Lecture Notes 2, Bonds (Ch 4) and Hedging (Ch 3)

K Foster, CCNY, Spring 2010

 

 

Learning Outcomes (from CFA exam)

Students will be able to:

§         interpret interest rates as required rate of return, discount rate, or opportunity cost;

§         calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding;

§         solve time value of money problems when compounding periods are other than annual;

§         calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;

§         draw a time line and solve time value of money applications (for example, mortgages and savings for college tuition or retirement).

§         describe the characteristics and calculate the gain/loss of forward rate agreements (FRAs);

§         calculate and interpret the payoff of an FRA, and explain each of the component terms;

§         define European option, American option, and the concept of moneyness of an option;

§         differentiate between exchange-traded options and over-the-counter options;

§         define intrinsic value and time value and explain their relationship;

 

 

 

On using these Lecture Notes:

 

We sometimes don't realize the real reason why our good habits work.  In the case of taking notes during lecture, this is probably the case.  You're not taking notes in order to have some information later.  If you took your day's notes, ripped them into shreds, and threw them away, you would still learn the material much better than if you hadn't taken notes. 

 

The process of listening, asking "what are the important things said?," answering this, then writing out the answer in your own words  that's what's important! 

 

So even though I give out lecture notes, don't stop taking notes during class.  Take notes on podcasts and video lectures, too.  Notes are not just a way to capture the fleeting sounds of the knowledge that the instructor said, before the information vanishes.  Instead they are a way for your brain to process the information in a more thorough and more profound way.  So keep on taking notes, even if it seems ridiculous.  The reason for note-taking is to take in the material, put it into your own words, and output it.  That's learning.

 

 

From Hull Chapter 4

 

Remember basic definitions: a "basis point" is one-hundredth of a percentage point.  So if the Fed cut rates by one half of one percent (say, from 4.25% to 3.75%) then this is a cut of 50 basis points (bp, sometimes pronounced "bip") from 425 bp to 375 bp.  Ordinary folks with, say, $1000 in their savings accounts don't see much of a change (50 bp less means $5) but if you're a major institution with $100m at short rates then that can get into serious money: $500,000.

 

A dollar today is not worth a dollar in the future, even without any inflation.  Remember that the true cost of something is its opportunity cost: what must be given up in order to get it.  To get a dollar in one year I don't need to give up a dollar today, not when I can put about 97 cents into the bank and, with 3% interest, get $1 in a year's time.

 

There are 2 basic issues that we must address: the rate of compounding and the fact that interest rates change over time.

 

Rate of Compounding

We often use continuously-compounded interest, so that an amount invested at a fixed interest rate grows exponentially.  Unless you've read the really fine print at the bottom of some loan document, you probably haven't given much thought to the differences between the various sorts of compounding  annual, semi-annual, etc.  Do that now:

 

If $1 is invested and grows at rate R then

annual compounding means I'll have

(1 + R) after one year.

If $1 is invested and grows at rate R then

semi-annual compounding means I'll have

 after one year.

"

compounding 3 times means I'll have

 after one year.

…

…

…

"

compounding m times means I'll have

 after one year.

"

…

…

"

continuous compounding (i.e. letting  ) means I'll have

eR after one year.

 

This odd irrational transcendental number, e, was first used by John Napier and William Outred in the early 1600's; Jacob Bernoulli derived it; Euler popularized it.  It is  or .  It is the expected minimum number of uniform [0,1] draws needed to sum to more than 1.  The area under  from 1 to e is equal to 1.

 

Sometimes we write eR; sometimes exp{R} if the stuff buried in the superscript is important enough to get the full font size.

 

Since interest was being paid in financial markets long before the mathematicians figured out natural logarithms (and computing power is so recent), many financial transactions are still made in convoluted ways. 

 

For an interest rate is 5%, this quick Excel calculation shows how the discount factors change as the number of periods per year (m) goes to infinity:

m per year

(1+R/m)^m

Discount Factor

1

1.05

0.952380952

2

1.050625

0.951814396

4

1.0509453

0.951524275

12

1.0511619

0.951328242

250

1.0512658

0.95123418

360

1.0512674

0.951232727

 

 

 

infinite

1.0512711

0.951229425

 

So going from 12 intervals (months) per year to 250 intervals (business days) makes a difference of one basis point; from 250 to an infinite number (continuous discounting) differs by less than a tenth of a bp.

