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:

- We don’t have to pay back the principal since we can always refinance; and
- 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:

- The total debt is exactly $11 trillion.
- 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.
- When any security matures, it is refinanced into an identical security with the same principal and interest rate. The refinancings continue indefinitely.
- 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.

Copyright 2012. All rights reserved.

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Net Present value would be zero

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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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$11 b. PV of future interest payments is value of loan. Interest payment / yield = current loan value

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

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

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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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