CFA Level -2 Derivatives - I

CFA Level - 2

Agenda

Forwards Markets and Contracts

Future Markets and Contracts

Pricing and Valuation Of Forward Contracts

A forward contract price is the fixed price or rate at which the transaction scheduled to occur at expiration will take place.

This price is agreed on the contract initiation date, and is commonly called the forward price or

forward rate.

The price here is different from value, which is what you can sell something for or what you must pay to acquire something.

Pricing means to determine the forward price or forward rate.

Valuation is the process of determining the value of an asset or service. It means to determine

the amount of money one would need to pay or would expect to receive to engage in the

Pricing and Valuation Of Forward Contracts

Arbitrage free Forward prices are given as: Where:

If we long the forward contract at time t=0, at forward's price F(0,T), the initial cash outlay would be zero

At time t=t, we have claim on the asset which is worth ST and an obligation to pay F(0,T) at time t=T

T f

T

S

R

F

=

×

(

1

+

)

At time t= T, we pay F(0,T) and receive the asset worth ST

Forward rates are rates of interest implied by the current zero rates for a period of time in the

t=0 _{t=t t=T}

Forward Contract - Example

You need a laptop, one year from now. The current price of a laptop is $1000.

You have 3 options, to fulfill your requirement:

1. Buy it now at $1000, and use it after a year

2. Buy it a year from now at the market price prevailing one year from now (unknown price).

3. Enter into an agreement with the vendor so as to purchase the laptop a year from now at a price of $1000.

In each of the three scenarios, what would be your profit/loss if the price of the laptop after one

In each of the three scenarios, what would be your profit/loss if the price of the laptop after one year is:

1. $500

2. $1000

Forward Contract - Example

Taking the Third Scenario, If you locked a fixed price of $1000 for a laptop after one year, your profit and loss would look like:

Agreed upon Price of Laptop (X=strike)

Price of Laptop (S=spot) Profit and Loss from the Buyer's Perspective (S-X)

1000 500 -500

1000 1000 0

1000 1500 500

The Profit and Loss from the vendor's point of view:

Agreed upon Price of Laptop (X=strike)

Price of Laptop (S=spot) Profit and Loss from the Buyer's Perspective (X-S)

1000 500 500

Forward Contract - Example

Payoff from Buyer's Perspective Payoff from Seller's Perspective

0 500 1000 1500 0 500 1000 1500 0 5 0 0 0 5 0 0 0 500 1000 1500 0 500 1000 1500 -5 0 0

Slope of the line is 1 -50

0

Slope of the line is -1

Example 2: Forward Contract Payoff

A person wants to buy a Machine a year from now. If the current forward price of one year at time (T=0) is $1000. The price can change over the period of time. If the actual price of that

laptop after one year is selling above the price of $1000. It will make profit for the long party, and equal loss for the short party and vice-versa.

Forward T=1 Spot T=0

Payoff Forward profit and loss

If the forward price at Time 1 is

Payoff for the forward position

For calculating the Present value of the payoff we will divide the amount by (1+Risk free rate) ^

Derivatives Payoff

Forward Contract Valuation

The price of a forward contract is given by the equation below:

F₀= S₀erT _{in the case of continuously compounded risk free interest rate, r}

F₀= S

0(1+r )T in the case of annual risk free interest rate, r

where,

F₀: forward price S₀: Spot price

t: time of the contract

Value of a long position in a forward contract during the life of the contract (V_t), where t < T is: Value of a long position in a forward contract during the life of the contract (V_t), where t < T is:

V_t = S_t - {F₀/ (1+r)T-t _}

The discounting period reduces with passage of time (T-t). Spot price(S_t ) also changes after initiation of the contract.

The value to a short position in a forward contract is negative of the long position value: V_t= {F₀/ (1+r)T-t _{} - S}

Example (Forward Price)

Spot Price of an Assets= $100, Risk Free Rate= 10%

Contract Maturity 1 year

Forward Contract Valuation

Example:

Suppose a person entered in a long position on a 6 months forward contract of a stock at $120. After four months the stock price has moved to $144. Calculate the value of the forward contract assuming a risk free rate of 5%.

Solution: V_t = S_t - {F₀/ (1+r)T-t _}

Value for the long position

120

Value for the short position is = - $25

Price And Value Of An Equity Forward Contract, Assuming

Dividends Are Paid Either Discretely Or Continuously

A stock or a equity index generally has expected dividends assigned to them.

To determine the price of such forward contracts, we can adjust the spot price for the present value of expected dividends (PVD) over the life of the contract or adjust the forward price for future value of the dividends (FVD) over the life of the contract.

