Although discounting has a critical role in funding and refunding decisions, it does not receive the attention it deserves.

For example, to analyze a refunding proposal, we need to determine the cost of the outstanding and the refunding bonds on a present-value basis.

Should tax-exempt and taxable bonds be discounted with the same rate or with different rates?

Earlier this year, I showed that tax-exempt and taxable bonds should be discounted with the same rate, namely with the municipality’s taxable borrowing rate.

Because the taxable rate is higher than the tax-exempt rate, the cost of a tax-exempt bond, as defined by the present value of its cashflow, will be lower than its market price.

Consider an optionless 30-year 3% bond when the municipality’s tax-exempt yield curve is 3% flat, and its taxable curve is 4% flat. The market price of this 3% bond is 100.00, while its cost to the municipality is 82.62. The 17.38 difference is the federal subsidy received by the municipality.

In light of the enthusiastic approval of the ‘taxable discount rate’ recommendation, we next investigate its implications regarding various bond structures. For example, instead of the 30-year 3% bond above, consider a 30-year step-down coupon bond which pays 3.50% for 10 years, and then 2.61% for the remaining 20 years.

At a 3% discount rate this step-down coupon bond would also be priced at 100. How about the municipality, whose discount rate is 4%? The reader can confirm that the cost of the step-down is 83.12, which is 0.50 point *higher* than 82.62 cost of the 3% single-coupon bond.

So the municipality would obviously prefer the single-coupon bond to the step-up.

But what is the relevance of step-down coupon bonds to callable bonds? Continuing with the example, suppose the current 30-year optionless rate is 3% and the 20-year rate 10 years from now is certain to be 2.61%. If the municipality wants to issue 30-year bonds callable at par in year 10, the market will set the coupon to 3.50%, i.e. 50 basis points above the 3% coupon of the optionless bond.

The combined cashflows of the 3.50% callable bond and the 2.61% refunding bond will be the same as that of the step-up coupon bond. Clearly the issuer should prefer the 3% optionless bond to the 3.50% callable bond.

The above example is admittedly simplistic, because interest rates are uncertain. What can we conclude under the realistic assumption that interest rates are in fact volatile?

This is a thorny problem — for starters, there are actually two relevant yield curves; taxable rates are needed for discounting and tax-exempt rates to value the call option.

Solving this problem requires heavy mathematical artillery. Assuming the taxable and tax-exempt rates are highly correlated (and in fact they are), we can show that the expected cost of a fairly priced callable bond exceeds that of an optionless bond. The proof can be found in ‘Tax Differentials and Callable Bonds’ by Boyce and Kalotay, published in the Journal of Finance (September 1979).

The task of the municipal debt manager is to minimize the present-value cost of the debt. The call options of the bonds are major contributors to this cost; the municipality pays for these options in the form of above-market coupons or reduced sale prices.

How does the expected savings from refunding compare to the cost of the call option? The discount rate is a critical consideration in analyzing this problem; the correct rate is the municipality’s taxable borrowing rate.

The correct rate directly implies that the cost of the call option of a tax-exempt bond exceeds its expected benefit.

Instead of issuing callable bonds, municipal borrowers should issue optionless bonds at higher prices, or with lower coupons.