利用者:加藤勝憲/債券価値評価

債券価値評価（さいけんかちひょうか）とは、投資家が債券の理論的な公正価値、つまり本質的な価値を見積もるプロセスのことである。他の証券や資本投資と同様に、債券の理論的公正価値は、それが生成すると予想されるキャッシュフローの流れの現在価値である。したがって、債券の価値は、債券の予想キャッシュフローを適切な割引率を用いて現在に割り引くことによって得られる^[1]^[2]。

実際には、この割引率は、類似の金融商品が存在する場合には、それを参照して決定されることが多い。その後、所定の価格に対して、関連する様々な利回り測定が計算される。債券の市場価格が額面価格を下回る場合、債券は割引価格で販売されている。逆に、債券の市場価格が額面価格より大きい場合、債券はプレミアムで販売されている。この価格と利回りの間の他の関係については、以下を参照のこと。

債券に組み込みオプションが含まれている場合、評価はより難しくなり、オプション価格と割引を組み合わせることになる。オプションの種類に応じて、算出されたオプション価格は、「ストレート」部分の価格に加算または減算される[3]。この合計が債券の価値となる。

If the bond includes embedded options, the valuation is more difficult and combines option pricing with discounting. Depending on the type of option, the option price as calculated is either added to or subtracted from the price of the "straight" portion.^[3] See further under Bond option. This total is then the value of the bond.

債券価値評価

「普通社債」（オプションが埋め込まれていない債券。債券（金融）#特徴§を参照）の公正価格は、通常、その期待キャッシュフローを適切な割引率で割り引くことによって決定される。この現在価値の関係は、債券の価値を決定するための理論的なアプローチを反映しているが、実際には、その価格は、（通常は）他の、より流動性の高い金融商品を参照して決定される。ここでいう2つの主なアプローチ、相対プライシングとアービトラージ・フリー・プライシングについては次に述べる。最後に、将来の金利が不確実であり、割引率が単一の固定数では適切に表せないことを認識することが重要な場合、例えば、問題の債券にオプションが書き込まれている場合などには、確率微積分を用いることができる。

The fair price of a "straight bond" (a bond with no embedded options; see Bond (finance)#Features § )

when an option is written on the bond in question—stochastic calculus may be employed.

現在価値アプローチ

債券の理論的な公正価値または本源的価値を計算する基本的な方法は、すべての期間のキャッシュ・フローを割り引くために単一の市場金利を使用し、以下に示す現在価値（PV）の公式を使用する。より複雑なアプローチでは、異なる期間のキャッシュ・フローに異なる金利を使用する^[2]^:294。以下の計算式は、クーポンの支払いが行われたばかりであると仮定している（他の日付での調整については以下を参照）。

The basic method for calculating a bond's theoretical fair value, or intrinsic worth, uses the present value (PV) formula shown below, using a single market interest rate to discount cash flows in all periods. A more complex approach would use different interest rates for cash flows in different periods. The formula shown below assumes that a coupon payment has just been made (see below for adjustments on other dates).

{\begin{aligned}TFV&={\begin{matrix}\left({\frac {C}{1+i}}+{\frac {C}{(1+i)^{2}}}+...+{\frac {C}{(1+i)^{N}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}\left(\sum _{n=1}^{N}{\frac {C}{(1+i)^{n}}}\right)+{\frac {M}{(1+i)^{N}}}\end{matrix}}\\&={\begin{matrix}C\left({\frac {1-(1+i)^{-N}}{i}}\right)+M(1+i)^{-N}\end{matrix}}\end{aligned}}

where:

F=

par value

i_{F}=

contractual interest rate

C=F*i_{F}=

coupon payment (periodic interest payment)

N=

number of payments

i=

market interest rate, or required yield, or observed / appropriate yield to maturity (see below)

