L4 - Time Value of Money

Question 1

What is the risk free rate?

Answer 1

• The risk-free rate of return is the theoretical rate of return of an investment with zero risk. The risk-free rate represents the interest an investor would expect from an absolutely risk-free investment over a specified period of time.
• The real risk-free rate can be calculated by subtracting the current inflation rate from the yield of the Treasury bond matching your investment duration.
• In theory, the risk-free rate is the minimum return an investor expects for any investment because he will not accept additional risk unless the potential rate of return is greater than the risk-free rate.
• In practice, however, the risk-free rate does not exist because even the safest investments carry a very small amount of risk. Thus, the interest rate on a three-month U.S. Treasury bill is often used as the risk-free rate for U.S.-based investors.

Question 2

How do you calculate future value?

Answer 2

• A dollar today is worth more than a dollar to be received in the future, because if you had it now, you could invest it and earn interest
• Future value = Present value (1+interest rate)^{time period}

FV= PV(1+r)^t

Question 3

How do you calculate Present Value?

Answer 3

PV = FV/(1+r)^t

where the discount factor 1/(1+r)^t < 1 denotes a present value of £1 received in the future

Question 4

What is the general formula for Future Value when compounded in different periods?

Answer 4

FV = PV (1 + r/m) ^t×m

References where r denotes annual interest rate, m number of compound periods per annum, and t number of years

Question 5

What is the formula for continuous compounding?

Answer 5

FV = PVe^t×r and PV = FV e^−t×r

lim_m–>∞ PV (1 + r/m) ^t×m = PVe^txr

Question 6

What is Effective Interest Rates?

Answer 6

• The effective interest rate refers to the actual annual interest rate paid on an investment or loan. It is used to compare the annual interest between investments, or loans with different compounding periods

EAR = (1 +r/m)^m

Question 7

What is Annuity?

Answer 7

A simple annuity is defined as a series of payments of a fixed amount for a specified number of periods.

• Cash flows occur at the end of each period (usual)- ordinary annuity/deferred payment annuity
• Cash flows made at the beginning of each period -annuity due
• Examples: Bonds, regular savings, regularloan payments

Question 8

What is Perpetuity?

Answer 8

Perpetuity is a special case in which cash flowslast indefinitely. • Examples: Consols, Shares (Gordon model), Realestate.

Question 9

How do you calculate Ordinary Annuity?

Answer 9

simple annuity is calculated by:

PV=C/r(1-(1/(1+r)^t)

Question 10

How do we calculate the Present Value of a Perpetuity?

Answer 10

• Present value is equal to discounted future values:

PV = C/(1 + r)¹ + C/(1 + r)² + C/(1 + r)³ + … (1)

To derive the formula for PV, multiply both sides by 1 + r:

PV (1 + r) = C + C/(1 + r)¹ + C/(1 + r)² + … (2)

• Subtract equation (1) from equation (2):

PV (1 + r) − PV = C + C/(1 + r)¹ + C/(1 + r)² + … - C/(1 + r)¹ - C/(1 + r)² - C/(1 + r)³ -…

Consequently all values cancel out till you get:

PV (1 + r) − PV = C => rPV = C => PV = C/r

Question 11

How do we derive the Annuity formula?

Answer 11

What we are looking for is the difference between the present value of Perpetuity A and present value of Perpetuity B:

• Present value of perpentuity A is given by:

PV_A=C/r

Present value of perpetuty B is given by:

PV_B= (C/r)/(1+r)^t = (C/r) x 1/(1+r)^t = C/r(1+r)^t

The formula for annuity is therefore given by:

PV = PV_A - PV_B= C/r - C/r(1+r)^t=C/r(1-(1/(1+r)^t))

Question 12

How do you calculate Future Value of Annuity?

Answer 12

The Formula for present value of annuity is given by:

PV =C/r(1-(1/(1+r)^t))

The formula for future value is given by:

FV=PV(1+r)^t

FV=C/r(1-(1/(1+r)^t)) x (1+r)^t=C/r ((1+r)^t-1)

Question 13

How to calculate Holding Period Return?

