Bond price from discount factors

What you will be able to do

After this lesson, you should be able to calculate a toy bond’s price from its promised cash-flow schedule and input discount factors.

Bond pricing is a present-value application

Use the bond-payment-index $\explain{bond-payment-index}{k}$ to select one of the number-of-bond-payments $\explain{number-of-bond-payments}{n}$. Write the promised bond-cash-flow as $\explain{bond-cash-flow}{CF_{\explain{bond-payment-index}{k}}^{\mathrm{bond}}}$, its payment-time as $\explain{payment-time}{t_{\explain{bond-payment-index}{k}}}$, and its matching discount-factor as $\explain{discount-factor}{D}(0,\explain{payment-time}{t_{\explain{bond-payment-index}{k}}})$. Write the bond-price as $\explain{bond-price}{P_0}$. The bond price is the present-value at valuation-time $0$ of the promised payments. So the bond price is the schedule sum (1.5.1), applied to the promised payments of a bond:

$$\explain{bond-price}{P_0}=\sum_{\explain{bond-payment-index}{k}=1}^{\explain{number-of-bond-payments}{n}}\explain{bond-cash-flow}{CF_{\explain{bond-payment-index}{k}}^{\mathrm{bond}}}\explain{discount-factor}{D}(0,\explain{payment-time}{t_{\explain{bond-payment-index}{k}}})$$

(2.2.1)

In this lesson, “price” means this present value. It does not mean a clean price, a quoted price, or a probability. Those concepts are outside this lesson.

Example 2.2.1 Bond prices from discount factors

Open the set, then choose a discounting case.

Two discount factors

The annual-coupon bond from the preceding lesson pays USD 50 after one year and USD 1,050 after two years. Suppose the matching discount factors are 0.96 and 0.91.

Payment index $\explain{bond-payment-index}{k}$	Cash flow, USD	Discount factor	Present value, USD
1	50	0.96	48.00
2	1,050	0.91	955.50

The bond price is the sum of the present values:

$$\explain{bond-price}{P_0}=50(0.96)+1050(0.91)=\text{USD }1{,}003.50$$

The price is above USD 1,000 because, with these two discount factors, the present value of the promised payments is larger than the face value. This lesson does not use a yield.

Unfamiliar schedule

Consider payments of USD 20 at 0.5 years, USD 20 at 1.0 year, and USD 1,020 at 1.5 years, with discount factors 0.98, 0.95, and 0.92. The bond price is:

$$\explain{bond-price}{P_0} =20(0.98)+20(0.95)+1020(0.92) =19.60+19.00+938.40 =\text{USD }977.00$$

Zero-coupon boundary

If every coupon is zero, the only payment is the face-value $\explain{face-value}{F}$ at maturity-time $\explain{maturity-time}{T}$. The bond price is then:

$$\explain{bond-price}{P_0}=\explain{face-value}{F} \explain{discount-factor}{D}(0,\explain{maturity-time}{T})$$

This is (2.2.1) with one payment. It is not a separate rule.

Error pattern: discount after adding

For the two-payment example, compare a wrong calculation, which applies one discount factor to the sum of the payments, with the calculation in (2.2.1):

$$\begin{aligned} (50+1050)(0.91) &= 1001.00\\ 50(0.96)+1050(0.91) &= 1003.50\\ 1001.00 &\ne 1003.50 \end{aligned}$$

(2.2.2)

One common discount factor is correct only when the model assigns that discount factor to every payment.

Check your understanding

This set includes one coupon-bond calculation and one transfer schedule. Each question stays collapsed until you open it; answers and explanations appear once you check.

Knowledge check 2.2.1 Bond price from discount factors

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

Model boundary and review note

The discount factors are inputs; they are not calibrated. This lesson has no credit risk, although the course is about credit products.

The pricing rule (2.2.1) follows Tuckman & Serrat[1]; Hull discounts each cash flow at its own zero rate the same way[2]. The lesson stays draft pending human confirmation of the printed locators.

References

1. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §1.2, each bond price as its cash flows times the matching discount factors (eqs. 1.1-1.3), checked against market prices in Table 1.4. draft ↩
2. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 4, “Bond Pricing”. draft ↩