Arbitrage free Forward prices are given for such contracts:

Example

Example:

Calculate the forward price for Neev Inc. on a 90 days forward contract which is currently quoted at $75 and is expected to pay a dividend of $2 in 30 days. Assume a risk-free rate of 5%.

Forward Price Of Different Assets Classes

Equity Commodity Currency

Benefit to the assets holder

Dividend Convenience yield N/A

Cost to the assets holder

Nothing Storage Cost N/A

Forward price _{S (1 R )}T _{FV of div.} f

Value Of A Equity Forward Contract

Value of long position in a forward contract paying dividends (V_t)

V_t = (S_t - PVD) – {FP / (1+r)T-t _}

Example: A person has entered into a 150-day forward contract on a stock at $80 and there is an

expected dividend on the stock of $2.5 payable in 90 days. After 30 days the stock price has moved to $86. What is the value of this forward contract given risk free rate of 5%.

Solution: The dividend payable has 60 days left before the stock becomes ex-dividend from the Solution: The dividend payable has 60 days left before the stock becomes ex-dividend from the current date and the contract will mature in 120 days.

= 2.48

Value of the contract = (86-2.48) – {80/(1.05)120/365 _}

Equity Forward Contract With Continuous Dividends

If the underlying asset on which the forward contract is entered into provides a continuously compounded yield q, then the forward contract would be valued as:

where q is continuously compounded % of return on the asset, such as a equity index

Example: The current value of Nifty is 5350 providing a continuous dividend yield of 2.5%.Calculate the

price of a 100 day contract given the continuously compounded risk free rate of 6%.

q)T (r 0

e

S

Price

Forward

=

×

Value of forward contract(for a long position) with continuous yield is derived:

Equity Forward Contract With Continuous Dividends

In the previous example suppose after 45 days the index moves from 5350 to 5430, calculate the value of a forward contract assuming the same risk free rate and dividend yield.

Value Of A Fixed Income Security (Example)

In a fixed income security there are cash flows in the form of coupons. So even for these cash flows we have to find the present value of coupons(PVC) or future value of coupons(FVC)

The forward price for a fixed income security:

Example:

Calculate the price of a 220 day forward contract on a 6% semi-annual coupon bond with a spot price of $106(includes accrued interest), which has just paid a coupon and will make its next payment in 180 days. The risk free is assumed to be 4%.

payment in 180 days. The risk free is assumed to be 4%.

Solution: Price of the forward contract

Price And Value Of A Fixed Income Security

Value of a forward contract for a long position:

Suppose a person had entered a 200 day forward contract on a bond at 107, 60 days have passed and the current price of the bond is 110. The interest due on this bond $3.5 which is payable 45 days from now. Calculate the value of the contract assuming a risk free rate of 5%.

Price and Value of a Forward Rate Agreement

Forward rates are rates of interest, implied by the current zero rates, for a period of time in the future

For example, if we have the zero rates for year 4 and year 5, the forward rate for the period of time between year 4 and year 5 would be known as the forward rate for that time period of 1 year.

Year 4

Year 5

F

F

= 4%

F

= 5%

Consider that you invest $100 for 4 years and then roll it forward for one year in the 5 year. Then the total amount would be given as:

100*e0.04*4_eF4,5*1

If the same $100 was invested for 5 years instead then it would grow to

100*e.05*5

Equating the two we get

Forward Rate Agreement (FRA)

A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period.

Forward contract to borrow (long) or lend (short) at a pre-specified rate.

A forward rate agreement is a forward contact on a short-term interest rate, usually LIBOR.

It is a forward contract to borrow / lend money at a certain rate at some future date,

Payment to the long at settlement:

days

(

rate)

FRA

settlement

at

(Rate

−

Numerator - the difference between the actual rate that exists in the marketplace on the expiration date and the agreed-upon rate at the beginning of the contract.

FRA Example

Example: FRA that settles in 30 days

• Notional Principal $1,000,000

• Based on 90-day LIBOR

• Forward rate of 5%, Actual 90-day LIBOR at settlement is 6%

Solution:

Payment to the long at settlement

Forward Rate Agreements Assumption

A forward rate agreement (FRA) is an over the counter agreement where the forward interest

rate, F_t1,t2,is fixed for a certain principal between times T1 and T2

The payer of the fixed interest rate is also known as the borrower or the buyer. The buyer hedges against the risk of rising interest rates, while the seller hedges against the risk of falling interest rates.

The potential borrower can lock in borrowing rate with FRA, a contract to enter into a loan at a future date.

Usually settled with cash payment, the amount of settlement makes the total interest rate (actual interest cost + FRA settlement) equal to the contracted interest cost in FRA

Currency Forward Contract. (Example)

The spot rate of EUR/USD foreign exchange is $1.1565 per EUR. USD LIBOR is 4% and EUR LIBOR is 2%. What is the price of a 180 day forward contract on USD per EUR exchange rate?

