Bootstrapping Short-Term OIS Discount Factors from Compounded Rates

Summary

The document asks why short-maturity overnight index swap discount factors are often calculated using a simple-rate expression instead of an annual-compounding expression. The answer relates the quoted swap rate to the compounded overnight rates on the floating leg. For a maturity under one year, the compounded floating return is represented by the swap rate applied over the total accrual period, so the discount factor is the reciprocal of one plus that rate times the accrual fraction.

This provides the short-tenor relationship used in OIS curve construction and explains why the floating leg’s underlying overnight rates compound even when the short swap rate is expressed with simple accrual. The document does not expand on the longer-tenor bootstrap formula or discuss day-count conventions, payment schedules, or market-specific details, so implementation requires those conventions from the relevant curve setup.

Key ideas

• Short-maturity OIS rates are linked to compounded overnight returns on the floating leg.
• The short-tenor discount factor uses the swap rate multiplied by the total accrual fraction.
• The discount factor is the reciprocal of the compounded return over the period.
• Curve construction also depends on conventions and schedules not covered in the response.

Tags

# OIS Discount Factor Bootstrapping - Do we assume simple interest?


# OIS Discount Factor Bootstrapping - Do we assume simple interest?












When I am reading papers (ie here and here) on bootstrapping discount curves they refer to obtaining discount factors from rates for swaps maturing less than a year with:

$$D(t, T_i) = \frac{1}{1+s_i(T_i-t)}$$

(where $s_i$ is the swap rate, $T_i-t$ is the time to maturity in years and $D$ is the discount factor)

In short my question is why not this?:

$$D(t, T_i) = \frac{1}{(1+s_i)^{(T_i-t)}}$$

Is the difference because we are assuming there is no compounding? Is this a correct assumption? It's quite a key question because the denominator in the bootstrapping formula for tenors > 1 year (because the ois swaps pay annually) depends on this:

$$ D(0,T_i) = \frac{1-s_i\sum_{j=1}^{i-1}(T_j-T_{j-1})D(0,T_j)}{1+s_i(T_i-T_{i-1})} $$

## Answer by Attack68 (score 2, accepted)

https://quant.stackexchange.com/a/36404

An OIS interest rate swap rate with annual-annual freq is determined under one year by: $$1 + d_i s_i = \prod_{j=1}^{n(i)}(1+ d_j r_j) \; , \quad \text{where} \quad d_i = \sum_{j=0}^{n(i)} d_j \;.$$ Each $r_j$ is a forecast overnight OIS rate which as you can see are compounded in the floating side. Therefore a discount factor in the future, for maturity $m_i<1Y$, which would be represented by: $$ \text{discount factor at }m_i = \frac{1}{\prod_{j=1}^{n(i)}(1+d_jr_j)}=\frac{1}{1+d_is_i}\;.$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.