Case 097Fixed income and cash managementCore

A tax-free bond with a 7.3% coupon and 8 years left trades at 108. What is its yield to maturity, and what taxable yield is it worth to a client at an illustrative 30% slab?

The situation

A client in a high tax slab is offered a tax-free bond on the exchange. It pays a 7.3% annual coupon on its Rs 100 face value, has eight years to maturity, and trades at Rs 108. The coupon is exempt from income tax; assume, for illustration, that the client's marginal rate on interest is 30%, and tell him to confirm his slab including any surcharge and cess.

The alternative on the table is a taxable bond of similar quality and term yielding 7.8%. Ignore trading costs and the tax on any capital loss at maturity.

Your task

What does the tax-free bond actually yield at 108, what is that worth against a taxable bond, and which is better for this client?

Quick check

Is the yield to maturity above or below the 7.3% coupon?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The yield to maturity is about 6.01%, and at a 30% slab that is worth a taxable yield of about 8.59%. Buying at 108 means losing 8 points of premium by maturity, so the yield sits below the 7.3% coupon. The taxable alternative at 7.8% leaves only 5.46% after tax, so for this client the tax-free bond is the better income, provided his slab is confirmed.

Step 1Why is the yield below the coupon?

Because the coupon is paid on Rs 100 and he pays Rs 108. Paying extra for a flat with a sitting tenant gets you the rent, but the extra is gone when you sell at the ordinary price. Each year he receives Rs 7.30 of tax-free coupon but loses about Rs 1 of the Rs 8 premium, so his return is roughly Rs 6.30 on about Rs 104 of average outlay, near 6.1%. The exact yield to maturityThe single annual rate that makes the present value of all remaining coupons and the final repayment equal to the price paid today. solves the price equation and gives 6.01%.

The relationship
108=∑t=187.3(1+y)t+100(1+y)8  ⇒  y≈6.01%108 = \sum_{t=1}^{8} \frac{7.3}{(1+y)^t} + \frac{100}{(1+y)^8} \;\Rightarrow\; y \approx 6.01\%
108price paid today per Rs 100 of face value
7.3annual tax-free coupon
100repaid at maturity
yyield to maturity
What it says in wordsThe yield is the discount rate at which eight coupons and the final repayment are worth exactly the Rs 108 paid.
Coupon, current yield, yield to maturity, and what it is worth: four numbersCoupon on face value7.30%, paid on Rs 100, not on Rs 108Current yield6.76%, coupon / priceYield to maturity6.01%, after losing the premiumTaxable equivalent at 30%8.59%, what a taxed bond must payBars drawn to scale from zero.
The bond's 7.3% coupon becomes a 6.76% current yield at a price of 108 and a 6.01% yield to maturity once the premium is lost, which is worth 8.59% taxable to a client at a 30% slab.
Step 2What is the tax-free yield worth to this client?

Gross it up at his marginal rate. A 6.01% tax-free yield divided by one minus 30% is a taxable-equivalent yield of about 8.59%: a taxable bond would need to yield that much to leave him the same after tax. The 7.8% taxable alternative leaves 5.46% after tax, well below 6.01%. For a client at a low slab the comparison reverses: at 10%, the taxable bond leaves 7.02% and wins. The same bond can be right for one client and wrong for another.

Step 3What happens to the price over the eight years?

If yields stay put, the price drifts down to par. At a constant 6.0% yield, the bond is worth about 104.5 after four years and exactly 100 at maturity, so a client who reads only the coupon will be surprised by the falling price even though nothing has gone wrong. Rising rates would push the price lower still in the meantime, which matters only if he sells before maturity. Say the limit: tax-free bonds trade thinly, so the price quoted may not be the price a large order gets.

Pull to par: the 8-point premium melts away by maturity100104108Yr 0Yr 1Yr 2Yr 3Yr 4Yr 5Yr 6Yr 7Yr 8108.0 paid today100 repaidYear 4: 104.5Each year the client earns Rs 7.30 of coupon but loses about Rs 1 of premium
At an unchanged 6.0% yield, the tax-free bond's price falls from 108 to about 104.5 by year four and to 100 at maturity, which is the premium the yield to maturity accounts for.

Where candidates lose it

The common error is quoting the 7.3% coupon as the yield, or grossing up the coupon to 10.4% taxable-equivalent. Both ignore the premium, which costs about a point a year.

The second miss is using current yield, 7.3 over 108, as the answer. It captures the higher price but not the loss of the premium at maturity, so it overstates the return by about three quarters of a point.

What the interviewer asks next

  • At what price would the tax-free bond yield exactly the same after tax as the 7.8% taxable bond for this client?
  • How would a rise in rates of one point change the bond's price today?
  • Why might a client at a 10% slab still prefer the tax-free bond?

Company names and figures are illustrative.