Connecting Continuous Yield and Discrete Income Forward Pricing

Summary

The document reconciles two forward-pricing expressions for an asset that pays income during the contract. One uses a continuously compounded risk-free rate and an average continuous yield; the other subtracts the present value of fixed income payments from spot and then accumulates the remainder at a force of interest. The answer shows the link by rewriting the yield-based expression as an interest-accrued value after subtracting the income component.

In that setup, the present value of income is represented by spot multiplied by one minus the exponential of negative yield times maturity. Substituting this amount into the second expression makes it agree with the first, assuming the rates and income description are aligned. The explanation supplies an algebraic identification and an interpretation in terms of dividends paid over the contract’s life, but does not address noncontinuous dividend schedules, taxes, transaction costs, or other carry components.

Key ideas

• Forward price can be expressed using either a continuous yield or the present value of income paid during the contract.
• The yield-based expression can be rewritten as an accumulated spot value less an income component.
• The matching present value of dividends is spot multiplied by one minus the yield discount factor.
• The equivalence depends on consistent rate conventions and an income stream represented by the assumed yield.

Tags

# Pricing of forwards contracts


# Pricing of forwards contracts












Of the courses I am taking in college this semester, two are Financial Mathematics and Derivatives. In each course, we learn different formulas to calculate the forward price of a forward contract. Obviously, the two formulas must equal each other, but I am not sure what the link is i.e. how to get from one formula to the other.

In my Derivatives course, I was taught that $$F_0 = S_0 e^{(r - d)T},$$ where

$F_0$ is the forward price,

$S_0$ is the spot price,

$r$ is the risk-free rate,

$d$ is the average yield per annum and

$T$ is the length of the contract (in years).

In my Financial Mathematics course, it is also given that $$F_0 = (S_0 - PV_I) e^{\delta T},$$ where

$F_0$ is the forward price,

$S_0$ is the spot price,

$PV_I$ is the present value of the fixed income payment(s) due during the term of the contract,

$\delta$ is the force of interest and

$T$ is the length of the contract (in years).

How do I reconcile the two formulas? In other words, how can I prove that $$F_0 = S_0 e^{(r - d)T} = (S_0 - PV_I) e^{\delta T}?$$

Any intuitive explanations will be greatly appreciated! :)

## Answer by Antoine Conze (score 5, accepted)

https://quant.stackexchange.com/a/61526

Decompose the first formula as $F_0=(S_0 - S_0(1-e^{-dT}))e^{rT}$ then let $PV_{I} = S_0(1-e^{-dT})$ which represents the present value of dividends (dividend rate = $d$) paid on the security during the life of the contract, and you obtain the second formula.

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