Chapter 13: Valuation of investments (2)

Question 1

Give 4 reasons why interest rate derivatives are more difficult to value than equity derivatives

Answer 1

• Behaviour of an individual interest rate more complicated than that of a stock price, as interest rates vary by term
• For many products, it is necessary to develop a model describing behavour of entire yield curve, whereas Black scholes model of share option prices is based on single share price only
• Volatilities of different points on yield curve are different
• Interest rates are used for discounting as well as for determining payoffs from derivatives

Question 2

State the Black’s formula for the price of a call option, definin all the notation used

Answer 2

c = P(0,T)[F₀Ŷ(d₁) - XŶ(d₂)]

Where

• Ŷ(x) is the culmative standard normal distribution function
• d₁ = [In(F₀/x) + (σ²T/2)]/[σT^1/2 and d₂ = d₁-σT^1/2
• F₀ is forward price of underlying asset
• σ is the voloatility of forward price
• X is option strike price

Question 3

State Black’s formula for the price of a put option

Answer 3

c = P(0,T)[XŶ(-d2) - F0Ŷ(-d1)]

Where

Ŷ(x) is the culmative standard normal distribution function

d1 = [In(F0/x) + (σ²T/2)]/[σT^1/2 and d2 = d₁-σT^1/2

F0 is forward price of underlying asset

σ is the voloatility of forward price

X is option strike price

Question 4

State a formula for the forward price of a coupon bearing bond

Answer 4

F₀ = (B₀ - i)/P(0,T) = (B₀ - I)e^rT

Where

• B₀ is dirty bond price at time zero
• I is present value of coupons that will be paid during the option
• P(0,T) is discount factor from time T back to 0

Question 5

Explain what are meant by each of the following in relation to a coupon bearing bond

• The clean price
• The dirty price
• Accured interest

Answer 5

Clean Price

• Equals dirty prices less accrued interest
• Is quoted price

Dirty price

• Include accrued interest
• Is price paid for bond Represents discounted present value of future cashflows paid by bond

Accrued interest

• Accrued interest is proportion of next coupon deemed to have accrued since last coupon was paid

Question 6

State the formula relating the price and yield volatilities of a bond

Answer 6

Price and yield volatilities of bond

σ = Dy₀σ_y

Where

• σ is forward price volatility
• σ_y is corresponding forward yield volatility
• D is modified duration of forward bond underlying option
• y₀ is initial (forward) yield on forward bond underlying option

Question 7

State a formula for the modified duration in terms of the duration for a fixed interest bond

Answer 7

D = Duration/(1+y/m)

Where m is the frequency per annumm with which y is compounded

Question 8

Describe how interest rate caps and floors work

Answer 8

Interest rate caps and floors

• Over the counter derivatives that can be purchased from investment bank
• In return for initial premium, interest rate cap provides payment each time floating interest rate R_k rises above fixed cap rate, R_x
• In contrast, buyer of interest rate floor recieves payment each time floating interest rate falls below fixed floor rate
• Can be used to hedge against movements in short term interest rates, or speculate on such movements

Question 9

Explain how each pay off is determined for a caplet

Answer 9

• In each sub-period of interest rate cap, interest payment is made under relevant caplet if floating interest rate in that sub - period exceeds cap interest rate. Otherwise no payment made in that sub - period
• Interest payment made at end of sub - period
• Interest payment based on interest rate that applies over sub - period at start of sub period
• Actual monetary payment based on paymet in interest rate terms, multiplied by both cap principal and length of sub - period and length of sub - period (or tenor)

Question 10

State the formulae for the payoff from a caplet and the payoff from a floorlet

Answer 10

Payoff from a caplet

L§_k max(R_k - R_X,0)

Payoff from a floorlet

L§_k max(RX - Rk,0)

• L is principal
• §_k = t_k+1 - t_k is tenor (time between resets)
• R_k is floating rate (compounding frequency = §_k)
• R_x is cap rate (compounding frequency = §_k)

Question 11

Outline in words how to value an interest rate cap and an interest rate floor

Answer 11

• Each interest rate caplet valued using black’s formula
• This values cap as call option on floating interest rate, with strike price equal to cap interest rate
• Value of interest cap is then sum of values of constituents caplets
• Likewise, floor isi valued as sum of values constituent floorlets, where each floorlet valued (using black formula) as put option on floating rate, with strike price equal to floor interest rate