 

A bond should be worth the present discounted value of its cash flows.  Actually any financial instrument (if the flows are known with certainty) ought to be, which is why we want to work out the discounting on simple bonds before getting to more complicated payoffs. 

 

What is present discounted value (PDV)?

 

Recall that we first determine the schedule of cash flows and then discount those back to the present.  The schedule of cash flows can be most easily represented as a timeline, so for example payments of $100 in 3 months, $200 in 6 months, and $300 in 9 months, then a payment of $5000 in 12 months, would be represented as:

But we need to be precise about the units.  The $100 in 3 months is not 100 of today-dollars but 100 of today-plus-3-months-dollars.  You might not usually make a precise distinction but, for a large institution, the difference is important.  Cash has an opportunity cost.  So we could write the present discounted value as:

 $T=.25 100 + $T=.5 200 + $T=.75 300 + $T=1 5000.

(Note that time is measured in years so 3 months is  of a year.)

 

But adding different units gives us a meaningless result  we need to get these values into common units.  We want to find some ratio (a discount factor) that gives  for each future date, T.

 

So the PDV would be calculated as these cash flows times some function that accounts for the time delay.  Of course we can't have just time  economics says that just a quantity is never enough, we need a price too!   What is the price of time?  Sounds like a deep philosophical issue but economists answer "interest rate."

 

So we include the quantity (time) and its price (the interest rate) into a function which we will (for now) label PDV(T,r).  This function gives the ratio, .

 

So the PDV of the bond with the timeline shown above is

 

We measure T in years so we can then substitute in for the different values, getting:

 

 

What is the function, PDV(T,r)?  It can take one of two basic forms, depending on whether we are considering continuous time or discrete time periods.  In continuous time , sometimes written as exp{-rT} so that we don't need to squint at the superscript.  We'll assume continuous time for a while here.

 

This would be simple if the interest rate, r, were constant over the entire time period.  We will start with that simple case but we want to work up to the more complicated version.   So for the example above, if the interest rates are unchanging over the next 12 months then r.25 = r.5 = r.75 = r1 = r and this function is:

=  

                = 100 e-.25r + 200 e-.5r + 300 e-.75r + 5000 e-r

 

If r = 0.06 (a 6% interest rate) then we can easily calculate this value using a table in Excel. 

A

B

C

D

      E

T=

r=

discount factor

cash flow

discounted cash flow

0.25

0.06

0.985112

100

98.511194

 

0.5

0.06

0.970446

200

194.08911

 

0.75

0.06

0.955997

300

286.79924

 

1

0.06

0.941765

5000

4708.8227

 

 

 

 

 

 

 

 

 

 

 

5288.2222

sum of discounted cash flows

 

First set a column that gives the T to use at each date when there is cash flow (column A).  Then figure out what interest rate applies for cash discounted from that date (column B), and the cash flow on that date (column D).  Then the discount factor (in continuous time) is exp(-rt) (column C).  The discounted cash flow is the product of the discount factor and the cash flow (column E); then the sum of the discounted cash flows is the present discounted value.

 

Changing Interest Rates over Time

If the interest rate changes over the period, then we have to account for the changing interest rates.  How do we do this?  We have to distinguish between two types of interest rates: those measuring the rate from today until some date in the future ("zero rate") and those measuring the rate from some date in the future to some date even farther into the future (forward rates).  The zero rate can be thought of as the average of all of the forward rates. 

 

If we graph the evolution of the interest rate then we might have a graph like this:

where the interest rate over the first quarter is 3% and then the interest rate over the second quarter (from the first quarter to the second quarter) is 4%.  The zero rate for the first-and-second quarters is 3.5%, the average, which is the dot-dash line in the picture.  If the interest rate in the third quarter rises to 5% then the zero rate for the first-second-and-third quarters (the 3-quarter zero rate) is 4%, which is the average of 3, 4, and 5 percent:

Then, finally, if the interest rate during the fourth quarter is 6%, the one-year zero rate is 4.5%, which is the average of 3, 4, 5, and 6.