Solution:

02

.

1

04

.

1

1565

.

1

$













=

F

Here F_T and S₀ are quoted in domestic currency per unit of foreign currency R_DC = domestic interest rate

Valuing A Currency Forward Contract

Before maturity, value of a forward currency contract(Vt) will depend on the spot rate at time t.

Credit Risk In A Forward Contract

After entering the forward contract any party can have positive value.

This party which can be long or short on a contract faces the credit risk from the opposite party as it owes the positive value.

It is the risk that the counter party will not pay when a positive amount is owed at settlement.

The larger the value of the forward to one party, the greater the credit risk to that party.

The contract value and simultaneously the credit risk may increase, decrease or can even change the values for either party over the term of the contract.

Credit risk can be reduced by marking to market.

Questions

1. Which of the following is TRUE about a forward rate agreement (FRA)?

A. It can be cash or physically settled

B. A borrower who intends to borrow cash at LIBOR in the future will hedge by receiving the fixed interest rate, R(k), in an FRA

C. A bank that intends to lend cash at LIBOR in the future will hedge by receiving the fixed interest rate, R(k), in an FRA

2. A company wants to borrow $10 million for 90 days starting in one year. To hedge the interest rate risk of the future

borrowing, the company enters into a forward rate agreement (FRA) where the company will pay a fixed rate, R(k), of 5.0%. The FRA cash settles in one year; i.e., in advance (T=1.0) not in arrears (T=1.25). All rates are expressed with quarterly compounding. If the actual 90-day LIBOR observed one year forward turns out to be 6.0%, what is the cash flow payment/receipt by the company under the FRA?

A. Company pays $24,631

A. Company pays $24,631

B. Company pays $25,000

C. Company receives $24,631

3. Assume the one-year spot (zero) interest rates is 3.0% and the fifteen month (1.25 years) zero rate is 4.0%, with continuous compounding. What is the value of a forward rate agreement (FRA) that enables the holder to earn 7.0%, expressed with quarterly compounding, for a 3-month period starting in one (1) years on a (principal) notional of $1,000,000?

A. -$2,701

B. -$2,570

Questions

4. Calculate the no-arbitrage forward price for a 90-day forward on a stock that is currently priced at $50.00 and is expected to pay a dividend of $0.50 in 30 days and a $0.60 in 75 days. The annual risk free rate is 5% and the yield curve is flat.

A. $50.31.

B. $48.51.

C. $49.49.

5. Calculate the price of a 200-day forward contract on an 8% U.S. Treasury bond with a spot price of $1,310. The bond has just paid a coupon and will make another coupon payment in 150

days. The annual risk-free rate is 5%.

A. $1,333.50.

B. $1,305.22.

Answers

1. C. A bank that intends to lend cash at LIBOR in the future will hedge by receiving the fixed interest rate,

R(k), in an FRA In the future, the bank’s cash flows will be: (+) receive LIBOR on the lent cash (+) receive fixed rate on the FRA (-) pay LIBOR on the FRA = net (+) receive fixed rate on the FRA

In regard to (a)FRA is cash settled. In regard to (b), to hedge the future LIBOR, the borrower wants to pay fixed and receive LIBOR (i.e., the gain/loss on LIBOR in the FRA offsets the future borrowing).

2. C. Company receives $24,631 The payoff to the company = $10 MM * (6.0% - 5.0%) * 0.25 = $25,000;

i.e., if LIBOR goes up, the companies borrowing cost will increase but the FRA will hedge by paying the company But the FRA settles at T = 1.0,

company But the FRA settles at T = 1.0,

such that payoff = $25,000 / (1 + 6.0% * 0.25) = $24,631

3. B. -$2,570 The forward rate (1.0, 1.25) = (4%*1.25 - 3%)/0.25 = 8.0% with continuous compounding.

The converts to 4*(exp(4%/4 )- 1) = 8.0805% with quarterly compounding. In this case, R(k) = 7.0% and R(f) = 8.0805%, and R(k) is earned by the holder such that:

-Answers

4. C. The present value of expected dividends is: $0.50 / (1.0530 / 365_{) + $0.60 / (1.05}75 / 365_{) = $1.092} Future price = ($50.00 − 1.092) × 1.0590 / 365_{= $49.49}

5. B

Coupon = (1,000 × 0.08) / 2 = $40.00 Coupon = (1,000 × 0.08) / 2 = $40.00

Present value of coupon payment = $40.00 / 1.05150/365 _{= $39.21}

Thank You !