# Probability of default ^ bond rating

* 1-Y Probability of default (denoted PD) is defined for a single issuer. * Rating (like AA) is defined for one bond among many by the same issuer.

Jon Frye confirmed that an AA rating is not an expression of the firm’s status/viability/strength/health.

I guess the rating does convey something about the LossGivenDefault, another attributes of the bond not the issuer.

# BUY a (low) interest rate = Borrow at a lock-in rate

Q: What does “buying at 2% interest rate” mean?

It’s good to get an intuitive and memorable short explanation.

Rule — Buying a 2% interest rate means borrowing at 2%.

Rule — there’s always a repayment period.

Rule — the 2% is a fixed rate not a floating rate. In a way, whenever you buy you buy with a fixed price. You could buy the “floating stream” …. but let’s not digress.

Real, personal, example — I “bought” my first mortgage at 1.18% for first year, locking in a low rate before it went up.

# factors affecting bond sensitivity to IR

In this context, we are concerned with the current market value (eg a \$9bn bond) and how this holding may devalue due to Rising interest rate for that particular maturity.

* lower (zero) coupon bonds are more sensitive. More of the cash flow occurs in the distant future, therefore subject to more discounting.

* longer bonds are more sensitive. More of the cashflow is “pushed” to the distant future.

* lower yield bonds are more sensitive. On the Price/yield curve, at the left side, the curve is steeper.

(I read the above on a slide by Investment Analytics.)

Note if we hold the bond to maturity, then the dollar value received on maturity is completely deterministic i.e. known in advance. There are 3 issues with this strategy:

1) if in the interim my bond’s MV drops badly, then this asset offers poor liquidity. I won’t have the flexibility to get contingency cash out of this asset.

1b) Let’s ignore credit risk in the bond itself. If this is a huge position (like 2 trillion) in the portfolio of a big organization (even for a sovereign fund), a MV drop could threaten the organization’s balance sheet, credit rating and borrowing cost. Put yourself in the shoes of a creditor. Fundamentally, the market and the creditors need to be assured that this borrower could safely liquidity part of this bond asset to meet contingent obligations.

Fundamental risk to the creditors – the borrower i.e. the bond holder could become insolvent before bond maturity, when the bond price recovers.

2) over a long horizon like 30Y, that fixed dollar amount may suffer unexpected higher inflation. I feel this issue tends to affect any long-horizon investment.

3) if in the near future interest rises sharply (hurting my MV), that means there are better ways to invest my \$9bn.

# CEV, very briefly

CEV has a formulation of sigma(t). It's a function of the current underlier price. A power function, to model the volatility smile…

SABR is related to CEV…

# Hull-White — learning notes

Hull white is simpler than LMM. Both Arbitrage free.

LMM has too many factors. Hull White can be one-factor or 2-factor.

# OIS instruments in YC bootstrapping – Jeff@UChicago

Jeff’s lecture notes has numerous practical tips, such as
*

# fwd libor rate =\= instantaneous rate

Fwd libor rate , like a 20Y forward-start 3-month Libor has an accrual period of 3 months.

Similarly, the instantaneous fwd rate has an underlying loan with accrual period shrunk to 1 picosecond…

# zero curve doesn’t mean discount curve

Zero curve usually means the zero bond yield curve. I think it’s usually upward sloping.

I think in STIRT/Sprite zero curve means the discount factor curve, so it’s decreasing with maturity and always below 1.0.

A YC can be interchangeably represented as zero curve, discount curve, fwd curve (i.e. inst fwd rate) etc.

# Gaussian HJM, briefly

… is a subset of HJM models.

An HJM model is Gaussian HJM if vol term is deterministic. Note “vol” term means the coefficient of the dW term. Every Brownian motion must always refer to an implicit measure. In this case, the RN measure.

How about the drift term i.e. the “dt” coefficient? It too has to be deterministic to give us a Gaussian HJM.

Well, Under the RN measure, the drift process is determined completely by the vol process. Both evolve with time, but are considered slow-moving [1] relative to the extremely fast-moving Brownian Motion of “dW”. Extremely because there’s no time-derivative of a BM

[1] I would say “quasi constant”

Language is not yet precise so not ready to publish on recrec…