If interest rates for the first quarter are 4% and the forward rate for the second quarter is 6%, then the forward rate over the third quarter is 3% and the forward rate over the fourth quarter is 5%, then the average rate is 4.5% -- this would be the rate over the whole year (the one year zero rate).

 

If we are interpolating rates over different periods then we need to worry about weighted averages.  Suppose that we have one forward rate going from T1 to T2, which we call RF.  The zero rate over T1 is R1.  To find the zero rate over T2, just take the weighted average, so that

                 so that

                 or, equivalently, .

 

To find a forward rate from the zero rates, the formula is .

 

In the language of micro, the forward rate is the marginal rate, which changes the zero rate.

 

If the path of the interest rate is more complicated then the formulas can get more complicated (each interest rate would be multiplied by a weight representing the fraction of the time period during which the interest rate had that value).  [In an extreme case where the interest rate is continuous then we would integrate, Rieman-style, over the intervals.] The instantaneous forward rate at time T is .

 

Generally for bond pricing, we want to use the zero rate for the appropriate time period in each PDV( ) function.

 

Forward Rate Agreement (FRA) is where company trades a fixed rate (for some future period of time) against the actual rate that occurs in that future period.  If the one-year rate is 5% and the 2-year rate is 5.5%, then that means that the forward rate (from one to two years out) is 6%.  Suppose that I don't have money to invest today, but I expect to have it in one year, so I can't buy a 2-year bond but I still want to lock-in that 6% rate.  The FRA solves this problem.  Define RK as the agreed forward rate, RM as the actual market rate, L is the principal amount, and the time goes from T1 to T2.  Then one company gets L(RK  RM)(T1  T2) and the other gets the negative.

 

Often theory tells us to use a "risk-free" interest rate in valuing some financial asset.  In the US, securities issued by the Federal government, such as T-bills and Treasury bonds are generally defined as risk-free; but traders commonly use LIBOR 1-mo, 3-mo, 6-mo, 12-mo.  This is from the Euro-currency market (often cited as "Eurodollar" market, even though it's not necessarily in Europe and they're quoted in dollars not euro.  The market began decades ago as a way for banks to circumvent banking regulation by offshoring some operations  its continued existence is a reminder of how technology allows disintermediation of regulators!).

 

If you remember your money-and-banking class, monetary policy is typically made by the Fed using Repos (repurchase agreements)  often overnight.

 

Let's go back to the example of the stream of cash flows:

 

But now with the interest rate changing (along the path that we showed in the graph above), so we need to find the zero rates.  The one-quarter zero rate is 3%.  The second quarter forward rate is 4% so the zero rate is 3.5%.  The third quarter forward rate is 5% and the three quarter zero rate is 4%.  Finally the fourth quarter forward rate is 6% so the one-year zero rate is 4.5%.  So we substitute these values into the equation:

 then use the continuous-time PDV function of exp{ } to get:

.

To again create a table we just change the values of the zero interest rate over each time period:

T=

r=

discount factor

cash flow

discounted cash flow

0.25

0.03

0.992528

100

 $     99.25

 

0.5

0.035

0.982652

200

 $   196.53

 

0.75

0.4

0.740818

300

 $   222.25

 

1

0.045

0.955997

5000

 $4,779.99

 

 

 

 

 

 

 

 

 

 

 

 $5,298.02

sum of discounted cash flows

So the steps required to calculate the present discounted value are:

1.       figure out at what time every amount of cash flows in or out,

2.      figure out whether the compounding is to be done continuously or over discrete time,

3.       figure out the zero rate for each of the relevant times when cash flows,

4.      calculate the discount factors

5.      multiply the cash times the discount factor to get the discounted cash flow

6.      sum the discounted cash flows.

 

The formula would be

                ,

where ci is the cash flow at date i, ri is the zero rate to that time period, and ti is the time (expressed in fractions of one year) and B is the price of the bond.