M=

value at maturity, usually equals par value

TFV=

theoretical fair value

Relative price approach

「Corporate bond #Valuation」および「Corporate bond#Risk analysis」を参照

この手法（上記の拡張または応用）では、債券はベンチマーク（通常は国債）に対して相対的に価格決定される（Relative valuationを参照）。ここでは、債券の満期までの利回りは、同様の満期またはデュレーションを持つ国債との相対的な債券の信用格付けに基づいて決定される（Credit spread (bond)参照）。債券の質が高いほど、必要リターンとベンチマークのYTMとのスプレッドは小さくなる。この要求リターンを用いて、債券のキャッシュフローを割引くことで、上式の 𝑖 を置き換えて価格を求める^[4]。

Arbitrage-free pricing approach

「Rational pricing#Fixed income securities」を参照

上記の2つの関連するアプローチとは異なり、債券は「キャッシュフローのパッケージ」（クーポンまたはフェース）と考えることができ、各キャッシュフローは、それが受け取られる日に満期を迎えるゼロ・クーポンzero-couponの商品とみなされる。したがって、単一の割引率を使用するのではなく、複数の割引率を使用し、各キャッシュ・フローをそれぞれの割引率で割り引くべきである。ここで、各キャッシュ・フローは、クーポン日に対応するゼロ・クーポン債zero-coupon bondと同じ利率で個別に割り引かれ、同等の信用力（可能であれば、評価対象の債券と同じ発行体から発行されたもの、そうでなければ、適切な信用スプレッドcredit spreadを付したもの）がある。

このアプローチでは、債券価格は「裁定取引のない」価格を反映するはずである。この価格からの乖離は利用され、債券はすぐに正しい水準に価格を戻すからである。ここでは、「キャッシュフローが同一である資産Assets with identical cash flows」に関する合理的な価格決定（rational pricing）ロジックを適用する。具体的には、(1)債券の利札日と利札額は確実に分かっている。したがって、(2)ゼロ・クーポン債の複数（または分数）を、それぞれ債券のクーポン日に対応させ、債券と同一のキャッシュ・フローを生み出すように指定することができる。したがって、(3) 今日の債券価格は、対応するZCBの値によって暗示される割引率で割り引かれた各キャッシュ・フローの合計に等しくなければならない。

Stochastic calculus approach

When modelling a bond option, or other interest rate derivative (IRD), it is important to recognize that future interest rates are uncertain, and therefore, the discount rate(s) referred to above, under all three cases—i.e. whether for all coupons or for each individual coupon—is not adequately represented by a fixed (deterministic) number. In such cases, stochastic calculus is employed.

The following is a partial differential equation (PDE) in stochastic calculus, which, by arbitrage arguments, is satisfied by any zero-coupon bond $P$ , over (instantaneous) time $t$ , for corresponding changes in $r$ , the short rate.

${\frac {1}{2}}\sigma (r)^{2}{\frac {\partial ^{2}P}{\partial r^{2}}}+[a(r)+\sigma (r)+\varphi (r,t)]{\frac {\partial P}{\partial r}}+{\frac {\partial P}{\partial t}}-rP=0$

The solution to the PDE (i.e. the corresponding formula for bond value) — given in Cox et al. — is:

$P[t,T,r(t)]=E_{t}^{\ast }[e^{-R(t,T)}]$

where

E_{t}^{\ast }

is the expectation with respect to risk-neutral probabilities, and

R(t,T)

is a random variable representing the discount rate; see also Martingale pricing.

To actually determine the bond price, the analyst must choose the specific short-rate model to be employed. The approaches commonly used are:

Note that depending on the model selected, a closed-form (“Black like”) solution may not be available, and a lattice- or simulation-based implementation of the model in question is then employed. See also Bond option § Valuation.

