Correcting the Synthetic Deposit Spread Formula

Summary

The document concerns the correction term used when constructing synthetic deposits for the short end of an interest-rate curve. The question describes applying a quadratic expression involving parameters alpha and beta and a time interval, but reports that the resulting values do not match those in a referenced curve-bootstrapping paper. It also says the paper’s own factors fail to reproduce the expected figures under that interpretation.

The answer identifies the issue as a misreading of the expression: the correction should be alpha plus beta multiplied by half the interval, rather than alpha times the interval plus beta times half the interval squared. This changes how the parameters are applied and removes the quadratic dependence described in the question. The exchange supplies no independent derivation, calibration details, or discussion of day-count conventions, so it is a focused formula clarification rather than a complete guide to synthetic deposit construction.

Key ideas

• Synthetic deposits can be used to improve the short end of a bootstrapped yield curve.
• The question’s proposed correction term applies the parameters with an incorrect formula.
• The answer states that the correction is alpha plus beta times half the time interval.
• The exchange does not provide a derivation or broader curve-construction guidance.

Tags

# Construction of synthetic deposits


# Construction of synthetic deposits












I'm looking at the paper "Everything You Always Wanted to Know About Multiple Interest Rate Curve Bootstrapping But Were Afraid To Ask". It describes how to construct synthetic deposits in order get a "better" yield curve at the short end of the curve (see 4.4.2). Despite I'm not sure about the OIS quotes the synthetic depo should be $OIS + \alpha\cdot\delta+\beta\cdot\delta^2/2$ with the parameters alpha and beta given in figure 17 and $\delta$ being the time-interval $T_2-T_1$ with a given day-count-convention.

In the following I will just talk about the 12M example in figure 17. Ignoring any day-count-convention, i.e. using the number of days between $T_1$ and $T_2$ I can replicate similar factors for $\alpha$ and $\beta$, namely $\alpha = 0.5091; \beta = -0.0003$. Using them to calculate the 12M "spread" ($\alpha\cdot\delta+\beta\cdot\delta^2/2$) I get values like 0.0506, 0.3431, 0.6597, ... -9.6754, 13.3500, -17.5124 (all given in %). So they do absolutely not match the ones given in the paper.

Also when using the factors in the paper, the numbers do not match. Can someone explain to me what's going on there?

Best

## Answer by Daniel (score 1)

https://quant.stackexchange.com/a/39796

I figured out that the correction term should not be $\alpha\cdot\delta + \beta\cdot\delta^2/2$ but rather $\alpha+\beta\cdot\delta/2$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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