 

Certain types of cash flows are discounted using continuous-time compounding while other sorts of cash flows are discounted at discrete intervals.  You can think of these schedule differences as just different units of measurement, like Celsius and Fahrenheit.

 

Discrete Compounding

Sometimes we work in discrete time units, so the interest paid to $1 at rate R, if compounded m times, is  .  So the discounted value of a dollar after 1 year is  (note the negative sign in the exponent) and the discounted value after T years is .  So if we are compounding at discrete intervals, m, then we have PDV(r,t,m) =  and we can consider PDV(r,t,∞) as exp(-rt) a limiting case.

 

So to take the example above, with

$T=.25 100 + $T=.5 200 + $T=.75 300 + $T=1 5000.

worth

=  

then we put in this different PDV function.  So if we put this into Excel, if the interest rate were constant and m=0.5 (semi-annual compounding) then the bond is worth:

t

r

m

PDV

cash

discounted cash value

0.25

0.06

0.5

0.998302

100

99.8302

0.5

0.06

0.5

0.996606

200

199.3212

0.75

0.06

0.5

0.994913

300

298.4740

1

0.06

0.5

0.993223

5000

4966.1167

 

 

 

 

 

 

 

 

 

 

 

5563.7420

The difference in valuation is not small, from 5564 to 5298, so you might ask why such a difference?  Which one is right?  The answer is that it depends on your opportunity cost  which valuation more accurately characterizes your next-best use of the funds?  (Alternately we rely on past precedent: certain bonds are typically valued in certain ways, so that's how we do it.)

 

Going Backwards

Having worked out how to value a bond from a given set of interest rates, then we can invert the function and ask how to find interest rates from bond prices.  This is the more typical case: bonds are traded but interest rates are not usually directly traded.  A bond that pays a fixed schedule of money has a present discounted value which is its price.  If we don't know what interest rates were used to get the market price of the bond, then we have to work backwards  from the present discounted value back to the interest rates that were used for the discounting. 

 

The simplest case is a bond that pays zero coupons but only makes a single repayment.  For instance if a bond makes a single payment of 7000 in one year then the present discounted value is the market price.  If we assume that there is discrete compounding only once per year then the market price, P, is 7000/(1+r).  If we knew r then we could find the valuation of the bond; if we know the market price then we know the implied interest rate.  This rate is the bond yield, the interest rate that makes the bond valuation equal to its price.  In the simple example above, where P = 7000/(1+r), we can see the inverse relationship between interest rates and bond prices.  When rates rise, bond prices fall and vice versa.  If instead it were continuously compounded then P=7000e-r.  Again you can see the inverse relationship.

 

Bonds typically are issued at some notional "par value" (sometimes the "face value") which is the notional amount that the interest is computed from.  For instance a bond might have a par value of 1000 and pay 50 per year forever  this is a five percent par yield.  But as interest rates change subsequent to the bond's issuance, its market price will also change and the par value will become just a relic of history.

 

The bond yield gets more complicated as the number of cashflow dates rises since we are trying to make more inferences.  If we have a bond that, say, returns $3000 after 1 year and $4000 after 2 years then with simple annual compounding the value is 3000/(1+r1) + 4000/(1+r2)2.  If we have only the market price then we can make no further progress  we have a single equation with two unknowns.  If we know either of the zero rates (or if we know that the interest rate remained constant, or some other relationship between r1 and r2) then we can solve for the implied bond yield.

 

Bond yield is the single constant discount rate, applied to the actual coupon payments, that makes the bond price equal to its market price.  (If there were no coupon payments, then this is just the zero rate that gives the current bond price.)

 

Par yield is the coupon rate that causes the bond price to equal its par value (the principal).

 

 

Examples:

Find the present discounted value of $1000 paid in six months, if the 7% interest rate is compounded continuously.

                = 1000 * PDV(0.5, 0.07) = 1000 e-.5*.07 = 1000 * 0.965605 = $965.61

We can, of course, work backwards as well.  If the market values a bond, that pays $1000 in six months, at $965.61, then what continuous-time interest rate is implied?