Question 12

State the formulae for valuing an interest rate caplet

Answer 12

c = Lς_kP(0,t_k+1)[F_kŷ(d₁) - R_xŷ(d₂)]

• d₁ = ([ln(F_k/R_x) + σ_k²t_k/2)]/(σ_kxt_k^0.5)
• d₂ = d₁ - σ_kt_k^0.5
• t is time to start of caplet strike deate
• σ volatility of forward rate

Question 13

State the formulae for valuing an interest rate floorlet

Answer 13

P = Lŵ_kP(0,t_K+1)[R_Xŷ(-d₂) - F_kŷ(-d1)]

• d₁ = (ln(F_k/R_x) + σ²_kt_k/2))/(σ_kt_k^0.5)
• d₂ = d₁ - σ_kt_k+1
• F_k is the forward rate between t_k and t_{k + 1}
• t_k is time to start of caplet (‘strike date’)
• σ_k volatility of forward rate

Question 14

State the put call parity relationship between swaps, caps and floors

Answer 14

Cap price = floor price +value of swap

Where

• cap interest rate and floor interest rate are same
• terms, principals, frequency of payments
• swap is agreement to recieve floating and pay fixed

Question 15

Explain what is meant by an interest rate collar

Answer 15

• Consists of long position in interest rate cap and short position in floor
• design to guaranteee that interest rate on underlying floating rate note always lies between two levels
• Usually constructed so that price of long position in cap initally equal to price of short position in floor, so that cost of entering into collar is zero

Question 16

Assuming that you hold a swaption, explain, with reference to the swap rate, how you would decide whether or not to exercise your option to enter into the swap

Answer 16

• Calculate net present value of swap to you on strike date
• If its positive, so that you expect to recieve more than you pay out, then you enter into it
• If it is negative, let option expire worthless
• in practice this would be done by comparing swap rate quoted on strike date with fixed rate previously agreed
• Recall that swap rate is fixed interest rate that would make swap have zero NPV at outset. It is equivalent to par yield

Question 17

State a formula for the value of a swaption, which gives you the right to enter into a pay fixed rate (R_x) and recieve floating rate swap

Answer 17

L/Mx[F₀ŷ(d₁) - R_xŷ(d2)] Σ^mn_{i = 1}p(0,t_i)

• d₁ = (ln(F₀/R_x) + σ²T/2))/(σT^0.5)
• d₂ = d₁ - σT^0.5
• L is principle amount
• m is number of swap payments per time period
• F₀ is forward swap rate and R_x is fixed interest rate
• P(0,t) is discount factor for cashflow at time t_i
• σ is volatility of swap rate
• T is the strike date of swaption

Question 18

Describe how to value a securitised bond

Answer 18

• Use deterministic discounted cashflow approach
  
  • Discount rate reflects overall riskiness of bond and should be similar to yield offered by equally risky bonds (similar to credit rating)
• Simulate and discount possible cashflows, allowing for
  
  • Probablility, timinig of any defaults, and likely recoveries
  • Probability, timing and likely extent of any resource to originating company and or any guarantees
  • potential treasury management issues
  • ranking and strucutre of different bond tranches
• Discount rates should allow for risks not captured in the cashflows

Question 19

State what the price of a plain vanilla credit default swap shoudl be if it purchased via a single premium and an annual premium

Explain what the value of a total return swap should be equal to

Answer 19

Price of single premium credit default swap

Equals expected default loss on reference bond

Price of annual premium credit default swap

Equals credit spread on reference bond

Value of total return swap

Equals difference between values of assets generating returns on each side of swap.

These ignore taxes, transaction costs, bank’s profit margin and default risk of bank

Question 20

Equal why the equally of a company can be considered as a call option on the company’s assets

Answer 20

• Suppose company has amount D of zero coupon bond outstanding that matures at time T and let V_t = value of company’s assets at time t.
• If VT < D, company will default on its debt at T, the value of equity is then zero
• If VT > D, company will repay debt and value of equity = V_t - D
• So, value of firm’s equity at time T is E_t = max(V_t - D,0)
• This is equivalent to payoff from a call option on company’s assets with a strike price equal to the amount of debt, D