Derivatives - Exam 2

[BASIS] Basis: definition

Spot price minus futures price: b = S - F.

[BASIS] What do b1 and b2 stand for?

b1 = S1 - F1 (basis when the hedge is set up, time t1). b2 = S2 - F2 (basis when the hedge is closed out / asset is bought or sold, time t2).

[BASIS] What do the subscripts 1 and 2 mean on S, F and b?

1 = time the hedge is SET UP. 2 = time the hedge is CLOSED OUT (asset purchased or sold).

[BASIS] Long hedge: when do you use it and what futures position do you take?

When you will BUY an asset later and want to lock in the cost. Take a LONG futures position.

[BASIS] Short hedge: when do you use it and what futures position do you take?

When you will SELL an asset later (or are expecting to receive it) and want to lock in the price. Take a SHORT futures position.

[BASIS] Example type: a company expects to RECEIVE foreign currency (yen) in a few months. Which hedge?

Short hedge (Hull 3.1: short Sep yen futures).

[BASIS] Example type: a company needs to PURCHASE crude oil in a few months. Which hedge?

Long hedge (Hull 3.2: long Dec crude futures).

[BASIS] Gain on futures: long hedge

F2 - F1

[BASIS] Gain on futures: short hedge

F1 - F2

[BASIS] Shortcut for gain on futures (any position)

Long = (price you sell at) - (price you bought at) = F2 - F1. Short = (price you sold at) - (price you buy back at) = F1 - F2.

[BASIS] How does each hedger close out the futures?

Long hedger closes out by SELLING the futures. Short hedger closes out by BUYING the futures back.

[BASIS] Long hedge: net amount paid

S2 - (F2 - F1) = F1 + b2

[BASIS] Short hedge: net amount received

S2 + (F1 - F2) = F1 + b2

[BASIS] Fill in the blank: both long and short hedges give an effective price of ____

F1 + b2

[BASIS] Why is the effective price F1 + b2 and not just F1?

Because the basis at close-out (b2) is not known when the hedge is set up. That uncertainty is basis risk.

[BASIS] When is a hedge perfect?

When b2 = 0, so the effective price is exactly F1.

[BASIS] Two ways to compute the effective price

(1) Spot price at close-out adjusted by the futures gain/loss: S2 - (F2 - F1) for long, S2 + (F1 - F2) for short. (2) F1 + b2.

[BASIS] [EXTRA, not on slides] Short hedger: does a strengthening or weakening basis help?

Strengthening (b rises). The short hedger receives F1 + b2.

[BASIS] [EXTRA, not on slides] Long hedger: does a strengthening or weakening basis help?

Weakening (b falls). The long hedger pays F1 + b2.

[BASIS] [EXTRA, not on slides] Sources of basis risk (ALL three)

1) Hedged asset differs from the asset underlying the futures (cross hedge). 2) Uncertainty about exactly when the asset will be bought or sold. 3) Hedge has to be closed out before futures expiration.

[HEDGE RATIO] Minimum variance hedge ratio h*

h* = rho x (sigma_S / sigma_F)

[HEDGE RATIO] Optimal number of contracts (units version)

N* = h* x (QA / QF)

[HEDGE RATIO] What is sigma_S?

Standard deviation of the CHANGE in the spot price (delta S).

[HEDGE RATIO] What is sigma_F?

Standard deviation of the CHANGE in the futures price (delta F).

[HEDGE RATIO] What is rho in the hedge ratio?

The correlation coefficient between changes in the spot price and changes in the futures price.

[HEDGE RATIO] What are QA and QF?

QA = size of the position being hedged (units). QF = size of ONE futures contract (units).

[HEDGE RATIO] What do h* and N* each represent?

h* = optimal hedge ratio. N* = optimal number of futures contracts.

[HEDGE RATIO] Percent-change version: formulas

h-hat = rho-hat x (sigma-hat_S / sigma-hat_F), using % daily changes in value. N* = h-hat x VA / VF.

[HEDGE RATIO] In the percent-change version, what are the inputs?

sigma-hat_S and sigma-hat_F = std dev of % daily changes in spot and futures. rho-hat = correlation between those % changes.

[HEDGE RATIO] What are VA and VF?

VA = value of the position being hedged (spot price x QA). VF = value of the futures position (futures price x QF), i.e. one contract.

[HEDGE RATIO] Which N* formula goes with which inputs?

Changes in PRICE and units: N* = h* x QA/QF. % changes in VALUE: N* = h-hat x VA/VF.

[HEDGE RATIO] When does h* = 1?

When rho = 1 and sigma_S = sigma_F (perfect correlation and equal volatility), the hedge is 1-for-1.

[HEDGE RATIO] N* comes out as a fraction (like 34.82). What do you do?

