Question

1. [10pts] Consider the following bond (assume: credit risk free, no embedded options, pays interest semiannually):

Coupon	9%
YTM	8%
Term (yrs)	5
Par	100.00

Compute the following

[1pt] The price value of a basis point;

[4pts] The modified duration and convexity;

[1pt] Calculate the exact price change for a 100bp increase in the bond yield;

[1pts] Using duration only, estimate the price of the bond for a 100bp increase in yields;

[1pts] Using duration and convexity, estimate the price change of the bond for a 100bp increase in yields;

[2pts] Without an actual calculation, indicate (“explain”) whether the duration of the bond would be higher or lower if the YTM were 10%, rather than 8%. Hint: recall that the price-yield curve is convex for risk- and option-free bonds Also, think in terms of Macauley duration as a weighted time average.

[10pts] Stratify the following Agency mortgages at a deal coupon of 4%. What are the initial pool, pass-through, and PO principal amounts, and what are the initial notional principal and coupon of the IO?

Gross	Net	Balance (mln $)
4.10%	3.852%	50.343
4.20%	3.934%	101.435
4.30%	4.071%	123.777
4.40%	4.153%	40.123

[20pts]

[18pts] Consider a 100,000,000 CMBS pass-through (PT) security consisting of fresh 15-year fixed rate loans that would fully amortize over a period of 30 years, and have a WAC of 7%, with fees amounting to 0.5%. For purposes of this HW, a CMBS is like a RMBS, except that all loans have a prepayment lock-out (e.g., defeasance, assume for the entire term), but default occurs according to the standard residential SDA function. Additionally, since the loans do not fully amortize, there is generally a balloon loss - quoted as a fraction of the outstanding balance at mortgage maturity. Assume that the PT is sold at 94.345. Allow for loans to default at the PSA’s SDA CDR, and allow for balloon risk. The current (corresponding maturity) Treasury yield is 5%. Create a table and graph of the spread (in bps) of cash flow yield to Treasury versus SDA, for 0% and 10% balloon loss. In the spreadsheet that you hand-in, show the situation for 100 SDA and 10% balloon loss. See notes below for further assumptions and hints.

[2pts] Why is there generally a large fraction of outstanding principal that defaults at maturity?

Notes for question 3:

Note that there is no messy tranching going on here; it is pass-through.

Allow for a general SDA; see the graph and spreadsheet in your lecture notes.

Here is one way to get the CDR (as there are better versions, feel free to use your own, extra points if you can show me that the formula below is wrong); “A13” refers to the month and “SDA” to the SDA factor: =(IF(A13<=30,A13*0.006/30,0)+IF(AND(A13>30,A13<=60),0.006,0)+IF(AND(A13>60,A13<=120),0.006-(0.006-0.0003)/(120-60)*(A13-60),0)+IF(A13>120,0.0003,0))*SDA/100

Assume that loss recovery is zero – any loan that defaults is a completely written down.

Defaults work much like prepayment, with one exception: the monthly default happens just before the payment (why pay interest and principal if you are defaulting anyway). So, take default = beginning balance × MDR and then amortize the “non-defaulted” balance over the remaining number of payments to get the realized “mortgage payment.”

Make sure you create a column of the PT cash flows, including the initial investment, so that you can calculate the CF yield.

Balloon losses are quoted as a fraction of the final outstanding principal.

In your graph, show a range of SDA’s of 0 to 1,000. You will find that you do not have to compute very many values in that range.

Display all cash flows to the nearest dollar (use format ? cell).

[30pts] Consider a 1,000,000 (for convenience) GNMA pass-through security consisting of fresh 30-year fixed rate, fully amortizing mortgages, with a WAC of 7%, and servicing/guarantee fees totaling 0.5%. Allow for a general PSA prepayment rate.

[10 pts] Construct an IO/PO stripped MBS from this MPT above, showing the total cash flows to investors of each piece. For simplicity, price each piece so that the IRR at PSA 165 equals the MPT coupon. You must show your amortization table at a 165 PSA.

[15 pts] Suppose you bought the tranches at the rice obtained in part a., but that mortgage rates (MR) suddenly change just after the purchase (just after t=0). Make a graph of the market value of the MPT and each piece as the MR varies from 2% to 12% in steps of 1%. To simplify, use MR less fees as the discount rate (DR) in your calculation. Also, to make things a little more realistic, assume that prepayment PSA changes with MR as

PSA = 165*max(1,WAC/MR).

Hand in your amortization table for the case when MR=7%.

[5pts] Explain the shape of each of the three curves you obtain in part b.

Answer 1

answer 1
price=	coupon(pvaf ytm,5)+redemption(pvif ytm,5)
=	103.9923
Modified duration and covexity
year	cash flow	pv @8%	pv of cash flow	year*pv
1	9	0.925926	8.333333	8.333333
2	9	0.857339	7.716049	15.4321
3	9	0.793832	7.14449	21.43347
4	9	0.73503	6.615269	26.46107
5	109	0.680583	74.18357	370.9178
			103.9927	442.5778
duration=	442.5778/103.9927
=	4.255854
Modified duration=		4.255854/1.08
		3.940606
convexity=	1/B(d*d(Br )/d*r*r)
B=bond price
r=interest rate
d= duration
c=	1/103.9923(3.940606*3.940606(103.9923*9%)/3.940606*9%*9%)
=	43.78451
Price change for 100 basis point increase in yield
duration=	3.940606
change=	duration * basis point change
=	0.039406	%
change in price=		4.097927
new price=	initial price-change
=	99.89437
price change=	4.097927
Using duration only price
price=	initial price -change in price
=	99.89437