                Now we set 965.61 = 1000 * PDV(0.5,r) = 1000 e-.5r so 0.96561 = e-.5r and ln(0.96561) = 0.5r so that r = 0.06999, which is 7% with a bit of rounding error.

 

 

Now do the same exercise if interest is semi-annual, quarterly, or monthly.  We have

                = 1000 * PDV(0.5, 0.07,m) where m=2, 4, or 12.

which is

.

For m=2, this is 1000/(1.035) =966.18; for m=4 this is 965.90; for m=12 this is 965.70.

 

The remainder is steadily working up new details.

 

Now to find the present discounted value of $1000 paid in 6 months and in 12 months, if the interest rate is still constant at 7% compounded continuously, this is 1000 * PDV(0.5, 0.07) + 1000 * PDV(1, 0.07) = 1000 e-.5*.07 + 1000 e-1*.07 = 965.604 + 932.39 = 1898. 

 

Working backwards again, if the market values this bond at 1898 then what interest rate is implied?  Again, set 1898 = 1000 * PDV(0.5, r) + 1000 * PDV(1, r) = 1000 e-.5r + 1000 e-1r.  Now we have to remember the rules of exponents: we can't just add the two terms together (just like x2 + x3 ≠ x5!).  So we can use a computer to figure out which values of r give the correct bond value.  You could experiment with some different values to figure out that, indeed (what a shock!) r = 0.07 does, indeed, work.  It's a numerical problem; there is no general analytic solution.

 

Note that the interest rate used for discounting has absolutely nothing to do with the par yield!  A bond might be sold with a $100,000 face value paying $5000 annually, implying a 5% par yield.  But if interest rates rise then that bond isn't worth as much (& vice versa)  the "5%" isn't relevant in any way once the bond has been issued.

 

Next with interest rates changing over time.  So the rate for three months forward is different from the rate six months forward, because the rate for 3-6 months ahead is different.

 

Consider two bonds that pay a quarterly coupon and, so, we use quarterly compounding to find their present value.  One bond, with face value of $100,000, pays its final quarterly coupon ($1200) in one quarter as well as the 100,000 of principal.  The other bond pays a quarterly coupon in one quarter and then, 2 quarters from today, makes its last quarterly payment and its par value.  From the market prices of those two bonds, find the implied zero rates for one quarter and for two quarters, as well as the second quarter forward rate.  If the market gives a price for the shorter-term bond of $100,074.17 and a price for the 2-quarter bond of $99,903.31 then we have two equations:

$100,074.17 =  and

$99,903.31 = .

To solve, we would first calculate the bond yield implied for the first bond to get r1 = 0.045 = 4.5%.  Then once we have r1 we can substitute into the second equation and find that r2 = 0.05.  So the forward rate over the second quarter must be 5.5% (enough of a rise to get the average of 4.5% and 5.5% up to the 5% zero rate).

 

If the same question were posed but with continuous compounding then the precise value of the interest rate would change.  Now we would have two equations that:

$100,074.17 = (100000+1200)exp(-.25r1) and

$99,903.31 = 1200 exp(-.25r1) + (100000+1200) exp(-.5r2).

Again these equations can be solved one at a time, knitting up the solutions from the first equation on  there is no reason we couldn't do this for every quarter for the next 30 years!  (That sounds like a good exam question, doesn't it?  I'm joking, of course, 30 years is too many…)

 

This method would work just as well if the first bond made no payments for a year (so it paid 101200 after 5 quarters) while the second bond made a payment of 1200 in 5 quarters and then 101200 in 6 quarters.  The discount factor would be larger but the basic solution strategy would be the same.

 

All of these interest rates are part of the yield curve, which shows the interest rates over different time horizons.  This set of zero rates (derived from bond prices) is then useful for discounting other cash flows.

 

For some bonds there may not be enough quoted prices to figure out every step of the yield curve so analysts would have various methods for filling in the holes.

