FM for Actuaries

Interest Accumulation and Time Value of Money 37
1.43 Assume that the force of interest is constant over time. Let the constant force
of interest be δ and the equivalent effective rate of interest and discount be i
and d, respectively.
(a) Express δ in terms of i.
(b) Express δ in terms of d.
(c) By using the Taylor’s expansion
∞∑ (−1) k−1 x k
ln(1 + x) =
k
k=1
= x − x2
2 + x3
3 − x4
4 + ··· ,
and the expressions in (a) and (b), show that
δ = 1 [ i + d
− i2 − d 2
+ i3 + d 3
+ i4 − d 4
2 1 2 3 4
]
+ ··· .
The formula above shows that δ is very close to the average of i and d.