The amount that must be invested at i% (0% < i% < 10%) to accumulate to Y at the end of three years at an annual rate of:

i) simple interest of i% is Q

ii) compound interest of i% is R

iii) simple discount of i% is S

iv) compound discount of i% is T

Determine which of the following is true.

• Q < R < S < T
• R < Q < S < T
• S < T < R < Q
• T < S < Q < R
• T < S < R < Q

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Given that the problem states that the inequality is true for all interest rates from [math]0 \%[/math] to [math]10 \%[/math] and all values of [math]Y[/math], it is sufficient to determine it for one set of values. Select [math]i=7 \%[/math] and [math]Y=121[/math].

Then,

[[math]] \begin{aligned} & Q=121 /(1+3(0.07))=100 \\ & R=121 /(1.07)^3=98 / 77 \\ & S=121(1-0.07(3))=95.59 \\ & T=121(0.93)^3=97.33 \end{aligned} [[/math]]

Hence,

[[math]] \mathrm{S}\lt\mathrm{T}\lt\mathrm{R}\lt\mathrm{Q} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.