【補足】6 つの係数の導出①:終価係数と原価係数

この記事は約6分で読めます。

この記事は「【補足】資金計画に関する 6 つの係数の導出」の連載です。直リンクでここに来た方向けに、以下の記事に概要が記されています:)


 さて、まずは資金係数に関する 6 つの係数のうち、最初の二つ:終価係数と原価係数の一般式を導出しましょう。ここでいう「一般」というのは「運用年数が何年であっても、利率がいくらであっても通用する式」という程度の意味です。ななみんの本にあった表には「運用年数 5 年」で「利率 1%-5%」の場合の係数しか載っていないので、一般の年数と利率に対してそれぞれの係数が計算できるような式が分かることは大きな進歩です。


終価係数

 終価係数とは、最初に用意されたある一定の額 \( M \) を一定の年利 \( r \) で \( n \) 年間運用した場合に、\( n \) 年後に得られる金額 \( v_n \) を求める際に用いられる係数のことであり、次の式によって定義される係数 \( k_1 \) のことを指します:

\[ M k_1 = v_n \qquad \text{or} \qquad k_1 = \frac{v_n}{M} \,\, . \]

日本語で書けば次のようです:

\[ \text{終価係数} = \frac{\text{n 年後に得られる金額}}{\text{最初に用意された金額}} \,\, . \]

大前提が単純運用型の資産運用なので、積み立て開始後の残高は下の図のように推移していくことになります:

図:終価係数の説明。\( n \) 年間の単純運用を行なった場合の残高の変遷。点線より上は各年に加算される利息分を表す。

 終価係数は 6 つの係数の中で最も簡単な係数なのでわざわざ漸化式を立てるまでもありませんが、これ以降他の係数の一般式を求めることも見据えて、最初から漸化式を立てる方針を採用することにします。

 さて \( v_n \) を求めるために、\( k \) 年後 の残高 \( v_k \) と \( k + 1 \) 年後の残高 \( v_{k + 1} \) の関係を考えます。下の図を見れば、

\[ v_{k+1} = v_k + r v_k = (1 + r) v_k \]

という漸化式が成り立つことが分かります。

図:終価係数を考える際の残高の変遷。

上の漸化式を初期条件:\( v_0 = M \)(これは 0 年後つまり最初に用意された積立金からまだ増えていない状態の残高を表します)の下で解くと

\[ v_n = (1 + r)^n M \]

を得ます。最後に、冒頭の終価係数 \( k_1 \) の定義と見比べれば

\[ k_1 = \frac{ v_n }{M} = (1 + r)^n \]

が分かります。これは一般に運用年数 \( r \) と年利 \( n \) の関数なので、本来は \( k_1 = k_1 (n, r) \) のように記すべきですが、以下では引数の表記を省略します。

 さて、以上で終価係数の一般式が求まったので、ここで運用年数を \( n=5 \) 年に固定して、年利 \( r \) に対する終価係数 \( k_1 \) の挙動を見てみましょう:

図:運用年数を \( n=5 \) 年に固定したときの、年利 \( r \) に対する終価係数 \( k_1 \) の挙動。

 綺麗な図ですね。表示範囲の都合上、直線のように見えなくもないですが、実際には上の \( k_1 \) の式を見れば分かるように冪乗関数の振る舞い(今回は特に \( n = 5 \) なので 5 次関数)です。

 特に年利 \( r = 1 \% – 5 \% \) に対応する \( k_1 \) がいくつなのかを出してみると、次の表のようになりました:

表:上のグラフの年利 \( r = 1 \% – 5 \% \) とそれぞれに対応する終価係数 \( k_1 \) のペア。

 これらの数値はななみんの本に与えられている表の「終価係数」の行の 5 つの値を正しく再現していますね。以上で「終価係数の一般式を導出すること」と「ななみんの本に与えられた終価係数の値が正しいことの確認」ができました


原価係数

 原価係数とは、一定の年利 \( r \) で \( n \) 年間の 資産運用を行ない、\( n \) 年後に \( v_n \) 円になって欲しいと思ったときにに、最初に用意しなければならない額 \( M \) を 求める際に用いられる係数として定義されます。式で書けば \( v_n k_2 = M \) の中の \( k_2 \) が原価係数です。つまり

\[ k_2 = \frac{M}{v_n} \,\, . \]

日本語で書けば

\[ \text{原価係数} = \frac{\text{最初に用意されるべき額}}{\text{n 年後に欲しい額}} \]

です。

 上でやった終価係数の話を思い出せば、原価係数 \( k_2 \) は終価係数 \( k_1 \) のちょうど逆の概念に対応する量であることが分かりますね。二つの定義を見比べると、原価係数 \( k_2 \) は終価係数 \( k_1 \) の逆数です:

\[ k_2 = \frac{M}{v_n} = \frac{1}{(1+r)^n} \,\, \left( = \frac{1}{k_1} \right) \,\, . \]

終価係数のときと全く同じ議論によって上の式を導出することももちろん可能ですが、ここではそれはやりません。

 終価係数のときと同様に運用期間を \( n = 5 \) 年に固定して、年利 \( r \) に対する原価係数 \( k_2 \) を表示すると下の図のようになります:

図:運用年数を \( n=5 \) 年に固定したときの、年利 \( r \) に対する原価係数 \( k_2 \) の挙動。

 再び同様に、特に年利 \( r = 1 \% – 5 \% \) に対応する \( k_2 \) がいくつなのかを出してみると、次の表のようになりました:

表:上のグラフの年利 \( r = 1 \% – 5 \% \) とそれぞれに対応する原価係数 \( k_2 \) のペア。

 やはりななみんの本に与えられている表の「原価係数」の行の 5 つの値を正しく再現していますね。以上で「原価係数の一般式を導出すること」と「ななみんの本に与えられた原価係数の値が正しいことの確認」ができました


まとめ!

 終価係数 \( k_1 \) および原価係数 \( k_2 \) の一般式を導出し、ななみんの本に与えられている表のうち、これらに対応する数値が正しいことを確認しました。


 引き続き、年金終価係数&減債基金係数、資本回収係数&年金原価係数についても、一般式の導出とななみんの本に与えられている表の数値の確認をやっていきます(一般式の結果だけでいいよ、という方は一番下のリンクまで!):

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