight bonds which are valued using spot rates, a callable or putable bond is valued using the one-period forward rates in the binomial tree framework and because the probability of the upside-downside movement of the interest rate is 50%, the value of the bond at any given node will be an average of the present value of the two possible values from the next period.
The above diagram depicts two-period binomial tree with multiple notes for period 1 and period 2. Important to note, a node is a point in time when the interest rates can take one of the two possible paths, i.e. the upper path and the lower path. For instance, in the above diagram, Sod is the interest rate when the spot rate follows the lower path to be Sod, and then follows the upper path to S0ud or Sodd. It is considerable that the interest rates at each node in the binomial tree is the one-period forward rates that correspond to the nodal period. However,each nodal period is related to the one and is 2 standard deviations apart from the adjacent nodel. For instance, the relationship between S0ud and Sodd can be expressed as:
S0ud= Sodd e2*standard deviation
Similarly, non-adjacent forward rates are related to the multiple of two standard deviations. For instance, For instance, the relationship between S0uu and Sodd can be expressed as:
S0uu= Sodd e4*standard deviation
Binomial interest rate tree assumes higher volatility at higher interest rates
Binomial interest rate tree only considers non-negative interest rates
-Valuation of callable bond with binomial interest rate tree
It is considerable that at the time of valuation of callable bond, the value at any nodes where the bond is callable must be either the price at which the bond issuer will call the bond or the computed value, whichever is lower. On the other hand, the value used at any nod corresponding to a put date must be either the price at which the investor will put the bond or the straight value of the bond, whichever is higher. To make the things understandable, we have performed some calculations.
Scenario: A 2-year, 7% annual pay, $100 par bond callable in one year at the price of $100. The interest rate tree under the given scenario is as follows:
Value of the bond at the upper node for year 1:
V= ½((100+7)/1.071826+ 100+7/1.071826)= $99.83
Value of the bond at the lower node for year 1:
V= ½((100+7)/1.053210+ (100+7)/1.053201)= $101.594
Now, since the price at the lower node is higher than the call price of $100. Therefore, in order to value this bond, we will disregard the value of $101.594 and will use the price of $100. Hence, the value of callable bond at current node will be:
V= ½ ((99.83+7) /1.045749+ (100+7) /1.045749) = $102.238
-Valuation of putable bond with binomial interest rate tree
Scenario: A 2-year, 7% annual pay, $100 par bond putable in one year at the price of $100. The interest rate tree under the given scenario is as follows:
Value of the bond at the upper node for year 1:
V= ½((100+7)/1.071826+ 100+7/1.071826)= $99.83
Value of the bond at the lower node for year 1:
V= ½((100+7)/1.053210+ (100+7)/1.053201)= $101.594
Now, since the price at the upper node is lower than the call price of $100, therefore, in order to value this bond, we will disregard the value of $99.83 and will use the price of $100. Hence, the value of putable bond at current node will be:
V= ½((100+7) /1.045749+ (101.594+7) /1.045749) = $103.081
Effective Duration
Modified duration, which measures the price sensitivity of the bond to change in the interest rate changes, assuming there is no change in the cash flows, cannot be applied to bonds with embedded options because any change in the interest rates will change the cash flows if the call or put option is exercised. Therefore, to overcome this problem, effective duration should be used as the same consider the impact of change in the interest rates on the cash flow amount. The formula for calculating effective duration is:
Effective Duration = (P- - P+ ) / [(2)*(P0)*(Y+ - Y-)]
Here:
P0 = Par value of the bond P- = Change in bond price on account of fall in the bond yield
P+ = Change in the bond price on account of rise in the bond yield
(Y+ - Y-) = Change in bond yield
Important to note, since the exercising of call and put options have the potential to reduce the life of the bond, the effective duration of these bonds will be less than the straight or option-free bonds.
Effective Convexity
Just like the modified duration, even the standard measures of convexity cannot be used for bonds with embedded options as the standard measure of convexity can be used to improve changes estimated from modified duration. Henceforth, to overcome this issue, effective convexity should be used:
Effective Convexity= (P- + P+ ) -2*P0/ [(P0)*(Y+ - Y-)2]
References
Kaplan . "Valuation and Analysis of Bonds with Embedded Options." Kaplan. Schweser Notes for CFA Exam- Fixed Income Investment. USA: Kaplan Inc., 2016. 185-195.
Leslie Vaaler, James Daniel. Mathematical Interest Theory. Pearson Education, 2009.
