Let's get down to Finance 101 - Basics of Forward Pricing.
What is a Forward Price?
A forward is an obligation to trade (i.e. buy or sell) an asset in the future at a particular price. What is a fair price to trade the asset in the future? - answer: the forward price.
Based on no-arbitrage, forward price K is the spot price of the asset, multiplied by our interest factor (exp(r*(T-t)). As long as interest rates are positive, the forward price will be greater than the spot price. Additionally, a forward further in the future will have a higher price than a forward traded nearer to the present date.
The above argument is true for a simple asset (i.e. one that doesn't pay dividends or have any other financial benefits of ownership). Also note, that the forward price does not depend on our expectation of the actual spot price at time T. Modifications are needed for different types of assets.
Also note here, that in the formula, rates are quoted in continuously compounded form, and time is measured in years. In other areas of finance (outside of derivatives) rates might be annually, semi-annually or quarterly compounded, and the quoted interest rate will be different. If rate is quoted in annual form, the equivalent continuously compounded rate will be lower (because the compounding is more frequent).
A continuously compounded rate is a rather fantastic concept if you think about. It represents the interest rate on cash compounded at a frequency approaching infinity. Option pricing and forward pricing are, in this sense, mathematically "advanced" concepts.
How do we get From Forward Prices to Forward Discount Factors?
To recap what a discount factor is - practically speaking, a discount factor is the multiplier that brings a future cash flow back to the present value. It's the basic component of Discounted Cash Flows. (Often, in financial markets, the discount factor comes from the zero-coupon yield curve and does not reflect inflation rates). Arithmetically, the discount factor is the ratio of the present value of a cash flow, over its future value. A discount factor can also be thought of as the price of a zero-coupon bond expiring at time T.
A forward discount factor as of today (t0) for the future interest accrual period (t1, t2) is the forward price (previously discussed) for a forward contract maturing at time t1 on a zero bond maturing at t2.
A forward discount factor, thus, is the discount factor that applies for an interest period interval in the future.
What is a forward rate?
A forward rate at time t (today, T0) for the future interest accrual period (T1,T2) is the interest rate R implied by the forward discount factor. In continuous compounding, R(t, T) = - (T-t)^(-1) ln (B(t,T)).
Spot Value Date and Discount Factors
Present values are often calculated with a spot value date; in which case discount factors will reflect discounting to the spot date rather than the current date.
What is a Forward Price?
A forward is an obligation to trade (i.e. buy or sell) an asset in the future at a particular price. What is a fair price to trade the asset in the future? - answer: the forward price.
Based on no-arbitrage, forward price K is the spot price of the asset, multiplied by our interest factor (exp(r*(T-t)). As long as interest rates are positive, the forward price will be greater than the spot price. Additionally, a forward further in the future will have a higher price than a forward traded nearer to the present date.
The above argument is true for a simple asset (i.e. one that doesn't pay dividends or have any other financial benefits of ownership). Also note, that the forward price does not depend on our expectation of the actual spot price at time T. Modifications are needed for different types of assets.
Also note here, that in the formula, rates are quoted in continuously compounded form, and time is measured in years. In other areas of finance (outside of derivatives) rates might be annually, semi-annually or quarterly compounded, and the quoted interest rate will be different. If rate is quoted in annual form, the equivalent continuously compounded rate will be lower (because the compounding is more frequent).
A continuously compounded rate is a rather fantastic concept if you think about. It represents the interest rate on cash compounded at a frequency approaching infinity. Option pricing and forward pricing are, in this sense, mathematically "advanced" concepts.
How do we get From Forward Prices to Forward Discount Factors?
To recap what a discount factor is - practically speaking, a discount factor is the multiplier that brings a future cash flow back to the present value. It's the basic component of Discounted Cash Flows. (Often, in financial markets, the discount factor comes from the zero-coupon yield curve and does not reflect inflation rates). Arithmetically, the discount factor is the ratio of the present value of a cash flow, over its future value. A discount factor can also be thought of as the price of a zero-coupon bond expiring at time T.
A forward discount factor as of today (t0) for the future interest accrual period (t1, t2) is the forward price (previously discussed) for a forward contract maturing at time t1 on a zero bond maturing at t2.
A forward discount factor, thus, is the discount factor that applies for an interest period interval in the future.
What is a forward rate?
A forward rate at time t (today, T0) for the future interest accrual period (T1,T2) is the interest rate R implied by the forward discount factor. In continuous compounding, R(t, T) = - (T-t)^(-1) ln (B(t,T)).
Spot Value Date and Discount Factors
Present values are often calculated with a spot value date; in which case discount factors will reflect discounting to the spot date rather than the current date.
