interpolation calc difficulties (help please)

[George]

Hi, I’m having a hard time calculating the following (exam 3 – mapping zero-coupon bonds)

“Suppose we wish to estimate the VaR of a mapped zero-coupon bond w/ 10 months to maturity,
and the nearest reference horizons in our mapping system are three months and 1 year.”

Given 3-mth and 12-month zero rates of 4.5% and 5%, the interpolated 10-mth zero is 4.889%.

1st rate x (2nd rate term – mapped term) + 2nd rate x (mapped term -1st rate term) / (2nd term – 1st term)
(0.045*(12/12-10/12)+0.05*(10/12-3/12))/(12/12-3/12)=0.04889

An ? of 1 bp on the 3-mth rate = 0.0451 interpolates to ->
(0.04501*(12/12-10/12)+0.05*(10/12-3/12))/(12/12-3/12)=0.048911109 or 4.891%

An ? of 1 bp on the 12-m rate = 0.0501 interpolates to ->
(0.045*(12/12-10/12)+0.0501*(10/12-3/12))/(12/12-3/12)=0.048966666 or 4.896%

The CF mapped to the 3-mth has a value sensitivity of $17.78 to a one bp change in the three-month rate.
The CF mapped to the 12-mth has a value sensitivity of $62.23 to a one bp change in the 12-month rate.

PV01 = PVBP = $1(e-10/12 x 0.04891109 – e-10/12 x 0.04889) = -16.89
PV01 = PVBP = $1(e-10/12 x 0.04896666 – e-10/12 x 0.04889) = -61.34
I get 16.89 and 61.34 (is this due to rounding?)

CF = -16.89/e-3/12 x 0.0451 – e-3/12 x 0.045) = $683,263.17
CF = -61.34/e-12/12 x 0.0501 – e-12/12 x 0.05) = $644,879.80

Textbook:

CFs are $719,217 at 3 mths and $654,189 at 12 mths. (big difference compared to my numbers)???

3-mth rate absolute vol. of 0.0125 × 4.5% = 0.0563% or 5.63 bps as a one std dev daily change.
12-mth rate absolute vol. of 0.01 × 5.0% = 0.05% or 5.0 bps as a one standard deviation daily change.
Assume an estimated ? of 0.85 bw 3 and 12mth returns.

Based on this info, the portfolio of mapped cash flows has a standard deviation of $399.62.
Hence the VaR at the 95% C.L. (taking Z0.05 = -1.645) is 1.645 x 399.62 = $657.31.
The derived one-day std dev of the 10-mth zero rate equals 4.995 bps.
Since the impact of a one bp change on the value of the bond is $80.003, the resulting VaR is 1.645 x 4.995 x 80.003 = 657.31.

How do they calculate this?

[Author]

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