 

 

Once we get accustomed to using interest rates to figure out bond prices, we want to then find a quick rule that avoids the complicated math (it's tough to figure e to decimal powers or natural logs in your head!).  This is duration  it approximates the percentage change in a bond price for a given change in the interest rate; so . Duration is the time at which the cashflows are paid, with the weighting as just the present discounted value of the cashflow, so

                .

If duration is in discrete time then we replace the exp{ } function with the appropriate PDV function and the term is called modified duration.

 

The rate at which duration changes, as the interest rate changes, is convexity (it is the second derivative with respect to changes in the interest rate). Don't worry about the formula for now  but later, if you do a fixed-income securities class or need it for your job, you can remember that you've heard of it.

 

 

 

 

 

 

 

 

 

 

Recall payoff graphs, nets to positions including call/put.

 

Chapter 3 concentrates on hedgers.

 

A hedge is basically locking in cash flows at an early date before the asset changes hands.  Consider a position, St, that will have value ST at some future date, T.  If that asset is hedged then a forward is sold at time T and bought at time t, so that the net asset position is ST  FT + Ft.  We expect that, by the time of expiration, the spot and futures price will be equal (or else there would be arbitrage opportunities) so we expect that, by date T, ST  FT = 0.  So the hedge is exchanging a volatile price (St becoming ST) for a known price, Ft.

 

Short Hedge: own an asset and short a forward to sell at a pre-specified price.

                examples: gold mines might sell the gold, that's still in the ground, at pre-determined prices at some date in the future to "lock in" a profit; farmer can sell the crop forward; exporter with short-term receivables might pre-sell (sell forward) to lock in a profit rate.  Consider an insurer selling annuities in Japan that doesn't want the business affected by FX fluctuations so it could sell forward contracts for 3, 6, 9, 12 months (based on expected sales over the year).  If these revenues are to be invested in, say, US Treasury securities, then these securities can be bought forward as well.

 

Payoff Graphs: a short forward position at strike price K pays off as:

While a long spot position (at S0) pays off as:

So if S0 = K then when we add together the two payoff graphs then the net payoff to the portfolio (of one long spot position plus one short forward) is zero everywhere:

If the price at which the initial asset was bought (S0) is not the same as the forward price (K) then the net payoff is a bit more complicated:

 

To find the net position graphically, just line up the two graphs of the base positions, find a convenient point (where one or the other is zero) and add.  Then move right or left, figuring out how much one payoff increases and the other payoff decreases (if there is a hedge).

 

So could do others: if K<S0, for non-hedge positions (both long or both short).

 

Hedge can be considered by comparing the money lost on the asset position with the money gained from the offsetting hedge.  For instance, if the insurer above is getting ¥100,000,000 in 3 months.  Right now the rate is 90¥/$ so this is worth $1,111,111 (=100m/90).  If the rate increases to 95¥/$ then this is worth only $1,052,632.  The movement of ¥5 in the FX rate meant a loss of $58,480 on the asset position.  If the forward price is also 90¥/$ then selling ¥90 forward (getting one dollar delivered in 3 months) would mean that, if the yen increased to 95 per dollar then the short forward position would mean that the company could sell ¥90 for $1 and still have ¥5 left over to buy dollars (0.0526 worth).  This 5-penny gain is small compared to the $58,480 loss  but the company could sell more than ¥90.  How many 90¥/$ contracts?  58,480/0.0526 =1,111,111 worth (which is exactly the number we discovered earlier).  This might seem like the long way to go about it but it is worth showing the basic method: one position loses a certain amount; it can be hedged if I can find some other position that would gain that same amount.  Most companies use hedges with much more complicated structures, but the basic idea remains: construct two offsetting positions so that, as one loses the other gains (and vice versa).

 

Long Hedge: will buy an asset in the future and buy a forward to but at a pre-specified price.

                examples: manufacturers that use mining products (gold, copper, etc) or plastics can buy in advance and lock-in their costs.  Southwest Airlines made huge profits, compared with some of their competitors, when they bought fuel forward for mid 2008 before the oil price rose drastically.  Their competitors had to pay higher prices while Southwest reaped the profits.  (Their competitors, of course, noted that by 2009, when fuel prices had fallen, then Southwest was paying extra for insurance.)

