［2008］Topic 31: The Science of Term Structure Models相关习题 principles, backward, interest, multiple, possible

Baran

 

AIM 3: Explain how the principles of arbitrage pricing of derivatives on fixed income securities can be extended over multiple periods.

 

1、With respect to interest rate models, backward induction refers to determining:

A) convexity from duration.

B) duration from convexity.

C) one portion of the yield curve from another portion.

D) the current value of a bond based on possible final values of the bond.

Baran

 

The correct answer is D

 

Backward induction refers to the process of valuing a bond using a binomial interest rate tree. For a bond that has N compounding periods, the current value of the bond is determined by computing the bond’s possible values at period N and working "backwards."

Baran

 

2、A binomial model or any other model that uses the backward induction method cannot be used to value a mortgage-backed security (MBS) because:

A) the cash flows for an MBS only depend on the current rate, not the path that rates have followed.

B) the prepayments occur linearly over the life of an interest rate trend (either up or down).

C) interest rates are irrelevant to the value of a mortgage-backed security.

D) the cash flows for the MBS are dependent upon the path that interest rates follow.

Baran

 

The correct answer is D

 

A binomial model or any other model that uses the backward induction method cannot be used to value an MBS because the cash flows for the MBS are dependent upon the path that interest rates have followed.

Baran

 

3、A bond with a 10 percent annual coupon will mature in two years at par value. The current one-year spot rate is 8.5 percent. For the second year, the yield volatility model forecasts that the one-year rate will be either eight or nine percent. Using a binomial interest rate tree, what is the current price?

A) 103.572.

B) 101.837.

C) 101.761.

D) 102.659.

Baran

 

The correct answer is D

The tree will have three nodal periods: 0, 1, and 2. The goal is to find the value at node 0. We know the value in nodal period 2: V2=100. In nodal period 1, there will be two possible prices:

V1,U=[(100+10)/1.09+(100+10)/1.09]/2= 100.917

V1,L=[(100+10)/1.08+(100+10)/1.08]/2= 101.852

Thus

V0=[(100.917+10)/1.085+(101.852+10)/1.085]/2= 102.659

Baran

 

4、A binomial interest-rate tree indicates a 6-month period spot rate of 3.5 percent. The price of the zero-coupon bond if rates decline is 97.25 and if rates increase the bond price is 95.875. If the bond’s market price is 94.5, the risk-neutral probabilities with a decline and increase in rates, respectively, are closest to:

A) 0.1/0.9.

B) 0.9/0.1.

C) 0.2/0.8.

D) 0.8/0.2.

Baran

 

The correct answer is C

Baran

 

5 、Patrick Wall is a new associate at a large international financial institution. His boss, C.D. Johnson, is responsible for familiarizing Wall with the basics of fixed income investing. Johnson asks Wall to evaluate the two otherwise identical bonds shown in Table 1. The callable bond is callable at par and exercisable on the coupon dates only.

Wall is told to evaluate the bonds with respect to duration and convexity when interest rates decline by 50 basis points at all maturities over the next six months.

Johnson supplies Wall with the requisite interest rate tree shown in Figure 1. Johnson explains to Wall that the prices of the bonds in Table 1 were computed using the interest rate lattice. Johnson instructs Wall to try and replicate the information in Table 1 and use his analysis to derive an investment decision for his portfolio.

Table 1 Bond Descriptions
	Non-callable Bond	Callable Bond
Price	$100.83	$98.79
Time to Maturity (years)	5	5
Time to First Call Date	--	0
Annual Coupon	$6.25	$6.25
Interest Payment	Semi-annual	Semi-annual
Yield to Maturity	6.0547%	6.5366%
Price Value per Basis Point	428.0360	--

Figure 1.

									15.44%
								14.10%
							12.69%		12.46%
						11.85%		11.38%
					9.75%		10.25%		10.05%
				8.95%		9.57%		9.19%
			7.91%		7.88%		8.28%		8.11%
		7.35%		7.23%		7.74%		7.42%
	6.62%		6.40%		6.37%		6.69%		6.54%
6.05%		5.95%		5.85%		6.25%		5.99%
	5.36%		5.17%		5.15%		5.40%		5.28%
		4.81%		4.73%		5.05%		4.83%
			4.18%		4.16%		4.36%		4.26%
				3.82%		4.08%		3.90%
					3.37%		3.52%		3.44%
						3.30%		3.15%
							2.84%		2.77%
								2.54%
									2.24%
Years	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5

Given the following relevant part of the interest rate tree, the value of the callable bond at node A is closest to:

	3.44%
3.15%
	2.77%

Corresponding part of the callable bond tree:

		$100.00
A ====>	-
		$100.00

Baran

The value of the bond at node A is closest to:

A) $101.53.

B) $103.56  

C) $100.00.  

D) $104.60.

 

 

The correct answer is C

The value of the callable bond at node A is obtained as follows:

Bond Value = the lesser of the Call Price or {0.5 x [Bond Value_up + Coupon/2] + 0.5 x [Bond Value_down + Coupon/2]}/(1+ Interest Rate/2)]

So we have

Bond Value at node A = the lesser of either $100 or  {0.5 x [$100.00 + $6.25/2] + 0.5 x [$100.00+ $6.25/2]}/(1+ 3.15%/2) = $101.52.  Since the call price of $100 is less than the computed value of $101.52 the bond price would be $100 because once the price of the bond reached this value it would be called.