Discount factors

What you will be able to do

After this lesson, you should be able to:

• explain the units and meaning of a discount factor;
• calculate one from a stated nominal rate, frequency, and time;
• distinguish discounting from compounding and from payment probability.

Undo accumulation

In this lesson, $\explain{payment-time}{t}$ denotes a payment-time without a row index. It is measured in years from valuation-time $0$. Write the compounding-frequency as $\explain{compounding-frequency}{m}$, the periodic-rate as $\explain{periodic-rate}{r_{\explain{compounding-frequency}{m}}}$, the accumulation-factor as $\explain{accumulation-factor}{A(0,t)}$, and the discount-factor as $\explain{discount-factor}{D(0,t)}$.

The accumulation factor $\explain{accumulation-factor}{A(0,t)}$ is the amount at time $\explain{payment-time}{t}$ from one unit invested at time zero. The discount factor is its reciprocal:

$$\explain{discount-factor}{D(0,t)}=\frac{1}{\explain{accumulation-factor}{A(0,t)}}=\left(1+\explain{periodic-rate}{r_{\explain{compounding-frequency}{m}}}\right)^{-\explain{compounding-frequency}{m}\explain{payment-time}{t}}$$

(1.4.1)

The discount factor, the periodic rate, and the compounding frequency keep their different meanings, also when they appear in one expression.

Meaning before arithmetic

Write the local discounted-unit-value as $\explain{discounted-unit-value}{V_0(1_{\explain{payment-time}{t}})}$: the value at valuation time of one deterministic unit paid at $\explain{payment-time}{t}$. If $\explain{discount-factor}{D}(0,2)=0.94$, the value at valuation time of one unit paid at $\explain{payment-time}{t}=2$ years is:

$$\explain{discounted-unit-value}{V}_0(1_2) =1\times \explain{discount-factor}{D}(0,2) =0.94$$

(1.4.2)

The number 0.94 is not an annual interest rate of 94%. It is also not a probability of 94% that the payment occurs.

Example 1.4.1 Discount factors from rates and curves

Annual compounding

With a nominal-annual-rate of 5%, $\explain{compounding-frequency}{m}=1$, and $\explain{payment-time}{t}=2$, the discount factor is:

$$\explain{discount-factor}{D}(0,2) =\left(1+\frac{0.05}{1}\right)^{-1\times2} =\frac{1}{1.05^2} \approx0.907029$$

Semiannual compounding

With a nominal annual rate of 6%, compounded twice per year, the discount factor for 1.5 years is:

$$\explain{discount-factor}{D}(0,1.5) =\left(1+\frac{0.06}{2}\right)^{-2(1.5)} =\frac{1}{1.03^3} \approx0.915142$$

The horizon of 1.5 years contains three compounding periods. The exponent 1.5 would be wrong, because it counts years, not half-year periods.

Read a small curve

With the same nominal annual rate of 6%, compounded semiannually, the discount factors are:

Time $\explain{payment-time}{t}$, years	Periods $\explain{compounding-frequency}{m}\explain{payment-time}{t}$	Discount factor $\explain{discount-factor}{D(0,t)}$
0.0	0	1.000000
0.5	1	0.970874
1.0	2	0.942596
1.5	3	0.915142

If the rate is non-negative, the discount factor does not increase with the payment time. If the rate is positive, the discount factor decreases with the payment time.

Check your understanding

These items test meaning, annual calculation, and a semiannual transfer case. Each question stays collapsed until you open it; answers and explanations appear once you check.

Knowledge check 1.4.1 Discount factors

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

3. DirectQuestion 3Not answered

4. TransferQuestion 4Not answered

Model boundary and review note

The rate is the same for all times, and the cash flows are deterministic. This lesson does not teach spot curves, interpolation, day counts, default, recovery, liquidity, or funding.

The discount-factor definition (1.4.1) follows Tuckman & Serrat[1]; Hull gives the same interpretation, the present value of one unit paid in the future[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, the discount factor as the value today of one unit paid at t; §2.4, eqs. 2.17-2.19, the semiannually compounded form. draft ↩
2. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 4, “Bond Pricing”. draft ↩