Additional Solutions: Chapter Three: Cash Flow Analysis
3S.1
Mr Dashwood must find the least value of
N
for which
1 500 < 100
(P/A,5%,N)
which is equivalent to finding
N
such that
(P/A,5%,N)
> 15
Consulting Appendix A, this first occurs when
N
= 29 years.
If the interest rate is 10%, we come to the end of the table in Appendix A before finding a value of
N
for which
(P/A,10%,N)
> 15. We therefore go on to look at the capitalized value of the annuity, which
is 100/0.1, or £1 000. Thus, if he can invest his money at 10%, Mr Dashwood can afford to support his
widowed half-sister indefinitely for a cost equivalent to a single present payment of £1 000.
*3S.2
There are several ways of tackling this.
One way is to convert the bi-annuity to an equivalent annuity:
A1 = A(A/F,i,2)
where ¥
A1
is the annual payment equivalent to getting a final payment of ¥
A
at the end of two years.
Then we can convert ¥
A1
to its present worth using a formula we already know:
P=A1(P/A,i,2N)=A(A/F,i,2)(P/A,i,2N)
(Don’t forget that it’s now 2
N
rather than
N
.)
Another solution is to calculate the effective biennial interest rate,
j
, from the equation
j
= (1 +
i
)
2
-1 = 2
i
+
i
2
and then to use a conversion factor based on
j
:
P = A(P/A, j, N)
A third alternative is to construct a formula based on the one at the back of the book,
(P/A, j, N)
= ((1
+
j
)
N
-1)/(
j
(1 +
j
)
N
)
and substitute in the value of
j
, the effective biennial interest rate, as calculated above. This is a correct
solution, but it’s more work. Also, I would advise against using these algebraic formulas, except when
you’re constructing a spreadsheet – evaluating them is more work than looking a number up in the
tables, and there are more opportunities for making a slip.