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Fixed Income > The Arbitrage-Free Valuation Framework > Flashcards

The Arbitrage-Free Valuation Framework Flashcards

1
Q

How do we calculate the number of possible paths in Binomial trees?

A

2^(n-1)

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2
Q

Interest rate tree i2LL referes to?

A

The one year forward rate at time 2, assuming the lower rates at time 1 and 2

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3
Q

In determining the appropriate level of volatility to use in modeling paths interest rates, we would most likely NOT use

A

Implied volatility based on observed prices of option-free Government bonds.

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4
Q

What does the log-normal random walk volatility capture?

A

The volatility of the one-year rate

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5
Q

When are Callable Bonds more valueable?

A

During a downward sloping yield curve

Call Option is valuable when yield curve flattens

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6
Q

When are Putable Bonds more valueable?

A

When the yield curve is upward sloping

Put Option is valuable when yield curve steepens

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7
Q

Formula for Value of issuer call opion

A

Value of stright bond - Value of callable bond

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8
Q

Formula for Value investors Putable bond

A

Value of putable bond - Value of stright bond

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9
Q

If volatility increases, what will happen to the value of callable bond

A

The new value will be lower than the previous price.

Value of Call = V Stright - V Callable

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10
Q

If volatility increases, what will happen to the value of Putable bond

A

The new value will be greater than the previous value

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11
Q

Explain the relationship of what will happen to the value of callable and putable if volatility increases

A

Callable bond value decreases

Putable bond value increases

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12
Q

If the OAS (Option Adjusted Spread) for a bond is higher than its peers, it is considered to be…

A

Undervalued
OAS for a bond is higher than the OAS of its peers, it is considered to be undervalued i.e. attractive
investment meaning it offers a higher compensation for a given level of risk (cheap).

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13
Q

If the OAS (Option Adjusted Spread) for a bond is Lower than its peers, it is considered to be…

A

Overvalued!!

bonds with
low OAS relative to peers are considered to be overvalued (rich) and should be avoided.

It offers lower compenstation for a given level of risk

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14
Q

What is the formula for Option Cost

A

Z-Spread - OAS

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15
Q

What is the formula for Z-Spread

A

OAS - Option cost

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16
Q

Z-Spread ≥ OAS

A

Callable bond

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17
Q

Z-Spread ≤ OAS

A

Putable bond

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18
Q

Option cost for Callable bonds when Volatility increases

A

Positive Option cost

Callable Bond = Z-Spread ≥ OAS =

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19
Q

Option cost for Callable bonds when Volatility Decreases

A

Negative option cost

Callable Bond = Z-Spread ≥ OAS

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20
Q

Option cost for Putable bonds when Volatility increases

A

Negative option cost

Putable Bond = Z-Spread ≤ OAS

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21
Q

Option cost for Putable bonds when Volatility Decreases

A

Positive option cost
Putable Bond = Z-Spread ≤ OAS

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22
Q

What is the most appropriate duration to use for bonds with embedded options?

A

Effective duration

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22
Q

How do we interpret 1.97 effective duration?

A

for a 100 bsp change in the interest rates, the bond price will change by 1.97% on average.

23
Q

What is effective duration?

A

A parallel shift in the yield curve. (benchmark yield curve)

  • assuming no change in the bond’s credit spread, but it is not an accurate.
  • measure of interest rate sensitivity to non-parallel shifts in the yield curve like those described by ‘Shaping Risk’.
  • Shaping Risk refers to changes in portfolio value due to changes in the shape of the benchmark yield curve. However, parallel shifts explain more than 75% of the variation in bond portfolio returns.
24
Q

Please explain deep in-the-money embedded option bonds

A

When the embedded option (call or put) is deep in the money, the effective duration of the bond with an embedded option resembles that of the straight bond maturing on the first exercise date,
reflecting the fact that the bond is highly likely to be called or put on that date.

25
Q

Please explain the relationshop of out-of -the-money embedded option bonds

A

Effective Duration Callable ≤ Effective Duration Straight
Effective Duration Putable ≤ Effective Duration Straight

Effective Duration ZCB ≈ Maturity of the Bond
Effective Duration Fixed Rate Coupon < Maturity of the Bond
Effective Duration Floater ≈ Time in Years to Next Reset

26
Q

Please explain the relationshop of At -the-money embedded option bonds

A

The effective duration of the callable bond shortens when interest rate falls, which is when the call
option moves into the money, limiting the price appreciation of the callable bond.