Clean and dirty price

詳細は「Clean price」および「Dirty price」を参照

When the bond is not valued precisely on a coupon date, the calculated price, using the methods above, will incorporate accrued interest: i.e. any interest due to the owner of the bond over the "stub period" since the previous coupon date (see day count convention). The price of a bond which includes this accrued interest is known as the "dirty price" (or "full price" or "all in price" or "Cash price"). The "clean price" is the price excluding any interest that has accrued. Clean prices are generally more stable over time than dirty prices. This is because the dirty price will drop suddenly when the bond goes "ex interest" and the purchaser is no longer entitled to receive the next coupon payment. In many markets, it is market practice to quote bonds on a clean-price basis. When a purchase is settled, the accrued interest is added to the quoted clean price to arrive at the actual amount to be paid.

Yield and price relationships

Once the price or value has been calculated, various yields relating the price of the bond to its coupons can then be determined.

Yield to maturity

The yield to maturity (YTM) is the discount rate which returns the market price of a bond without embedded optionality; it is identical to $i$ (required return) in the above equation. YTM is thus the internal rate of return of an investment in the bond made at the observed price. Since YTM can be used to price a bond, bond prices are often quoted in terms of YTM.

To achieve a return equal to YTM, i.e. where it is the required return on the bond, the bond owner must:

buy the bond at price $P_{0}$ ,
hold the bond until maturity, and
redeem the bond at par.

Coupon rate

The coupon rate is the coupon payment $C$ as a percentage of the face value $F$ .

{\text{Coupon rate}}={\frac {C}{F}}

Coupon yield is also called nominal yield.

Current yield

The current yield is the coupon payment $C$ as a percentage of the (current) bond price $P$ .

{\text{Current yield}}={\frac {C}{P_{0}}}.

Relationship

The concept of current yield is closely related to other bond concepts, including yield to maturity, and coupon yield. The relationship between yield to maturity and the coupon rate is as follows:

Relationship between yield to maturity and the coupon rate
Status	Connection
At a discount	YTM > current yield > coupon yield
At a premium	coupon yield > current yield > YTM
Sells at par	YTM = current yield = coupon yield

Price sensitivity

The sensitivity of a bond's market price to interest rate (i.e. yield) movements is measured by its duration, and, additionally, by its convexity.

Duration is a linear measure of how the price of a bond changes in response to interest rate changes. It is approximately equal to the percentage change in price for a given change in yield, and may be thought of as the elasticity of the bond's price with respect to discount rates. For example, for small interest rate changes, the duration is the approximate percentage by which the value of the bond will fall for a 1% per annum increase in market interest rate. So the market price of a 17-year bond with a duration of 7 would fall about 7% if the market interest rate (or more precisely the corresponding force of interest) increased by 1% per annum.

Convexity is a measure of the "curvature" of price changes. It is needed because the price is not a linear function of the discount rate, but rather a convex function of the discount rate. Specifically, duration can be formulated as the first derivative of the price with respect to the interest rate, and convexity as the second derivative (see: Bond duration closed-form formula; Bond convexity closed-form formula; Taylor series). Continuing the above example, for a more accurate estimate of sensitivity, the convexity score would be multiplied by the square of the change in interest rate, and the result added to the value derived by the above linear formula.

For embedded options, see effective duration and effective convexity; more generally, see Corporate bond#Risk analysis § .

Accounting treatment

In accounting for liabilities, any bond discount or premium must be amortized over the life of the bond. A number of methods may be used for this depending on applicable accounting rules. One possibility is that amortization amount in each period is calculated from the following formula:^[要出典]

$n\in \{0,1,...,N-1\}$

$a_{n+1}$ = amortization amount in period number "n+1"

