Discount curve and forward discounting
What you will be able to do
Section titled “What you will be able to do”After this lesson, you should be able to read a deterministic discount curve, state its interpolation boundary, and calculate a discount factor between two future dates.
A curve is a dated collection, not one yield
Section titled “A curve is a dated collection, not one yield”In the earlier lessons, the discount-factor gave the value at valuation-time of a deterministic payment at one payment-time. A discount curve gives the discount factor for many payment times. Each node of the curve is one payment time with its discount factor. Every node must have the same currency, the same valuation date, and the same collateral and funding basis. Nodes with different bases do not form one consistent curve.
The curve in this lesson accepts only positive discount factors at its nodes, and it sets . Between two nodes, the logarithm of the discount factor is interpolated linearly. The curve is not extrapolated beyond its final node. Log-linear interpolation is a convention of this lesson. The lesson does not claim that it is the only correct interpolation.
A discount factor can be larger than one. For example, implies a negative continuously compounded one-year zero rate. The curve requires positive discount factors, but it does not require discount factors below one.
Discount between future times
Section titled “Discount between future times”Let the local forward-discount-start-time be earlier than the local forward-discount-end-time . The forward-discount-factor is defined as the ratio of the two discount factors from valuation time:
Its unit is currency at time per unit of currency at time . If and , the forward discount factor from year 1 to year 3 is:
The forward discount factor has its own symbol, . The symbol keeps its meaning: a discount factor from valuation time.
The ratio in (3.1.1) is needed for multi-period valuation. Each lattice node needs the discount factor over its next period. A discount factor from time zero is not the correct discount factor for that period.
Scope boundary
Section titled “Scope boundary”The curve is an input. It is deterministic and not calibrated. Later lessons with state-dependent discounting give a one-period discount factor at each node. They do not assume that today’s forward discount factor is realized in every future state.
Knowledge check 3.1.1 Discount curve and forward discounting
Link to Knowledge check 3.1.1: Discount curve and forward discountingWhat does a deterministic curve node D(0,3) = 0.94 mean?
Check your answer to reveal the explanation.
Can a valid input curve contain D(0,1) = 1.01 in this lesson?
Check your answer to reveal the explanation.
Given D(0,1) = 0.98 and D(0,3) = 0.94, calculate the forward discount factor Z(1,3).
Check your answer to reveal the explanation.
Given D(0,2) = 0.95 and D(0,5) = 0.88, calculate the forward discount factor Z(2,5).
Check your answer to reveal the explanation.
Sources
Section titled “Sources”Discount-factor pricing and the law-of-one-price interpretation follow Tuckman and Serrat, Chapter 1 §§1.2–1.4 and Chapter 2 §§2.1 and 2.4 [1]. The later derivatives use of dated factors is consistent with Hull’s interest-rate and forward-pricing chapters [2].
References
Section titled “References”Credit default swap, the credit derivative the CDS lessons define and value.
The name of a family of standard credit default swap indices, each a standard portfolio of single-name contracts.
Duration times spread, a spread-risk measure for bonds.
Financial Industry Regulatory Authority.
International Money Market. In the CDS lessons, IMM dates are the standard maturity dates on the twentieth of March, June, September, and December.
International Swaps and Derivatives Association.
International Organization for Standardization.
Jump to default, the loss on an immediate default of the reference entity.
Coordinated Universal Time, the time standard the date arithmetic counts calendar days in.