Are you keeping up with the Early Bird “homework?”

• Early Bird homework from #1 (financial returns).
• Early Bird homework from #2 (volatilities). Since Hull’s volatility chapter is assigned, this is decidedly relevant.

Last Saturday’s Early Bird #3 webinar had one big theme: the first partial derivative. It’s the utility player in risk measurement because we are generally concerned with sensitivities; e.g., how sensitive is a position to a change in interest rates? how sensitive is VaR to a change in the position (marginal VaR)? how sensitive is the portfolio’s variance to a change in the market’s equity risk premium?

If Portfolio Value (P) is a function of several risk factors [rf1, rf2, rf3…, rfn], then abstractly P = function[rf1, rf2, rf3…, rfn].

The first partial derivative with respect to, say, the second risk factor gives the instantaneous rate of change of the portfolio with respect to a (unit) change in the second risk factor. Notice it may be a mental shift to think in terms of risk factors rather than input variables. For example, call option model price is a function of inputs (stock, strike, volatility…). But we can also view model price as a function of these same risk factors. Volatility is then viewed as risk factor. This was an ‘aha’ moment for me because suddenly it made sense that, under a multi-factor model, excess returns are compensation for exposure to various risk factors.

I thought these questions would give nice practice on the first partial derivative:

1. A typical bond price/yield curve plots (maps) the bond price (the dependent variable on the y-axis) to the bond yield (independent variable on the x-axis). For a plain-vanilla bond (without embedded options), what is the proper name for the first derivative of the bond’s price with respect to yield? Can we say anything about its value as yield decreases/increases (e.g., positive increasing)?
2. A callable bond exhibits negative convexity at low yields (why?). We visually recognize negative convexity by its arching back; but how is negative convexity characterized in mathematical (derivative) terms?
3. Under continuous discounting, the price of a bond with face value (F) and maturity of (n) years is a function of yield (y): Price (P) = F*EXP[-y*n]. What is the first partial derivative of this bond’s price with respect to yield? If we divide this quantity itself by price(y), what do we get?
4. Under the Black-Scholes-Merton, the price of a European call option (c) is a function of several risk factors (note I called them ‘risk factors’) including stock price (S), strike price (K), volatility (sigma), interest rate (r), and term to maturity (t). What is called the first partial derivative of the call price with respect to stock price? with respect to volatility? with respect to interest rate? with respect to term?
5. The bond plot (above) maps price (the dependent variable) to yield (the independent variable). What are the units of the first derivative, in the case of the bond? Under the assumption of continuous discounting, what are the units of a modified/Macaulay duration?
6. In the case of the European call option delta, if calculate option delta = 0.7, what are the units? What does it mean to assert that the delta of a call option is 0.7?
7. How can a short position in 1,000 options be made delta neutral when the delta of each option is 0.7? [Hull 14.2]
   You may need the following background:
   1. position delta = delta multiplied by number of options; e.g., 1,000 options with a delta of 0.4 each implies a position delta of 400.
   2. A share has a delta of 1.0, and
   3. to make the position delta neutral is to make the total position delta equal to zero.
8. Why does a share have a delta of 1.0? Can you visualize that with a graph (even if it’s a little silly)?
9. If we succeed in making the short position above delta neutral, what does that imply about the position?
10. Is the delta neutral position perfectly hedged? If not, why not? Can you say that in mathematical (derivative) terms?
11. The CAPM says Expected [return] = Riskless rate + (beta)*(equity risk premium). What is the first derivative of the expected return with respect to to a change in the equity risk premium (ERP)? With respect to a change in beta?
12. In the bond example above, dollar/Macaulay/modified durations are all first derivatives (or functions of the first derivative). What does it mean when we insist that a limitation of duration is that it is only locally accurate?

Answers are here (but first try yourself, please!)

Comments

1. Ignacio 26 Mar 2009

   Sorry, David, where can I find the answers to these questions???

   Than you very much,
   Ignacio

2. David Harper, CFA, FRM, CIPM 07 Apr 2009

   Ignacio - Apologies, I just posted at the link above. Thanks, David

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