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Spot, Forward and Par Rates

4 important questions on Spot, Forward and Par Rates

What about a definition of spot rates?

Spot rate is the rate on a spot loan, an agreement in which a lender gives money to the borrower at the time of the agreement to be repaid at some single, specified time in the future.

We can get the discount rate from a spot rate by: d(t) = 1/(1+ r(t)/2)^2t

What about a definition of par rates?

What rate C(T) make that the present value is equal to par (or 100).

Coupon rate/2 * (sum of all discount factors) + 1*discount factor for maturity term = 1

Sum of all discount factors is called an annuity factor.

How can we quote prices based on discount factor, spot, forward and par?

c =coupon rate

Discount factors:
P = c/2*A(T) +(d(T)

Spot rates:
P = c/2*(1/(1+spot(0,5)/2) + 1/(1+spot(1)/2)^2 + ..... ) + 1/(1+spot rate(T)/2)^2t

Forwards:
P = c/2*(1/(1+forward(0,5)/2) + 1/((1+forward(0,5)/2)*(1+forward(1)/2) + .....) + 1/spot (but based on series of forwards)

Par rate:
P = 1 + (c-C(T))/2 * A(T)

What about an abnormally downward sloping 10s-30s curve?

Due to uncertainty in Europe on Greece, the 10 to 30 forward curve was downward sloping. Important for financial institutions to match maturity of assets and liabilities to prevent high capital requirements based on solvency II.

What does flattening mean? (problem: not necessarily flatten)
Long-term rates fall by more than shorter-term rates or
Short-term rates rise by more than longer-term rates

What does steepening mean? (problem: not necessarily steepen)
Longer-term rates rise by more than shorter-term rates or
Shorter-term rates fall by more than longer-term rates

The question on the page originate from the summary of the following study material:

Part 4: Valuation and Risk Models

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