Bond Discount or Bond Premium = $|F-P|$ = $a_{1}+a_{2}+...+a_{N}$

Bond Discount or Bond Premium = $F|i-i_{F}|({\frac {1-(1+i)^{-N}}{i}})$

脚注

^ Malkiel, Burton G. (1962). “Expectations, Bond Prices, and the Term Structure of Interest Rates”. The Quarterly Journal of Economics 76 (2): 197–218. doi:10.2307/1880816. ISSN 0033-5533. JSTOR 1880816. https://www.jstor.org/stable/1880816.
^ ^a ^b Bodi, Zvi; Kane, Alex.; Marcus, Alan J. (2010). Essentials of Investments (eighth ed.). New York: McGraw-Hill/Irwin. ISBN 978-0-07-338240-1
^ Kalotay, Andrew J.; Williams, George O.; Fabozzi, Frank J. (1993). “A Model for Valuing Bonds and Embedded Options” (英語). Financial Analysts Journal 49 (3): 35–46. doi:10.2469/faj.v49.n3.35. ISSN 0015-198X.
^ Jones, E. Philip; Mason, Scott P.; Rosenfeld, Eric (1984). “Contingent Claims Analysis of Corporate Capital Structures: An Empirical Investigation”. The Journal of Finance 39 (3): 611–625. doi:10.2307/2327919. ISSN 0022-1082. JSTOR 2327919. https://www.jstor.org/stable/2327919.

Selected bibliography

Guillermo L. Dumrauf (2012). “Chapter 1: Pricing and Return”. Bonds, a Step by Step Analysis with Excel. Kindle Edition
Frank Fabozzi (1998). Valuation of fixed income securities and derivatives (3rd ed.). John Wiley. ISBN 978-1-883249-25-0
Frank J. Fabozzi (2005). Fixed Income Mathematics: Analytical & Statistical Techniques (4th ed.). John Wiley. ISBN 978-0071460736
R. Stafford Johnson (2010). Bond Evaluation, Selection, and Management (2nd ed.). John Wiley. ISBN 978-0470478356
Mayle, Jan (1993), Standard Securities Calculation Methods: Fixed Income Securities Formulas for Price, Yield and Accrued Interest, 1 (3rd ed.), Securities Industry and Financial Markets Association, ISBN 1-882936-01-9
Donald J. Smith (2011). Bond Math: The Theory Behind the Formulas. John Wiley. ISBN 978-1576603062
Bruce Tuckman (2011). Fixed Income Securities: Tools for Today's Markets (3rd ed.). John Wiley. ISBN 978-0470891698
Pietro Veronesi (2010). Fixed Income Securities: Valuation, Risk, and Risk Management. John Wiley. ISBN 978-0470109106
Burton Malkiel (1962). "Expectations, Bond Prices, and the Term Structure of Interest Rates". The Quarterly Journal of Economics
Mark Mobius (2012). Bonds: An Introduction to the Core Concepts. John Wiley. ISBN 978-0470821473

外部リンク

Bond Valuation, Prof. Campbell R. Harvey, Duke University
A Primer on the Time Value of Money, Prof. Aswath Damodaran, Stern School of Business
Bond Price Volatility Investment Analysts Society of South Africa
Duration and convexity Investment Analysts Society of South Africa

Template:Bond market Template:Bond market [[Category:未査読の翻訳があるページ]]

[1] Malkiel, Burton G. (1962). “Expectations, Bond Prices, and the Term Structure of Interest Rates”. The Quarterly Journal of Economics 76 (2): 197–218. doi:10.2307/1880816. ISSN 0033-5533. JSTOR 1880816. https://www.jstor.org/stable/1880816.

[bodi-2] Bodi, Zvi; Kane, Alex.; Marcus, Alan J. (2010). Essentials of Investments (eighth ed.). New York: McGraw-Hill/Irwin. ISBN 978-0-07-338240-1

[3] Kalotay, Andrew J.; Williams, George O.; Fabozzi, Frank J. (1993). “A Model for Valuing Bonds and Embedded Options” (英語). Financial Analysts Journal 49 (3): 35–46. doi:10.2469/faj.v49.n3.35. ISSN 0015-198X.

[4] Jones, E. Philip; Mason, Scott P.; Rosenfeld, Eric (1984). “Contingent Claims Analysis of Corporate Capital Structures: An Empirical Investigation”. The Journal of Finance 39 (3): 611–625. doi:10.2307/2327919. ISSN 0022-1082. JSTOR 2327919. https://www.jstor.org/stable/2327919.

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