CFA 5_Fixed Income and Derivatives

Question 1

Duration

Answer 1

• % change in bond price / % change in yield

price if yields decline - price if yields rise) / (2 * initial price * change in yield in decimal

Question 2

Treasury bill / note / bond

Answer 2

1 year or less, zero-coupon / 2-10 years, semi-annual / 20-30 years, semi-annual

STRIP: a STRIPS security is a zero-coupon bond with no default risk and therefore represents the appropriate discount rate for a cash flow certain to be received at the maturity date for the STRIPS. (Yield on a STRIP is the Treasury spot rate)

Question 3

Four theories of the term structure of interest rates (yield curve)

Answer 3

1) Pure expectations: if ST rates are expected to rise, yield curve will slope up / 2) Liquidity preference: in addition to expectations, require risk premium for longer term / 3) Market segmentation: yields determined by supply+demand for bonds / 4) Preferred habitat: market segmentation, but investors can move segments if yield is high enough

Question 4

Absolute yield spread

Answer 4

yield on higher bond - yield on lower bond

Question 5

Relative yield spread

Answer 5

absolute yield spread / yield on benchmark bond

AYS = yield on higher bond - yield on lower bond

Question 6

Yield ratio

Answer 6

subject bond yield / benchmark bond yield1 + relative yield spread

(RYS = absolute yield spread / yield on benchmark bond)

Question 7

After-tax yield

Answer 7

taxable yield * (1 - marginal tax rate)

taxable yield - (taxable yield * marginal tax rate)

Question 8

Taxable-equivalent yield

Answer 8

tax-free yield / (1 - marginal tax rate)

Question 9

Current yield

Answer 9

annual cash coupon PMT / bond price [PV]; this measure only looks at annual interest income.

Question 10

Bond Equivalent Yield of a monthly CF instrument

Answer 10

BEY = [(1 + monthly CF yield)^6 - 1] * 2

Question 11

BEY (bond equivalent yield) of an annual-pay bond

Answer 11

[(1 + annual YTM)^1/2 - 1] * 2

Question 12

EAY (equivalent annual yield) of a semi-annual pay bond

Answer 12

(1 + semi-annual YTM/2)^2 - 1

Question 13

Calculating forward rates given spot rates

Answer 13

(1 + S4)^4 = (1 + S3)^3 * (1 + F3)

Question 14

Calculating spot rates given forward rates

Answer 14

(1 + S3)^3 = (1 + F0 [ie S1])(1 + F1)(1 + F2)

equivalent to…S3 = [(1 + F0 [ie S1])(1 + F1)(1 + F2)}^1/3 - 1

Remember to pay attention to semi-annual vs. annual, and half the forward rates if they are given in annualized basis but are supposed to use 6-month.

Question 15

Valuing a bond using forward rates

Answer 15

2 year bond value = [PMT / (1+F0)] + [PMT + FV / (1+F0)*(1+F1)]

note that the denominator has no exponent because it is already taken into account in the forward ratesOr, calculate spot rates for each period and discount:

(1 + S2)^2 = (1 + F0 [ie S1])(1 + F1)

Question 16

Nominal spread

Answer 16

YTM of bond - YTM of similar Treasury

Question 17

Zero-volatility spread (Z-spread)

Answer 17

The amount that must be added to each rate on the Treasury spot yield curve to make the PV of the risky bond’s CFs equal to the risky bond’s market price

NO DIFFERENCE with nominal spread when spot yield curve is flat. The main factor causing any difference between the nominal spread and the Z-spread is the shape of the Treasury spot rate curve. The steeper the spot rate curve, the greater the difference.

Question 18

Option-adjusted spread (OAS)

Answer 18

Removes option yield component from Z-spread measure.

Calls: require more yield on callable, therefore OAS < Z-spread (call increased Z-spread)

Puts: require less yield on puttable bond, therefore OAS > Z-spread (put reduced Z-spread)

Question 19

Bootstrapping a theoretical Treasury spot rate curve

Answer 19

1 year semi-annual bond: [PMT / (1+S.5)] + [PMT + FV / (1+S1 / 2)^2 = 100solve for S1 (the only unknown)

Question 20

Effective duration

Answer 20

(price when yields fall - price when yields rise) / (2 * inital price [PV] * change in yield in decimal form)

change in yield in decimal form = eg 60 basis points will be 0.0060.

Effective duration is best duration measure for bonds with embedded options.