Round to the nearest whole contract (35).

[HEDGE RATIO] Beef producer buying cattle in a month (Problem 3.16): long or short?

Long futures (buying the asset later is a long hedge).

[HEDGE RATIO] Airline buying jet fuel using heating oil futures (Hull 3.3): long or short, and what kind of hedge is it?

Long. It is a cross hedge (jet fuel hedged with a different asset, heating oil).

[HEDGE RATIO] Where do hedge ratio inputs (sigma, rho, h) come from?

Historical data. They are estimates, so the hedge is a best guess for the future.

[EQUITY] What risk does a stock index futures hedge target?

Systematic (market) risk in the portfolio, measured by beta from CAPM.

[EQUITY] Number of index futures contracts to hedge a portfolio

N* = beta x VA / VF

[EQUITY] To hedge a long stock portfolio with index futures, long or short?

SHORT index futures.

[EQUITY] Definition of beta (from the slides)

Slope of the line of best fit when the portfolio's excess return over the risk-free rate is regressed against the index's excess return over the risk-free rate.

[EQUITY] VA and VF in the equity hedge

VA = value of the portfolio. VF = value of ONE futures contract = futures price x multiplier.

[EQUITY] Example: index futures at 1,010 with $250 multiplier. VF?

250 x 1,010 = $252,500.

[EQUITY] CAPM: expected portfolio return

Rf + beta x (Return on index - Rf)

[EQUITY] Hull equity example: the index return used includes what?

The price change PLUS the dividend yield (for the period). Index 1000 to 900 with a 1% annual dividend over 3 months = -10% + 0.25% = -9.75%.

[EQUITY] Annual rates over a 3-month horizon

Divide by 4 (risk-free 4% per year is 1% for 3 months; dividend yield 1% per year is 0.25%).

[EQUITY] Hull equity example: what do you add to the portfolio's expected value to get the hedged position's value?

The gain on the short futures position (portfolio value after 3 months + futures gain).

[EQUITY] Changing portfolio beta from beta to beta*: if beta > beta*

Take a SHORT position in (beta - beta*) x (VA/VF) contracts.

[EQUITY] Changing portfolio beta from beta to beta*: if beta < beta*

Take a LONG position in (beta* - beta) x (VA/VF) contracts.

[EQUITY] Memory trick for changing beta

LOWER the beta means SHORT. RAISE the beta means LONG. Size = (difference in betas) x VA/VF.

[EQUITY] How do you drive a portfolio's beta to zero?

Short beta x (VA/VF) contracts (beta* = 0).

[EQUITY] "Carve out your ability to outperform the market": steps

1) Pick stocks you believe will outperform the market. 2) Short a beta-adjusted number of index futures: beta x (VA/VF). Goal: outperform the beta-adjusted market index.

[EQUITY] In the carve-out strategy, what is beta and VF?

Beta = beta of your stock portfolio relative to the index futures used to hedge. VF = value of one futures contract.

[EQUITY] E-mini example: stock rises more than index (beta-adjusted). What happens?

Gain on the long stock (+$100,000) exceeds the loss on the short futures (-$60,000), net +$40,000. The hedge strips out market moves so you keep the outperformance.

[EQUITY] E-mini S&P 500 contract size

$50 times the index (example: futures at 2,500, VF = 125,000).

[STACK AND ROLL] What is stack and roll?

A strategy used when the hedger wants to hedge to an expiry farther out than liquid futures contracts are trading.

[STACK AND ROLL] How does it work (steps)?

1) Enter "stacks" of liquid, shorter-dated futures as the hedge. 2) Before they mature, close them out and take a position in liquid shorter-dated futures with a later delivery date (roll forward). 3) Repeat until the desired hedge expiry is matched.

[BONDS] Bond: what is it?

A debt security issued by a borrower to obtain funds, bought and held by investors to earn a return. Most pay regular coupons plus face value at maturity.

[BONDS] Bond price formula

P = sum from t=1 to n of C/(1+r)^t + M/(1+r)^n

[BONDS] Bond price formula: variables

P = price, C = coupon payment ($), r = periodic interest rate (decimal), M = maturity value, n = number of periods, t = period when the payment is received.

[BONDS] Why is price inversely related to yield?

A bond's price is the present value of its cash flows. Raise the discount rate and PV falls.

[BONDS] Shape of the price/yield curve

Nonlinear, convex, inverse (downward sloping and curved).

[BONDS] Yield to maturity (YTM)

The internal rate of return: the rate that makes the PV of the cash flows equal the price.

[BONDS] Yield to call

The IRR when the cash flows terminate on the call date.

[BONDS] Current yield

Coupon / Price

[BONDS] Problem with YTM

It assumes all cash flows are reinvested at the same rate (the YTM), i.e. a flat yield curve.