 

A long forward pays off as

 

Can work an example in reverse (as above): how to hedge a short position today with a long forward.

 

Plenty of individuals hedge, even though they might not realize it.  Property owners choosing mortgages must choose between fixed-rate (where the interest rate paid is constant for the life of the loan) or variable or varieties in-between (sometimes a rate is fixed for a few years at first and then varies more often).  A business, that employs a person at a fixed salary even though the employee's productivity might vary, is, in some way, hedging.  Most insurance companies pass along risks through re-insurance, which are then shared among a wide net of different financial companies.

 

Why do so many companies hedge?  Wouldn't their investors want exposure to certain risks?  For instance investors might buy shares in both Exxon/Mobile (that does well when oil prices rise) as well as Ford (which does worse as oil prices rise).  If both companies hedge their positions, then that risk-diversification is lost.  (An investor would have to buy shares in the counter-parties.)  Insurers take a hit from hurricanes (like Katrina) but many pass along the risks as they hedge their positions (there are catastrophe bonds that are linked to occurrences of natural disasters).

 

But the reality is that many companies forecast their earnings, go through a great deal of effort to communicate this "guidance" to the analysts following their shares, and they know that their share price falls when they don't meet expectations; it's difficult to communicate the many sources of risk that might be faced by a global company with revenues in many different currencies and costs paid for many different goods.  Even internally, a company might want to sort out whether a particular division made money by luck (a favorable FX move) or skill (even after hedging they still out-performed).  A hedge means that the company can set its benchmarks and make profits only in its particular areas of comparative advantage.  Return to the example of the gold mine: by selling forward they commit that they will make profit if they are efficient at extracting gold; they will lose money if they are not efficient at that.  Random fluctuations in gold prices will not drive their results; their profits instead come only from their own efficiency. 

 

Competitive pressures can also be important and so every industry (even every firm) must make decisions based on their own particular needs. 

 

Finally, these hedges might allow a company to spread the risk more broadly to willing investors.  An individual company might hedge in order to pass the risk on to the global financial markets.  Instead of a small number of companies losing a lot of money, a large number of investors around the world can each lose a small amount.

 

Most hedging is not perfect  the real world is messier.  Basis = St  Ft.  If the asset that is held and the futures contract are the same then at expiration the Basis should be zero.  So if the position is opened at time t and closed out at time T, then we would like ST=FT.  Define bt = St  Ft and bT = ST  FT.  Then if the company has assets St, it could choose not to hedge, in which case it would have ST at the end of the period.  If it hedges then it would still get the return of ST at the end but would then accrue profits to the forward positions, Ft  FT, so the net position would be [this is the same formula as at the beginning, just with a slightly different interpretation]

ST + Ft  FT = Ft + bT. 

If it is a perfect hedge then the basis is zero at time T and the value is known at time t; if the basis is not zero then there is residual risk  basis risk.  This is generally common when the asset position is not one of the standard contracts traded on exchanges. 

 

This basis risk could have many sources: perhaps a local oil company would like to hedge its costs.  Heating oil is traded on NYMEX but the price paid by an oil company (at its local delivery) is not quite the same as the NYMEX price (delivered to NY harbor).  Or sales are not known exactly so the company hedges a substantial fraction of its anticipated sales (but not all).  Or a company hedges individual stock holdings (on a company with a high beta) with, say, S&P500 index futures (to save trading costs).  Or a company's bonds could be hedged with forward Treasuries and options on the company stock.

 

If we consider a distinction between St (the asset held) and St* (the asset that is traded on a market) then the position will be ST  FT + Ft.  If we add and subtract ST* then we can rearrange that to get Ft + (ST* - FT) + (ST  ST*)  the first term in parentheses is the basis risk between the "ideal" asset and its forward price; the second term in parentheses is the basis from the difference in assets.  For instance, a bank or insurance company might want to hedge positions, where customers are guaranteed some rate of return, using mixtures of Treasuries and private debt.