Question 21

% change in price based on duration and convexity

Answer 21

% change in price = duration effect + convexity effect

= [(-duration * change in yield) + (convexity * change in yield^2)] * 100

Question 22

100 basis points

Answer 22

1%, ie .011 basis point is .01%, or .0001

Question 23

Return impact (on changes in credit spread)

Answer 23

return impact = - modified duration * change in spread + 1/2(convexity) * (change in spread)^2

or approximate with:return impact = - modified duration * change in spread

Question 24

Payment to the long at settlement on a FRA

Answer 24

(notional principal) * [((market rate - FRA rate) * (days of loan/360) / (1 + (market rate * days of loan/360))]

The payment is discounted to the PV of the interest savings that would be realized at the end of the hypothetical loan term.

Question 25

Forward rate agreement: who pays who

Answer 25

Long “borrows” money, and therefore will received payment if actual rates are higher than FRA rate at settlement

Short “lends” money, and therefore will receive payment if actual rates are lower than FRA rate at settlement

Question 26

Computing amount owed on forward contract for bonds/loans quoted at IR

Answer 26

(actual rate at settlement - forward contract rate) * (days of bond / 360) * principal

Question 27

Margin in futures market

Answer 27

Initial margin: must be deposited before any trading.Maintenance margin: amount that must be maintained

Variation margin: amount that must be deposited to return balance to initial margin if it falls below maintenance margin*In securities markets only have to return balance back to maintenance margin

Question 28

Computing profit on long position on Eurodollar/Treasury futures contract

Answer 28

.01 change in price is worth $25. If price increases, long profits.

Question 29

Index options

Answer 29

Based on a market index. Settled in cash Long payoff per contract = Index level at expiration - exercise index level * contract multiplier

Question 30

Interest rate options

Answer 30

Exercise price is IR, underlier is reference rate. Settled in cash. Payoff = notional amount * difference between market rate and option contract rate. Long receives payment when market rate is greater than option contract rate.

Long position in an FRA (long “borrows”) is equivalent to a long interest rate call combined with a short interest rate put. Long will receive payment if market rate is greater than contract rate, and will pay when market rate is less.

Long payoff = notional amount * (market rate - contract rate) * (days of reference rate / 360)The payoff is made not at expiration, but at a future day corresponding to the term of reference rate.

Question 31

Option value (premium)

Answer 31

intrinsic value (amount option is in the money) + time value

Question 32

Lower bound for option price

Answer 32

Call/Put: Zero or intrinsic value. No option will sell for less than its intrinsic value (moneyness), and no option can take a negative value. Same for American and European.

Question 33

Upper bound for option price

Answer 33

Call option: Price of the underlying asset. No one will pay more for right to buy asset than what asset is worth. Same for American and European.Put option: American: Exercise price. European: Exercise price discounted at RFR (can’t exercise until exercise date).

Question 34

Lower bound for option price (expanded)

Answer 34

The greater of 0 or intrinsic value.

European/American call: Call price = greater of 0 or (Underlying price - Exercise price / (1 + RFR)^T) ie (S - X / (1 + RFR)^T)

In this case the same because American option must be worth at least as much because it has early exercise feature, but Euro must be at least theoretically equal because it’s discounting of exercise price means it is a larger intrinsic value.

European put: Put price = greater of 0 or (Exercise price / (1 + RFR)^T - Underlying price) ie (X / (1 + RFR)^T - S)

American put: Put price = greater of 0 or (Exercise price - Underlying price) ie (X - S)*Put doesn’t benefit if Exercise price is discounted

Question 35

Effect of exercise price on option value

Answer 35

Call option: inverse relationship b/w exercise price and option value. A higher exercise price will result in lower option value, because the underlying can’t be purchased for profit until an even higher market price. A lower exercise price will increase option value, because underlying could be purchased for profit at a lower market price.Put option: direct relationship b/w exercise price and option value. A higher exercise price will result in higher option value, because the underlying can be sold for profit at a higher price level. A lower exercise price will result in lower option value, because the underlying cannot be sold for profit until the market price decreases past that level.

Question 36

Effect of time to expiration on option value

Answer 36

Longer time to expiration will increase option value except for some instances of European puts.European puts discount the exercise price, and a lower exercise price means lower intrinsic value (Put value = exercise price - market price).

Question 37

Put-call parity

Answer 37

c + X / (1 + RFR)^T = S + p

(Call premium + PV of Exercise price) = (Stock [underlying] price + Put premium)

Demonstrates that the payoffs of a portfolio of European fiduciary call and portfolio of protective put must be equal.

