Fixed-Income Valuation: Trading at a Premium or Discount

Would you rather purchase a bond with a 2% coupon or a 7% coupon, assuming the same issuer, face value, and maturity?

No, you would undoubtedly rather have a 7% coupon bond, all else equal, but that's not really enough information to indicate the best purchase.

Good answer! The price of each bond has the power to make either one a better deal, or to make you indifferent between the two. Price is the equalizer.

Suppose these two bonds are both $1,000 three-year bonds. The price of bond A, the 7% annual coupon bond is: $$\displaystyle P_A = \frac{\$ 70}{1+r_A} + \frac{\$ 70}{(1+r_A)^2} +\frac{\$ 1,070}{(1+r_A)^3} $$

If investors decide that a fair discount rate is 5% for this bond, what would you expect them to decide about the price?

No, consider that if they thought the bond paid "too much," then it would be a really good deal.

Exactly! If the bond pays 7% when investors demand 5%, then this looks like a good deal. Investors will bid the price of the bond up above par, which means that the bond is __trading at a premium__ to par value. In fact, the price of this bond would be: $$\displaystyle P_A = \frac{\$ 70}{1 + 0.05} + \frac{\$ 70}{(1 + 0.05)^2} +\frac{\$ 1,070}{(1 + 0.05)^3} $$ $$\displaystyle = \$ 66.67 + \$ 63.49 + \$ 924.31 + = \$ 1,054.47 $$

No, if investors decide on a 5% discount rate, that's the expected return they demand, and this 7% coupon bond pays "too much" relative to this threshold.

No, that's not really enough information to make the best decision.

What's the price of this bond?

No, this is also a premium of par, but consider that this bond pays "too little" compared to the discount rate. So the price will need to be less than par.

No, it's possible to obtain this value by discounting each cash flow one period too little.

That's it. $$\displaystyle P_B = \frac{\$ 20}{1 + 0.05} + \frac{\$ 20}{(1 + 0.05)^2} +\frac{\$ 1,020}{(1 + 0.05)^3} $$ $$\displaystyle = \$ 19.05 + \$ 18.14 + \$ 881.11 + = \$ 918.30 $$ So this bond is trading at a discount to the par value of $1,000. [[calc: 3 n 5 iy 20 pmt 1000 fv cpt pv , 3 n 5 i 20 pmt 1000 fv pv]]

To summarize this discussion: [[summary]]

However, investors are using the same 5% discount rate for bond B, which is also a $1,000 three-year bond, but with a 2% annual coupon. $$\displaystyle P_B = \frac{\$ 20}{1+r_B} + \frac{\$ 20}{(1+r_B)^2} +\frac{\$ 1,020}{(1+r_B)^3} $$

What's nice is that, by choosing price, investors can get whatever return they require from a bond, no matter the coupon paid. If a bond pays a little "too much," investors will pay a premium. If it pays "too little," they will pay a discount. Either way, price is again the great equalizer, making sure that each bond is a fair deal.

2% coupon

7% coupon

It depends on the prices

This bond pays "too much" relative to the expected return, and so the price will be below par

This bond pays "too much" relative to the expected return, and so the price will be above par

This bond pays "too little" relative to the expected return, and so the price will be below par

$918.30

$964.22

$1,029.41

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