An investment firm analyzes two retirement portfolios, Portfolio P1 and Portfolio P2, each… — SAT

An investment firm analyzes two retirement portfolios, Portfolio P1 and Portfolio P2, each established with an initial deposit of $D$ dollars at time $t = 0$ years. The value of Portfolio P1 increases linearly by a constant amount each year. The value of Portfolio P2 increases exponentially at a constant annual rate. At $t = 10$ years, the value of both portfolios is equal. At $t = 20$ years, the value of Portfolio P2 is exactly $\frac{9}{8}$ times the value of Portfolio P1. Which of the following expressions represents the value of Portfolio P2, in dollars, at $t = 30$ years in terms of $D$ ?

A
$\frac{5}{2}D$
B
$\frac{9}{4}D$
$\frac{27}{8}D$ Answer
D
$\frac{9}{2}D$

Answer

The expression that represents the value of Portfolio P2 at

t = 30

years is

\frac{27}{8}D

To find the value of Portfolio P2 at

t = 30

, we model the linear portfolio as

V_1(t) = D + ct

and the exponential portfolio as

V_2(t) = D \cdot g^t

. At

t = 10

D + 10c = D \cdot g^{10}

, which gives

c = \frac{D}{10}(g^{10}-1)

. At

t = 20

V_2(20) = \frac{9}{8}V_1(20) \implies D \cdot g^{20} = \frac{9}{8}(D + 20c)

. Substituting

c

into the equation and simplifying yields the quadratic equation

8(g^{10})^2 - 18(g^{10}) + 9 = 0

. Solving for

g^{10}

and selecting the value greater than 1 (since the portfolio is growing) gives

g^{10} = 1.5

. Thus, at

t = 30

, the value of Portfolio P2 is

D \cdot g^{30} = D \cdot (g^{10})^3 = D \cdot (1.5)^3 = \frac{27}{8}D

Step-by-Step Solution

Write the models for both portfolios in terms of the initial deposit

D

For Portfolio P1 (linear):

V_1(t) = D + c \cdot t

, where

c

is the constant annual increase.
For Portfolio P2 (exponential):

V_2(t) = D \cdot g^t

, where

g

is the annual growth factor.

Establishing the mathematical definitions of linear and exponential growth models is necessary to set up equations representing the given relationships.

Use the condition at

t = 10

to express the linear growth rate

c

in terms of

D

and

g^{10}

V_1(10) = V_2(10) \implies D + 10c = D \cdot g^{10} \implies 10c = D(g^{10} - 1) \implies c = \frac{D}{10}(g^{10} - 1)

This allows us to eliminate the linear growth rate variable

c

and express all equations in terms of the exponential growth term

g^{10}

Set up the equation for

t = 20

using the given relationship and substitute

c

V_2(20) = \frac{9}{8} V_1(20) \implies D \cdot g^{20} = \frac{9}{8} (D + 20c)

. Substituting

c

gives:

D \cdot g^{20} = \frac{9}{8} \left( D + 20 \cdot \frac{D}{10}(g^{10} - 1) \right) \implies D \cdot g^{20} = \frac{9}{8} D \left( 1 + 2(g^{10} - 1) \right)

. Since

D > 0

, we divide both sides by

D

to get:

g^{20} = \frac{9}{8} (2g^{10} - 1)

This sets up a solvable quadratic equation for the variable

g^{10}

Solve the quadratic equation for

g^{10}

Let

x = g^{10}

. The equation is

x^2 = \frac{9}{8}(2x - 1) \implies 8x^2 - 18x + 9 = 0

. Factoring the quadratic yields

(2x - 3)(4x - 3) = 0

, giving

x = 1.5

x = 0.75

. Since the portfolios are growing, the growth factor

g > 1

, so

g^{10} > 1

. Therefore, we choose

g^{10} = 1.5

Determining the value of the 10-year growth factor

g^{10}

is the key step to finding the value at future time intervals.

Calculate the value of Portfolio P2 at

t = 30

years.

V_2(30) = D \cdot g^{30} = D \cdot (g^{10})^3 = D \cdot (1.5)^3 = D \cdot \left(\frac{3}{2}\right)^3 = \frac{27}{8}D

Using the properties of exponents, we express the value at 30 years in terms of the 10-year growth factor cubed.

Key Concept

Linear vs. Exponential Growth Models

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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