Question 38

Synthetic relationships resulting from put-call parity

Answer 38

Put-call parity: c + X / (1 + RFR)^T = S + p

Stock price = Call premium - Put premium + PV of exercise price

S = C - P + X / (1 + RFR)^T

Put premium = Call premium - Stock price + PV of exercise price

P = C - S + X / (1 + RFR)^T

Call premium = Stock price + Put premium - PV of exercise price

C = S + P - X / (1 + RFR)^T

PV of exercise price = Stock price + Put premium - Call premium

X / (1 + RFR)^T = S + P - C

Question 39

How positive CFs of underlying asset affect put-call partiy and lower bounds of option prices

Answer 39

The market price of the asset can be reduced by the PV of CFs on that asset in both these calculations. (S) therefore becomes (S - PVcf)

Question 40

Effect of changes in interest rates on options

Answer 40

Call options: Direct relationship. Increase in IRs increases the value of call option (a smaller discounted exercise price is subtracted from the market price)Put options: Inverse relationship. Increase in IRs decreases the value of a put option (the market price is subtracted from a smaller discounted exercise price)

Question 41

Effect of volatility on options

Answer 41

Increases value of both puts and calls.

Question 42

Short call / Short put

Answer 42

Short call: writer of call option. Has obligation to sell underlying asset.

Short put: writer of put option: Has obligation to buy underlying asset.

Question 43

Long call / Long put

Answer 43

Long call: buyer of call option. Has right to buy underlying asset.

Long put: buyer of put option. Has right to sell underlying asset.

Question 44

Currency swap

Answer 44

Eg. fixed-for-fixed currency swap to finance operations in foreign country.

1) Notional principal is swapped at initiation (A receives JPY and B receives USD)
2) Full interest payments are made at each settlement date, with A paying rate on JPY and B paying rate on USD. (not netted). NB payments are made in arrears, and therefore are based off the interest rate at the end of the prior period.
3) At termination, notional principal is swapped back.

Question 45

Interest rate swap (plain vanilla, fixed-for-floating)

Answer 45

net fixed-rate payment = (swap fixed rate * days/360) - (swap floating rate * days / 360) * notional principal

which is the distributed version of:

net fixed-rate payment = (swap fixed rate - swap floating rate) * (days/360) * notional principal

1) Notional principal is not swapped at initiation (purpose is to hedge IRs).
2) Only net interest payments are made to the one who is owed it. Payments are made in ARREARS, and therefore are based off the interest rate at the end of the prior period.
3) No notional principal to swap back at conclusion.

NB: “net fixed-rate payment” value is NEGATIVE if fixed-rate payer is RECEIVING payment; and POSITIVE if fixed-rate payer OWES payment. This is important because it effects equation if solving for other variables.

Question 46

Equity swap

Answer 46

The positive return on a portfolio is paid by one party in exchange for a fixed or floating-rate payment. Computed on a quarterly basis (divide annual rate by 4). The % increase in index each quarter is netted against the fixed rate to determine the payment to be made. (quarterly return - or + quarterly fixed rate)..- if index return payer is paying (increase in portfolio). + if index return payer is receiving (decline in portfolio). If the portfolio return is negative, the fixed/floating-rate payer must also pay the % decline in the portfolio to the portfolio manager. However, if positive return on portfolio exceeds the fixed/floating-rate payment, the fixed/floating-rate payer will profit.

Subtract the portfolio return in % from the fixed/floating-rate % payment to get the total % of notional amount payment due to either party.

Question 47

Contango / Backwardation

Answer 47

Contango: when commodities futures price is above spot price (producers concerned about risk of rising prices). Results in negative roll yield.

Backwardation: when commodities futures price is below spot price (produces paying for protection against price declines). Results in positive roll yield.

Question 48

Sources of return on commodities

Answer 48

1) Collateral yield: yield on collateral (T-bills) deposited as collateral on position in future.
2) Price return: change in spot rates and convergence of future prices to spot prices as contract nears expiration.
3) Roll yield: Gains or losses on rolling over the position as derivative contracts expire and the position must be maintained.

Question 49

What is the long position in an FRA equivalent to?

Answer 49

Long position in an FRA (long “borrows”) is equivalent to a long interest rate call combined with a short interest rate put. Long will receive payment if exercise rate is greater than contract rate, and will pay when exercise rate is less.

Short position in an FRA (“lends”) is equivalent to a short interest rate call and a long interest rate put. Will be profitable when interest rate falls.

Question 50

Enterprise value

Answer 50

EV = market capitalisation (share price * shares outstanding) + debt - cash and cash equivalents

Enterprise value = common equity at market value + preferred equity at market value + minority interest at market value, if any + debt at market value + unfunded pension liabilities and other debt-deemed provisions - associate company at market value, if any - cash and cash-equivalents.