There is the further complication of when the forward should mature.  Often traders do not want a forward that expires at the same time  they are planning on closing out the position for cash and don't want to be bothered with actual delivery!  So they choose a contract that matures as short a time afterward as is possible.  (After, since they don't want to accidentally take delivery!)  Since short-term markets have the greatest liquidity, someone hedging a large position might use a series of short contracts (again this rolling hedge does not deliver complete hedging).

 

Cross Hedging is used if there is no contract traded forward that is exactly what the firm desires.  High-paid professionals exert a great deal of ingenuity to figure out how to hedge various positions that their companies enter.  The hedge ratio, h, is the number of forwards that must be bought per unit of the asset.  If the asset and forward are the same thing, then the hedge ratio is one.  But generally it will be different because there are more underlying cash flows than traded contracts.

 

Can think of this as an econometrics problem: want to explain the variation in some Y variable (ΔS) as a linear combination of X variable, (F).  The minimum-variance hedge is the best-fitting coefficient, b, in the equation

.

 

When the position expires at T, the value of each asset is ST and the profit from the forward position is FT  Ft.  Define NA as the number of assets held and NF as the number of forwards so that h = NF/NA.  So the total position value is

= NAST  NF(FT  Ft). 

Add and subtract NASt and get

= NASt + NA(ST  St)  NF(FT  Ft).

= NASt + ΔSNA  ΔFNF.

= NASt + NA(ΔS  hΔF).

To minimize the variance of this term, since NA and St are known, means minimizing the variance of (ΔS  hΔF). 

 

Recall from your stats course that a term, aX + bY has variance of , where ρ (rho) is the correlation coefficient.  (Go back to your stats book, or integrate it yourself, or just trust me).  So the variance of (ΔS  hΔF) is .  To minimize this term with a choice of h, we set the first derivative equal to zero so  and h* = .

 

This should seem sensible: if ρ=1 and σS = σF then the hedge ratio is 1 (just like it should be).  If the forward price were twice as volatile as the spot price, then you would get h=  since then you would need only half as many positions to hedge.  If the forward and spot price are not perfectly correlated (two assets that generally move together but not perfectly, so perhaps ρ = 0.75 then again fewer forward positions would be needed.  The hedge ratio, h, can be found from historical data as the slope of a regression line of ΔS against ΔF.  (This assumes that past performance gives information about future returns.)

 

The value of the total hedge should be h*NA.  If we're using futures contracts that are each QF units, then .

 

A stock index is often used as a hedge.  Since they weight by market capitalization, however, a hedge can gradually erode as weights change slightly.  A stock bubble can lead to distortions.

 

Recall, from your previous coursework, the discussion of a stock or portfolio β (Beta)  it measures how sensitive the stock or portfolio is relative to movements in the whole market; in the simplest case it can be found by regression of the excess returns on the stock (over the risk-free rate) upon the excess market returns (again, over the risk-free rate).  A stock with a high beta will track the market closely; a stock with a near-zero beta will be uncorrelated with the market.  As the econometrics example from cross-hedging discussed, there should be a link between hedging and market beta.  In general,

h* = β.

 

A non-perfect hedge can change the beta of a portfolio as well, so a portfolio might be incompletely hedged in order to take on more risk or shed risk (as the portfolio manager desires).  If the original portfolio has β, and desires to change to β', then instead of taking a hedge ratio h = β, should take either h = (β  β') {if β > β'} or h = (β' - β) {if β < β'}.

 

A fully-hedged portfolio in the stock market will grow at the risk-free rate.  (Since a fully-hedged portfolio is riskless, this makes sense  two riskless assets should have the same return.)  Why hedge, then?  This allows the company to earn returns entirely from its ability to pick stocks, for example: a company with a meticulously-chosen portfolio that is fully hedged against an index will earn the risk-free rate plus the differential return accruing to its stock-picking skill.  Many other companies want exposure to other asset baskets and want to minimize their exposure to aggregate market risks.

 

Note that one person's hedge is sometimes another person's speculation.  Hedge funds were originally set up to take positions that were well hedged (thus the name) but gradually moved into assets where the basis risk got larger and larger, until they were essentially speculating.  (Long Term Capital Management was the best known failure in the past).