[BONDS] What should the cash flows be discounted at instead of the YTM?

The rates at which each cash flow could actually have been invested out to that date.

[BONDS] [EXTRA, not on slides] Coupon vs yield: par, premium, discount

Coupon = yield: price 100 (par). Coupon > yield: premium. Coupon < yield: discount. (See Duration Movements table: price = 100.00 when coupon = yield.)

[BONDS] Macaulay duration: meaning

The average weighted time until cash flow payments are received.

[BONDS] Macaulay duration of a zero-coupon bond

Equal to its maturity.

[BONDS] Macaulay duration formula

D = sum[t x Ct/(1+r)^t] / sum[Ct/(1+r)^t] (PV-weighted average of payment times)

[BONDS] Cash flow / duration diagram: what does the triangle represent?

Duration is the balance point (fulcrum) of the PV-weighted cash flows. The big final payment pulls it toward maturity.

[BONDS] Duration calc example (slide): 4% coupon, 10 years, 3% yield. Total PV and total t x PV?

Total PV of cash flows = 108.5302. Total of t x PV = 923.4499.

[BONDS] Duration calc example: duration?

923.4499 / 108.5302 = 8.5087 years.

[BONDS] Effects on duration: maturity UP

Duration UP

[BONDS] Effects on duration: yields UP

Duration DOWN

[BONDS] Effects on duration: coupons UP

Duration DOWN

[BONDS] Effects on duration: coupon frequency UP

Duration DOWN

[BONDS] Memory trick for duration

Only MATURITY moves duration the same direction. Yield, coupon and coupon frequency all move it the OPPOSITE direction.

[BONDS] ALL factors that INCREASE duration

Longer maturity, lower yields, lower coupons, less frequent coupon payments.

[BONDS] ALL factors that DECREASE duration

Shorter maturity, higher yields, higher coupons, more frequent coupon payments.

[BONDS] Modified duration: meaning

Percentage change in the FULL price of a bond for a 100 bp change in yield. A measure of percentage price volatility.

[BONDS] Modified duration: what kind of derivative?

First-order derivative; based on the slope at a particular point on the price/yield curve.

[BONDS] Modified duration formula

D_mod = D / (1 + y/f) (D = Macaulay duration, y = yield in decimal, f = coupon payments per year)

[BONDS] Mod duration example (slide): D = 8.256, yield 4.03742%, semiannual

8.256 / (1 + 0.0403742/2) = 8.093

[BONDS] Mod duration practice: D = 7.5, y = 5%, semiannual

7.5 / (1 + 0.05/2) = 7.5 / 1.025 = 7.317

[BONDS] Duration Movements table: patterns (ALL)

At a given maturity, mod duration falls as yield rises and falls as coupon rises. Longer maturity means higher mod duration. Price = 100.00 whenever coupon = yield.

[BONDS] Basis point value (BPV): definition

Change in the FULL price of a bond from a 1 bp (.01%) change in yield to maturity.

[BONDS] Other names for BPV

"Price value of a basis point", "dollar value of a .01", DV01 (DV'01).

[BONDS] BPV formula

BPV = Mod Dur / 100 x Full Price

[BONDS] BPV units: how to plug in mod duration

Write mod duration as a percent, then as a decimal: 8.093% = 0.08093. So BPV = 0.08093 / 100 x Full Price. Shortcut: ModDur x Full Price / 10,000.

[BONDS] BPV example (slide): mod dur .08381 (decimal), full price 90.204484

0.08381 / 100 x 90.204484 = 0.07560 per $100 face. This matches Bloomberg DV01 of 756 per $1MM.

[BONDS] Converting BPV per $100 face to DV01 per $1MM face

Multiply by 10,000 (0.0756 becomes 756; 0.07451 becomes 745).

[BONDS] Flat price vs full price

Full (dirty) price = flat (clean) price + accrued interest.

[BONDS] Which price does BPV use?

The FULL price.

[BONDS] Accrued interest formula (semiannual Treasury)

(Annual coupon / 2) x (days since last coupon / days in the coupon period)

[BONDS] Practice Problem 1 (slide): accrued interest for 2 7/8% note, 157 of 184 days

2.875/2 x (157/184) = 1.226563

[BONDS] Treasury quote 90-27 in decimal

90 + 27/32 = 90.84375

[BONDS] Practice Problem 1: full price

90.84375 + 1.226563 = 92.070313

[BONDS] Practice Problem 1: BPV

8.093% / 100 x 92.070313 = 0.07451 per $100 face

[BONDS] Practice Problem 1: all steps in order

1) Flat price (convert quote). 2) Accrued interest. 3) Full price = flat + accrued. 4) Mod duration = D / (1 + y/2). 5) BPV = mod dur / 100 x full price.