Present Value and Discount Factors

Valuing a bond, or managing how its value changes when interest rates move, means calculating the present value of its future payments. A change in value can then be explained as a change in interest rates and the resulting change in discount factors.

Present value is what future money is worth today

Consider a hypothetical example in which the interest rate is 2%. A deposit of 100 yen today grows to 102 yen in one year. Put another way, 102 yen received a year from now is worth 100 yen today. Converting future money into its value today is called discounting, and the converted amount is the present value (PV).

How to find a discount factor

The value today of 1 yen in the future is called the discount factor (DF)¹. With interest compounded once a year, the discount factor for one year from now is 1 divided once by "1 + the interest rate", and for two years from now it is 1 divided twice. At 2%, the one-year factor is 1 ÷ 1.02, or about 0.9804, and the two-year factor is 1 ÷ 1.02 ÷ 1.02, or about 0.9612. How the compounding frequency changes the result is covered on the Compounding and Discounting page.

To find the present value of a future cash flow (a payment received or paid), multiply the amount by the discount factor for the date it is received. When there are several cash flows, find the present value of each one and add them up².

Deep diveFormulas for the discount factor and present value

With an annually compounded rate $r_t$, the discount factor for $t$ years from now is:

$$DF(t) = \frac{1}{(1 + r_t)^t}$$

Adding up the present values of the cash flows $C_t$ at each date gives:

$$PV = \sum_{t=1}^{n} C_t \times DF(t)$$

$r_t$ is the interest rate for the period up to year $t$ (the spot rate), and it can take a different value for each period. If the same rate is used for every period, $r_t = r$.

Worked example: present value of a 3-year bond

Take a bond with 10 billion yen of principal, interest of 2% paid once a year (200 million yen each year) and a maturity of 3 years, and value it at a discount rate of 2%. The rate is assumed to be the same for every period³.

Figure description

The bondholder receives 200 million yen of interest in years 1 and 2, and in year 3 receives 10.2 billion yen: 200 million yen of interest plus the 10 billion yen principal. All of these are receipts, so every arrow points up. Arrow lengths are only a guide to the size of each amount: the interest arrows are drawn longer than their true proportion so that they remain visible.

Multiplying each cash flow by the discount factor for its date gives the following table.

Because the coupon rate and the discount rate are both 2%, the total present value equals the 10 billion yen principal.

When rates rise, present value falls

The higher the discount rate, the smaller the discount factor. When the rate is positive, the discount factor also gets smaller the further away the payment date is. Discounting the same bond at 5% gives smaller discount factors at every date than at 2%, and a total present value of 9,183 million yen⁴. The rise from 2% to 5% lowers the present value by about 817 million yen, even though the future cash flows have not changed.

Over the 1 to 3 years in this example, the discount factor falls more for later dates. When the rate goes from 2% to 5%, the one-year discount factor falls by 0.0280, while the three-year factor falls by 0.0785. Conversely, if the discount rate falls, discount factors rise and the present value of cash flows you receive goes up.

Comparing a single receipt of 100: at 2%, the 10-year discount factor is 0.8203 and the present value is 82.03. At 5%, the 10-year discount factor drops to 0.6139.

How it is used in practice

The Bank of Japan's training document calls one approach the "present value approach" (our translation)⁵. The approach values future cash flows such as interest and principal at their present values and examines how those values change when interest rates and other factors move. The factors that move present value (risk factors) include stock prices and exchange rates as well as interest rates. Financial institutions calculate how present value changes when risk factors move and manage the size of that change as risk⁶.

Assumptions and limits

The worked example discounts every period at the same rate and uses whole numbers of years. In actual markets the rate differs by period (1 year, 2 years, 3 years), and each cash flow is discounted at the rate for its period (the spot rate)⁸. How spot rates are found is covered on the Zero Rates and Forward Rates page.

Notes

1. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅰ．市場リスクの計測手法 ― 現在価値アプローチ―" (Market risk measurement methods: the present value approach, March 2009, in Japanese), page 7. The same center's "Ⅱ．市場リスクの把握と管理" (Understanding and managing market risk, June 2012, in Japanese) also uses the term discount factor (ディスカウント・ファクター) on page 5. ↩

2. "Ⅱ．市場リスクの把握と管理" (June 2012), "現在価値の計測方法" (Measuring present value), page 5. ↩

3. The conditions are those of "具体例① 債券投資" (Example 1: bond investment), page 6, and "具体例② 債券投資" (Example 2: bond investment), page 8, of the same document. ↩

4. Present values are calculated from unrounded discount factors and then rounded to the nearest million yen. The rounded values in the table add up to 9,182 million yen. ↩

5. "Ⅱ．市場リスクの把握と管理" (June 2012), "現在価値アプローチの考え方" (The present value approach), page 4. ↩

6. The same document, "（参考）リスクファクター：現在価値の変動をもたらすもの" (Risk factors: what moves present value), page 11, and "（参考）金融商品とリスクファクター" (Financial products and risk factors), page 12. The same approach is on page 20 of "Ⅰ．市場リスクの計測手法 ― 現在価値アプローチ―" (March 2009). ↩

7. "Ⅱ．市場リスクの把握と管理" (June 2012), "（１）ＢＰＶ（ベーシス・ポイント・バリュー）" (BPV), page 13. ↩

8. The same document, "（参考）割引率（スポットレート）の定義" (Definition of the discount rate (spot rate)), page 10. ↩