Answer 13

• Holding period return is the rate of return that investors obtain over a specific period of time.

r= D₁ +P₁-P₀/P₀

where: P₀ denotes the value of the investment at the beginning of the holding period, P₁ denotes the value of the investment at the end of the holding period, and D₁ denotes dividend or interest payments received during that period.

Question 14

How do you calculate the nominal interest rate?

Answer 14

Gross real interest rate x Gross inflation rate

• Nominal interest rate is the real interest rate including the inflation rate
• Gross means taxes are included, Net usually discounts them
• Net interest rate is just in the percentage of the interest rate, so if the interest rate is 8% it is just 0.08, Gross interest rate is 1 + the interest rate
• The same applies for Gross inflation

Question 15

What is Fisher’s Equation?

Answer 15

The generalized relationship between real rates of return and nominal (or market or money) rates of return and inflation is expressed in Fisher (1930) equation:

(1 + R) = (1 + r) (1 + Eπ)

where: R denotes nominal interest rate; r denotes real interest rate; π denotes inflation rate, Real interest rate:

r = [(1 +R)/(1 + Eπ)] - 1 = [(R -Eπ)/(1 + Eπ)] ≈ R - Eπ

Question 16

What is the Ex-ante nominal interest rate?

Answer 16

R_t ≈ r + Eπ

Ex-ante = based on forecasts rather than actual results

Question 17

What is the Ex-post nominal interest rate?

Answer 17

R ≈ r + π

Ex-post = based on actual results rather than forecasts.

Question 18

What should you never do when calculating returns?

Answer 18

1 Discount money cash flows with the real discount rate

2 Discount real cash flows with the money discount rate

Question 19

When will compensation be required to induce people to make consuption sacrifices?

Answer 19

• impatience to consume - the price of time - need to give consumers an incentive to save this is usually the pure interest rate with no inflation or risk
• inflation - need compensation for the loss of purchasing power over time
• Risk - account for futurns returns having a range of possible values

Question 20

How do you calculate the Required Return rate?

Answer 20

Required return = Risk Free Rate + Risk premium

the Risk premium accounts for the expected compensation for all three elements of the time value of money

Question 21

What is the Interal Rate of Return?

Answer 21

• IIR also is also referred to as the yield of a project
• Alternatively the internal rate of return r can be viewed as the discount rate at with NPV is equal to zero
• Recommended way to calculate IIR is by guess values:

1. plus a random value of r into your NPV calculation
2. keep plugging one in till you get close to zero
3. After you plug a number in that gives you a negative yield you know that the IIR sits inbetween those to percentage values
4. IIR = MIN + (MIN distance to 0)/(MIN distance to MAX)*(MAX - MIN)
   
   1. where MIN and MAX are the repective small and larger discount rates surrounding 0

The rule for IIR is the opportunity cost of capital (k) is greater than the IIR reject the project

Question 22

Problems with IRR?

Answer 22

• if the relationship between NPV and the discount rate isnt linear –> IRR uses an interpolation formula which assumes a straight line so could give you incorrect results?
• can give multiple solutions –> if cashflows are constantly changing (some years positive, some years negative) can give multiple values of IIR
• unlike NPV cannot provide any ranking - while one project may have a higher IIR it doesnt necessarily mean it will generate greater returns (may have had a high initial cost) –> NPV are measured in absolute amounts whereas IIR is a % value

Question 23

What is the Modified Internal Rate of Return?

Answer 23

QUESTIONS IN BOOK

• the main problems with the IRR is the assumption that the obtained positive cash flows are reinvested at the same rate at which they were generated.
• Alternatively, the MIRR considers that the proceeds from the positive cash flows of a project will be reinvested at the external rate of return. Frequently, the external rate of return is set equal to the company’s cost of capital.
• Also, in some cases, the calculations of IRR may provide two solutions. The fact creates ambiguity and unnecessary confusion regarding the correct outcome. Unlike the IRR, the MIRR calculations always return a single solution.
• The common view is that the MIRR provides a more realistic picture of the return on the investment project relative to the standard IRR. Due to such a reason, the MIRR is commonly lower than the IRR.

m = N^thrt(F/P) -1

• where F is the future terminal value and P is the intial outlay)
• the future terminal value is the sum of all the future values compounded at the investment or NPV rate