## Puzzle #6: Treasury Debt

Last week was historic for U.S. Treasury debt.  The yield on the 10-year Treasury dipped below 1.44% to its lowest level ever, and the total debt held by the public* rose to $11,006,251,094,374.26. Some might argue that$11 trillion in public debt is not that material for two reasons:

1. We don’t have to pay back the principal since we can always refinance; and
2. The interest payments are not significant since interest rates are so low.

This suggests a puzzle: assuming that the principal never has to be repaid because of refinancings, and that the interest rates never change, what is the total present value of all the future interest payments?

Make the following assumptions:

1. The total debt is exactly $11 trillion. 2. All debt securities pay interest at fixed rates. Treat T-Bills and inflation-protected securities as if they make fixed interest payments on fixed principal amounts. 3. When any security matures, it is refinanced into an identical security with the same principal and interest rate. The refinancings continue indefinitely. 4. The discount rate for each security (and its successors) is the same as its current interest rate. Oh, and you have five minutes. Please show your work. Good luck! *This excludes more than$4.7 trillion held by government accounts such as the Social Security trust fund.

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### 8 Responses to Puzzle #6: Treasury Debt

1. Saravanan says:

Net Present value would be zero

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• Win Smith says:

Thanks, but I’m asking for the present value of a positive stream of interest payments. I’m not really asking for an NPV, but would you explain your result?

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2. jeff says:

$11 b. PV of future interest payments is value of loan. Interest payment / yield = current loan value Like • Win Smith says: Jeff, That’s correct, except I think you mean$11 trillion. You give the formula for the price of a perpetual bond, which fits this scenario. Thank you for looking at the puzzle.

Win

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• jeff says:

Great; I haven’t forgotten everything I learned, except a few zeros.

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3. Giancarlo says:

I made the assumption of a continuous stream of payments because I like integrals better than series. So, setting N= \$11 trillion,denoting by r the fixed interest rate and by t the time in years from the calculation date, each payment is equal to N*r*dt and it’s present value(discounting continuously by coherence) is N*r*dt*exp(-r*t). Integrating the last expression from t=0 to +\infty gives you the present value of this perpetual stream of payments, i.e. N.

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• Giancarlo says:

oops, errata: … N*r*dt and its present … 🙂

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• Win Smith says:

Giancarlo, that is an elegant solution. I hadn’t thought about the continuous version of the problem. It makes sense since perpetual bond pricing relates to the geometric series, and your integral is the continuous analogue. Thank you, Win

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