Basic economic theory based on convertible government bonds (GCB) - a concise theory based on mathematical formulas -
October 18, 2026
Summary
- Purpose
This paper uses a formula to explain how the government’s demand for funds changes at maturity using the Government Convertible Bond (GCB).
GCB is a government bond that allows you to choose whether to receive a face value redemption at maturity or convert it to a government stock.
Convertible government bonds and national capital funds are treated as independent systems.
- Basic variables
B: Total face value of convertible government bonds that are mating
Α: Conversion rate to government stocks (0≤α≤1)
R: Face value redemption amount at maturity
C: Conversion amount to government shares
- Basic equation
The total face value B of convertible government bonds is divided into face value redemption amount R and conversion amount C.
B = R + C
If the conversion rate to government shares is α, the conversion amount is expressed in the following formula.
C = B x α
The face value redemption amount is as follows.
R = B × (1 - α)
Therefore, the basic equations are the following three.
B = R + C
C = B x α
R = B × (1 - α)
- Changes in demand for funds due to conversion
In the case of full repayment of convertible government bonds, the principal redemption fund at maturity required by the government is B.
If the conversion to government shares is made at a conversion rate of α, the principal redemption fund at maturity will be R.
If the decrease amount is ΔR,
ΔR = B - R
If you substitute R = B × (1 - α),
ΔR = B - B × (1 - α)
Therefore,
ΔR = B × α
Also, because C = B × α,
ΔR=C
Therefore, the amount of conversion to government shares and the amount of reduction in cash expenditure required for principal redemption at maturity are the same.
- The relationship between conversion rate and capital demand
The relationship between the conversion rate α and the face value redemption amount R is
R = B × (1 - α)
It is.
If B is constant, the more the conversion rate increases, the more the face value of the redemption decreases.
This relationship can also be expressed in the following formula.
dR/dα = -B
In other words, when the conversion rate increases, the face value redemption at maturity decreases proportionally.
- Future payment obligations of government shares
Even if cash expenditure at maturity decreases due to conversion to government shares, future expenditures will occur if future dividend obligations and principal redemption obligations are set on government shares.
If the conversion amount is C and the annual dividend rate is d, the annual dividend amount D is
D = C × d
If the dividend is paid in T years and the annual dividend amount is constant, the cumulative dividend amount, which does not take into account the discount,
Cumulative dividend amount = C × d × T
It will be.
Therefore, the conversion to government shares reduces the cash expenditure required for principal redemption at maturity, while creating future payment obligations according to the terms of the government shares contract.
- Conclusion
The basic structure of convertible government bonds (GCB) is summarized in the following formula.
B = R + C
C = B x α
R = B × (1 - α)
ΔR = B × α = C
From these formulas, it is induced that the higher the conversion rate to government shares, the less cash expenditure required for principal repayment at maturity.
GCB is a system that can express government funding and demand for funds at maturity by variables called conversion rates.
The decrease in cash expenditure shown here is not synonymous with a decrease in the overall debt burden of the government. The dividend obligation, principal redemption obligation, and accounting classification of government shares are determined based on the terms of each contract.