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### How to calculate the weightings of VSB and VSC given a desired allocation of government and corporate bonds?

Let’s say I want to allocate 10% of my investment portfolio in government bonds and 5% in corporate bonds. For that I want to use short-term bonds ETFs.

After some research, I narrowed my choices of ETFs to these:
VSB is composed approximately of 72% government bonds and of 28% corporate bonds. On the other hand, VSC is composed roughly of 8% government bonds and of 92% corporate bonds.

Question: given the description above, what would be the percentages of VSB and VSC in our portfolio?

At this point, I am going to throw a couple of formulas at you. You can use them to calculate the weightings of VSB and VSC. I came up with these formulas by myself. I can share the algebraic demonstration…If you want it, drop a comment below.

% VSC = S * [ ( X - Y*R ) / ( Z*R Y*R + X W )  ]
% VSB = S * [ (Z*R - W) / (Z*R Y*R + X - W)  ]

Where:
• X is the percentage of VSB dedicated to government bonds; in our example would be 72.
• Y is the percentage of VSB dedicated to corporate bonds; in our example would be 28.
• W is the percentage of VSC dedicated to government bonds; in our example would be 8.
• Z is the percentage of VSC dedicated to corporate bonds; in our example would be 92.
• S is the sum of the weightings of government and corporate bonds in our portfolio; in our example S would be 15 (10 + 5), because we want to have 10% in government bonds and 5% in corporate.
• R is the ratio of government bonds to corporate bonds; in our example R would be 2 (10 / 5).

Finally, let’s evaluate each formula:

% VSC = 15 * [ ( 72 - 28*2 ) / ( 92*2 – 28*2 + 72 – 8 )  ] = 1.25%
% VSB = 15 * [ (92*2 - 8) / (92*2 – 28*2 + 72 – 8)  ] = 13.75%

Answer: if we want to hold 10% in government bonds and 5% in corporate bonds; then 1.25% of our portfolio should be allocated in VSC and 13.75% should be allocated in VSB.

The usefulness of the formulas lies in the ability to change your target weightings for government and corporate bonds. This will of course result in new values for R and S that can be used to revaluate the formulas, allowing you to calculate the new percentages for VSB and VSC.

Finally, notice that the values of X, Y, Z and W do not change often. They represent the weighting of the different bonds within these ETFs. Refer to “Sector weighting (% of net asset value)” within the corresponding ETFs fact sheets if you want to determine these values yourself.

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VSB fact sheet: