# 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

## 1 Answer

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