Vol. 1 LM1 Present Value

Question 1

The Present Value of a Lump Sum

An insurance company has issued a Guaranteed Investment Contract (GIC) that promises to pay $100,000 in six years with an 8 percent return rate. What amount of money must the insurer invest today at 8 percent for six years to make the promised payment?

p. 18

Answer 1

= $100,000 [ 1 / (1.08)⁶
= $100,00 (0.6301696)
= $63,016.96
We can say that $63,016.96 today, with an interest rate of 8 percent, is equivalent to $100,000 to be received in six years.

Question 2

Describe

The Present Value of a Series of Equal Cash Flows

p. 21

Answer 2

• Recall that an ordinary annuity has equal annuity payments
• the first payment starting one period into the future, at t = 1
• the annuity makes N payments

Question 3

Formula

The Present Value of a Series of Equal Cash Flows

p. 21

Answer 3

Question 4

The Present Value of an Ordinary Annuity

Suppose you are considering purchasing a financial asset that promises to pay €1,000 per year for five years, with the first payment one year from now. The required rate of return is 12 percent per year. How much should you pay for this asset?

p. 22

Answer 4

We use the formula for the present value of an ordinary annuity
A = €1,000, r = 0.12, N = 5
PV = €1,000 [(1 - 1/(1.12)⁵)/0.12]
= €1,000 (3.604776) = €3,604.78

Question 5

Given a 5 percent discount rate, find the present value of a four-year ordinary annuity of £100 per year starting in Year 1 as the difference between the following two level perpetuities:
Perpetuity 1 £100 per year starting in Year 1 (first payment at t = 1)
Perpetuity 2 £100 per year starting in Year 5 (first payment at t = 5)

p. 28

Answer 5

If we subtract Perpetuity 2 from Perpetuity 1, we are left with an ordinary annuity of £100 per period for four years (payments at t = 1, 2, 3, 4). Sub- tracting the present value of Perpetuity 2 from that of Perpetuity 1, we arrive at the present value of the four-year ordinary annuity:

PV0(Perpetuity1) = £100 / 0.05 = £2,000
PV4(Perpetuity2) = £100 / 0.05 = £2,000
PV0 (Perpetuity 2) = £2, 000 / (1.05) 4 = £1, 645.40
PV0 (Annuity) = PV0 (Perpetuity 1) − PV0 (Perpetuity 2)

Question 6

Calculating a Growth Rate (1)

Hyundai Steel, the first Korean steelmaker, was established in 1953. Hyundai Steel’s sales increased from ₩14,146.4 billion in 2012 to ₩19,166.0 billion in 2017. However, its net profit declined from ₩796.4 billion in 2012 to ₩727.5 billion in 2017. Calculate the following growth rates for Hyundai Steel for the five-year period from the end of 2012 to the end of 2017:

p. 29

Answer 6

g = (FV_N / PV) ^{1/N - 1
We denote sales in 2012 as PV and sales in 2017 as FV₅}

Question 7

Calculating a Growth Rate (2)

Toyota Motor Corporation, one of the largest automakers in the world, had consolidated vehicle sales of 8.96 million units in 2018 (fiscal year ending 31 March 2018). This is substantially more than consolidated vehicle sales of 7.35 million units six years earlier in 2012. What was the growth rate in number of vehicles sold by Toyota from 2012 to 2018?

p. 30

Answer 7

Question 8

Calculating the Size of Payments on a Fixed-Rate Mortgage

You are planning to purchase a $120,000 house by making a down payment of $20,000 and borrowing the remainder with a 30-year fixed-rate mortgage with monthly payments. The first payment is due at t = 1. Current mortgage interest rates are quoted at 8 percent with monthly compounding. What will your monthly mortgage payments be?

p. 33

Answer 8

The bank will determine the mortgage payments such that at the stated periodic interest rate, the present value of the payments will be equal to the amount borrowed (in this case, $100,000).

Question 9

Concept

When dealing with uneven cash flows, we take maximum advantage of the principle that dollar amounts indexed at the same point in time are additive

p. 33

Answer 9

cash flow additivity principle

Question 10

To see how a lump sum can fund an annuity, assume that we place $4,329.48 in the bank today at 5 percent interest. We can calculate the size of the annuity payments by using Equation 11. Solving for A, we find

p. 37

Answer 10

Question 11

Calculate

general annuity formula

p. 16

Answer 11

FV_N = A [ (1+r)^N - 1] / r

Question 12

Concept

a set of level never-ending sequential cash flows, with the first cash flow occurring one period from now.

p. 15

Answer 12

perpetuity

Question 13

Define

annuity due

p. 15

Answer 13

has a first cash flow that occurs immediately (indexed at t=0)

Question 14

Define

ordinary annuity

p. 15

Answer 14

has a first cash flow that occurs one period from now (indexed at t=1)

Question 15

Concept

has a first cash flow that occurs one period from now (indexed at t=1)

p. 15

Answer 15

ordinary annuity

Question 16

Define

annuity

p. 15

Answer 16

is a finite set of level sequential cash flows

Question 17

Concept

is a finite set of level sequential cash flows

p. 15

Answer 17

annuity