The effective duration of the putable bond shortens when interest rates rise, which is when the put option moves into the money, limiting the price depreciation of the putable bond.

While effective duration of straight bonds is relatively unaffected by changes in interest rates.

27
Q

What kind of relationship does call option value have with interest rates?

A

Inverse relationship

Effective convexity of the callable bond turns negative when the call option is near the money
which indicates that the upside for a callable bond is much smaller than the downside. When rates are high, callable bonds are unlikely to be called and will exhibit positive convexity.

28
Q

What kind of relationship does Put option value have with interest rates?

A

Direct relationship

Putable bonds always have positive convexity.

When the option is near the money, the upside for a putable bond is much larger than the downside
because the price of a putable bond is floored by the price of the put option, if it is near the exercise
date.

29
Q

Which type of bonds can experience negative convexity?

A

Callable bonds.

30
Q

Convertible Bonds: Conversion Value

A

Share price x Conversion Ratio

31
Q

Convertible Bonds: Conversion Ratio

A

Bond Price / Conversion Price

32
Q

Convertible Bonds: Conversion Price

A

Par or issue price / Conversion ratio

33
Q

Convertible Bonds: Market Conversion premium per share

A

( Market value of convertible bond / Conversion Ratio ) - share price

34
Q

Convertible Bonds: Market Conversion premium per share RATIO

A

[ ( PV convertible bond / Conversion Ratio ) / Share price ] -1

35
Q

one-sided durations

A

Effective durations when interest rates go up or down, which are better at capturing the interest rate sensitivity of bonds with embedded options that do not react symmetrically to positive and negative changes in interest rates of the same magnitude

36
Q

Effective Duration indicated the sensitivty of the bonds full price to a 100 bsp shift in the government

A

Par Curve

37
Q

Downward sloping yeild curve -> Upward sloping yield curve

A

Put option: Increases
Call option: Decreases

38
Q

Upward sloping yeild curve -> Downward sloping yield curve

A

Put option: Decreases
Call option: Increases

39
Q

What will happen to OAS when volatility increases

A

OAS decreases

40
Q

What will happen to OAS when Volatility Decreases

A

OAS increases

41
Q

On-sided duration for Callable bonds

A

On-sided up duration > on-sided down duration

42
Q

On-sided duration for Putable bonds

A

On-sided Down duration > on-sided up duration

43
Q

Which combination will lead to lower Call option value

A
  • Lower Volatility
  • Higher call price (strike)
  • Low cupon price
44
Q

Which combination will lead to lower put option value

A
  • Lower Volatility
  • Lower put price (strike)
  • Higher coupon
45
Q

What is OAS (Option Adjusted Spread)

A

Option Adjusted Spread (OAS) is the constant spread, when added to all one-period forward rates on the interest rate tree which makes the arbitrage-free value of the bond (calculated value) equal to its market price.

The option-adjusted spread (OAS) is the measurement of the spread of a fixed-income security rate and the risk-free rate of return, which is then adjusted to take into account an embedded option.

Typically, an analyst uses Treasury yields for the risk-free rate. The spread is added to the fixed-income security price to make the risk-free bond price the same as the bond.

46
Q

Value of floored floater bond

A

Value of stright bond + Value of embedded floating floor

47
Q

Value of convertible bond

A

Value of stright bond + value of call option on issuer stock

48
Q

Explain why putable bond will be trading higher than callable bonds when rates decrease?

A

When rates drops, callable bonds will increase in value only to a certain limit, 100$.
Stright bonds and putable bonds will always have higher value than callable bonds because of more oppertunites to exercise the option.

49
Q

The minimal value of a convertible bond is equal to the greater of …

A

Conversion value and the value of the option free underlying bond

50
Q

Convertible bond

Premium over stright

A

[Convertible bond price / Option free bond price] - 1

51
Q

Formula for conversion price

A

Issue Price / Conversion Ratio

52
Q

If we only know the 2 year par rate, and we know the 1 year, par, spot, and forward rates, how to we find the 2 year spot rate and forward rate?

A

First we need to find the 2 year spot rate.
Use the 1 year spot rate as the discount factor, and use the 2 year par rate as the payment.

Solve for spot rate at year 2 as the discount factor.

That will provide us the 2 year spot rate which we can use to solve the forward rate

53
Q

Value of a floor floater

A

Value of stright bond + Value of embedded floor

54
Q

Value of callable bond

A

Value of stright bond - value of issuer call option

Fixed Income (6 decks)

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