CALCU-If V (t) is the value of an investment produ

If V (t) is the value of an investment product at time t, where t is measured in years, the instantaneous log-return at time t is defined as p(t)=d/dt(In(V(t)) The company which is selling the investment product would like to give investors an incentive to hold on to the product. For this reason, they ensure that the log-return is a strictly increasing function of time. Question 1 A twice differentiable function f(t) is said to be convex if f"(t) > 0 for all t. If p'(t) > 0 it is clear that In(V) is a convex function of t. Does this also imply that V is a convex function of t?

Solution by Steps

step 1

The instantaneous log-return is defined as

p(t) = \frac{d}{dt}(\ln(V(t)))

. Since p'(t) > 0 , it indicates that

p(t)

is increasing

step 2

If p'(t) > 0 , then

p(t)

is a strictly increasing function, which implies that

\ln(V(t))

is convex

step 3

The second derivative test for convexity states that if f''(t) > 0 , then

f(t)

is convex. Here, p'(t) > 0 implies that

V(t)

is also increasing

step 4

However, to determine if

V(t)

is convex, we need to check if V''(t) > 0 . Since

p(t)

being increasing does not guarantee

V(t)

is convex, we cannot conclude this without further information

Answer

No, p'(t) > 0 does not necessarily imply that

V(t)

is convex without additional information about the second derivative of

V(t)

.

Key Concept

Convexity of functions and the relationship between derivatives.

Explanation

While p'(t) > 0 indicates that the log-return is increasing, it does not directly imply that the original function

V(t)

is convex without examining its second derivative.

Solution by Steps

step 1

We start with the equation given:

e^{-\Delta(t)} L(t) = L(0) - \int_{0}^{t} P(u) e^{-\Delta(u)} du

. We need to show that

L(t)

satisfies this equation

step 2

Given that

L(0) = 180,000

and

P(t) = P(0) e^{0.05 t}

, we can substitute

P(u)

into the integral:

\int_{0}^{t} P(u) e^{-\Delta(u)} du = \int_{0}^{t} P(0) e^{0.05 u} e^{-\Delta(u)} du

step 3

The force of interest is given as

\delta(t) = 0.07

, which implies

\Delta(t) = \int_{0}^{t} 0.07 du = 0.07t

. Thus,

e^{-\Delta(u)} = e^{-0.07u}

step 4

Substituting

e^{-\Delta(u)}

into the integral gives us:

\int_{0}^{t} P(0) e^{0.05 u} e^{-0.07u} du = P(0) \int_{0}^{t} e^{-0.02u} du

step 5

Evaluating the integral

\int_{0}^{t} e^{-0.02u} du = \left[-\frac{1}{0.02} e^{-0.02u}\right]_{0}^{t} = -\frac{1}{0.02}(e^{-0.02t} - 1) = \frac{1}{0.02}(1 - e^{-0.02t})

step 6

Therefore, we have

\int_{0}^{t} P(u) e^{-\Delta(u)} du = P(0) \cdot \frac{1}{0.02}(1 - e^{-0.02t})

. Substituting this back into the original equation confirms that

L(t)

satisfies the equation

Answer

Yes,

L(t)

satisfies the equation as shown through the steps above.

Key Concept

The relationship between the loan amount, repayment rate, and the force of interest in continuous time.

Explanation

The steps demonstrate how the repayment function

L(t)

can be derived from the initial loan amount and the repayment rate, considering the exponential growth of repayments and the constant force of interest.

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Solution by Steps

step 1

To find the initial repayment rate

P(0)

such that the loan is repaid in 32 years, we need to set

L(32) = 0

. Using the equation from Question 5, we have:

e^{-\Delta(32)} L(32) = L(0) - \int_{0}^{32} P(u) e^{-\Delta(u)} du

step 2

We know

L(0) = 180,000

and

\Delta(32) = \int_{0}^{32} 0.07 du = 2.24

. Thus,

e^{-\Delta(32)} = e^{-2.24}

step 3

Setting

L(32) = 0

gives us:

0 = 180,000 - \int_{0}^{32} P(u) e^{-\Delta(u)} du

. This implies

\int_{0}^{32} P(u) e^{-\Delta(u)} du = 180,000

step 4

Substituting

P(u) = P(0) e^{0.05 u}

into the integral:

\int_{0}^{32} P(0) e^{0.05 u} e^{-0.07u} du = P(0) \int_{0}^{32} e^{-0.02u} du

step 5

Evaluating the integral

\int_{0}^{32} e^{-0.02u} du = \left[-\frac{1}{0.02} e^{-0.02u}\right]_{0}^{32} = -\frac{1}{0.02}(e^{-0.64} - 1)

step 6

Therefore, we have

P(0) \cdot \frac{1}{0.02}(1 - e^{-0.64}) = 180,000

. Solving for

P(0)

gives

P(0) = \frac{180,000 \cdot 0.02}{1 - e^{-0.64}}

Answer

The initial repayment rate

P(0)

should be approximately $5,000.

Key Concept

The calculation of the initial repayment rate based on the loan amount and the repayment schedule.

Explanation

The steps illustrate how to derive the initial repayment rate needed to ensure the loan is fully repaid after 32 years, taking into account the exponential growth of repayments and the constant force of interest.

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