Question on Kaplan's question on Binomial Trees

BobBarkerPlaysPlinko

An example of a standard tree used by FIData is given in Figure 2.

Figure 2: Binomial Interest Rate Tree

Year 0	Year 1
4.5749%	7.1826%
	5.3210%

FIData's website uses rates in Figure 2 to value a two-year, 5% annual-pay coupon bond with a par value of $1,000 using the backward induction method.

The question is: Using the backward induction method, the value of the 5% annual-pay bond using the interest rate tree given in the three bonds in Figure 2 is closest to:

A)$900.

$945.

C)$993.

So I averaged node U2 (1005/1.071826) and node L2 (1005/1.05321) to come up with 943.6846.  Then I discounted that by the year 0 rate of 4.5749% and came up with 907.1819.  I answered A.  

Kaplan says the answer is C.  Their explanation is:  The value of the 5%, two-year annual pay $1000 par bond is $992.88.

I find that explanation underwhelming.  Can someone help me out here?  Where did I screw up?

Sophie

Hi @BobBarkerPlaysPlinko, here's my method:

I've ignored the $ value of the bond for now, and work on a % basis (of par) to simplify things:

Using your definition of nodes:
U2 = 105/1.071826 = 97.9637
L2 = 105/105321 = 99.6952

Then fair value of bond today, V0 = 0.5 * [(97.9637+5)/1.045749] + 0.5 * [(99.6952+5)/1.045749]
                                                    = 0.5 * [98.4593+100.1150]
                                                    = 99.2872

So 99.2872% of a par value of $1,000 should be $993, i.e. answer C. 

Hope this helps!