## Monday, July 18, 2011

### Pricing Bonds

In a previous post, we discussed the key characteristics of bonds, one of the two primary investment alternatives available to investors (the other one being equity or stocks). In that post, we discussed the following features:
• Par or face value - the amount paid by the bond issuer (borrower) to the bond holder (lender or investor) on the bond's maturity date, when the bond expires or is redeemed
• Coupon rate - the percentage of the par value that is annually paid by the issuer to the bond holder as interest
• Price - how much a bond sells for
• Yield to maturity - the percentage return an investor would earn every year if he or she holds on to the bond up to maturity
Typically, investors would be interested in periodic price quotations of outstanding bonds in the market since price is the primary indicator of how well an existing investment is doing and how attractive available bonds are for potential investment. Unfortunately, some data sources quote yield to maturity in lieu of price, as what one of our readers, Zeny, discovered browsing the government securities board of the Philippine Dealing and Exchange Corporation (PDEX). Fortunately, all is not lost since we can easily convert yield to maturity to price given the available information.

First, we need to understand the relationship between these bond features by redefining yield to maturity as the discount rate which equates the present value of future cash flows of a bond to its price. A bondholder is entitled to future cash flows in the form of coupon payments and the par value; the price of the bond owned by the bond holder, given the bond's yield to maturity (YTM), is therefore

Price = PV(coupon payments at YTM) + PV(par value at YTM)

Where PV means present value, from the time value of money concept we discussed before.

We can easily compute for the price of a bond using Excel, specifically the "PV" function and the formula for the present value of a single cash flow. To illustrate, let's use the FXTN 25-05 bond quoted on the government securities board of PDEX. We find the following relevant details of the bond on the board:
• Coupon rate = 8.5%
• Years to maturity = 21.38
• Yield to maturity = 8%
Assuming a par value of 100, we use the following Excel expression to compute for the price of the bond

=PV(8%/2,21.38*2,(-8.5%/2)*100)+100/(1+8%/2)^(21.38*2)

and get 105.05

A few notes for the Excel formula above
• Dividing the YTM and coupon rate by 2 and multiplying the years to maturity by 2 "semiannualizes" everything, something that we need to do since coupons are paid semiannually
• The negative sign for the coupon payment argument is needed since Excel treats amortization payments as negative cash flows by default.
• The "PV" portion of the expression gives us the present value of the coupon payments, and the latter part is for the present value of the par value
• Note that since the YTM is less than the coupon rate, the price of the bond is greater than 100, meaning the bond is selling at a premium
• Finally, the following approach does not apply to treasury bills. Unlike bonds, T-bills don't provide coupon payments; investors earn by paying a price that is lower than the par value (discount)