Interest Rates

Question 1

What is an Interest Rate (r)?

Answer 1

A rate of return which reflects the difference between differently dated cashflows

Question 2

If £9,500 today and £10,000 in 1.0 years are equivalent in value, what is the interest rate?

Answer 2

r = (10,000-9,500) / 9,500

= 0.0526 / 5.26%

Question 3

What are three interchangeable names for interest rates?

Answer 3

1.) Required rate of return
2.) Discount rate
3.) Opportunity cost

Question 4

What are the 4 premiums above risk free rate of interest?

Answer 4

1.) Inflation premium
2.) Default risk premium
3.) Liquidity premium
4.) Maturity premium

Question 5

What is the inflation premium?

Answer 5

Compensation for expected inflation over the duration of the debt. Risk free rate + Inflation premium = Nominal risk free rate

Question 6

What is the liquidity premium?

Answer 6

Compensation for the risk of loss relative to fair value, if an investment has to be quickly converted to cash

Question 7

What is the default risk premium?

Answer 7

Compensate investors for the possibility that the borrower will - default/make incorrect/miss - payments.

Question 8

What is the maturity premium?

Answer 8

Compensate investors for increased sensitivity of the market value of debt over time - longer debt = > risk and premium.

Question 9

What is the equation for the future value of £100 invested for 1yr @ 5%

Answer 9

FV1 = PV(1+r) //// FV1 = £100(1+0.05)

Question 10

What is the equation for the future value of £100 for 100yrs @ 5%

Answer 10

FVn = PV(1+r)^n //// FV100 = £100(1+0.13) ^100 = £20,316,287.

Question 11

What is the future value of £5m for 5 yrs @ 7%

Answer 11

FVn = PV(1+r)^n ////// FV5 = £5m(1+0.07)^5 = £7,012,758

Question 12

If the stated interest (rs) is 8% compounded periodically, how does that impact the FV formula?

Answer 12

FVn = PV (1 + rs / no. annual compounds) ^ no.annual compounds * n

Question 13

What is the formula for FV on a continuously compounded rate?

eg., £5,000 - 8% - 2yrs

Answer 13

FVn = PVe^rsn

FV2= PV * e ^ 0.08 * 2

= £5,867.55

Question 14

How do you calculate the Effective Annual Rate where the stated interest is 8% but the security compounds monthly?

Answer 14

EAR = ( 1 + periodic rate) ^m -1

= (1 + 0.08/12)^12 -1

= 0.08299951

Question 15

Calculate the stated interest rate where the EAR (effective annual rate) is 8.299951% compounded monthly

Answer 15

EAR = ( 1 + periodic rate ) ^m -1
0.08299951 = (1+pr)^12 -1
1.08299951 = (1+pr)^12
1.08299951^1/12=1+pr
1 - (1.08299951) ^1/12 = pr
periodic rate = 0.0066667
rs = pr *12
= 8%

Question 16

If you are asked to calculate the FV of a sum which has many or continuous compounding periods within a year, what is the process?

Answer 16

Calculate the EAR -
(continuous) EAR = e^rs(n) -1
(periodic) EAR = (1 + periodic rate e.g., 0.08/12) ^12 -1

Then do FV

FVn = PV(1+r)^n

Question 17

What is an annuity?

Answer 17

A set of level/identical and sequential cashflows

Question 18

What is an ordinary annuity?

Answer 18

An annuity which has its first cashflow 1 period from now - indexed at t = 1

Question 19

What is an annuity due?

Answer 19

An annuity due has its first cashflow immediately - indexed at t = 0

Question 20

What is a perpetuity / annual perpetuity?

Answer 20

A set of level/identical, never-ending cashflows where the first cashflow is 1 period from now - indexed at t = 1

Question 21

What is the equation to simply calculate the future value of an ordinary annuity over e.g., 20 years, 20k pa, @ 9%

Answer 21

FVn = A ((1+r)^n -1) / r)
FVn = 20,000 (1+0.09)^20 -1 / 0.09)

Question 22

What is the formula for PV?

Answer 22

PV = FVn (1/(1+r)^n)

Question 23

What is the formula for PV with periodic compounding?

Answer 23

PV = FVn (1 + rs/m)^-m * n

Question 24

The manager of a Canadian pension fund knows that the fund must make a lump-sum payment of C$5 million 10 years from now. She wants to invest an amount today in a GIC so that it will grow to the required amount. The current interest rate on GICs is 6 percent a year, compounded monthly. How much should she invest today in the GIC?

Answer 24

PV = FVn (1 + rs/m)^-m*n

PV = 5,000,000 ( 1 + 0.06/12)^-12*10

PV = 2,748,163.67

Question 25

What the equation for the present value of an annuity?

Answer 25

PV = A ( 1 - 1/(1+r)^n / r)

Question 26

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?

Answer 26

PV = A (1- 1/(1+r)^n / r)
PV = 1000 (1 - 1/(1+0.12)^5 / r)
PV = £3,604.78

Question 27

You are retiring today and must choose to take your retirement benefits either as a lump sum or as an annuity. Your company’s benefits officer presents you with two alternatives: an immediate lump sum of $2 million or an annuity with 20 payments of $200,000 a year with the first payment starting today. The interest rate at your bank is 7 percent per year compounded annually. Which option has the greater present value? (Ignore any tax differences between the two options.)

Answer 27

$2m today = $2m today

It’s an annuity due, 1st payment is today.

Therefore £200k lump sum + PV of 19 yrs $200k ordinary annuity

PV = A (1 - 1/(1+r)^n / r)
PV = 200,000 (1 - 1/(1+0.07)^19 / 0.07)
PV = £2,267,119.05

Therefore annuity payments are superior

Question 28

A German pension fund manager anticipates that benefits of €1 million per year must be paid to retirees. Retirements will not occur until 10 years from now at time t = 10. Once benefits begin to be paid, they will extend until t = 39 for a total of 30 payments. What is the present value of the pension liability if the appropriate annual discount rate for plan liabilities is 5 percent compounded annually?

Answer 28

Draw a timeline.
t 0——9————-39
$ 0——1————30

Calculate the annuity of the 30 payments of 1m from yr 10 to 39

PV=1,000,000 (1- 1/(1.05)^30/0.05)
PV @ yr 9 = $15,372,451
Then PV to yr0
PV = FVn(1+r)^-n
PV = $15,372,451 (1.05)^-9
= £9,909,219

Question 29

What is the equation for a perpetual annuity?

Answer 29

PV = (A / r)

Question 30

Consider a level perpetuity of GBP100 per year with its first payment beginning at t = 5. What is its present value today (at t = 0), given a 5 percent discount rate?

Answer 30

Calculate the £100 in perp from yr 5 =
1/r * 100
step 2:
Calculate PV as if it is its own Ordinary annuity therefore PV from yr 4 -

PV = 2000 ( 1/(1+r)^4 = £1,645.40 @ t = 0.

Question 31

What is the equation for a growth rate?

Answer 31

g = (FVn / PV) ^1/n

Question 32

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?

Answer 32

g = (FVn/PV)^1/n
g= (8.96/7.35)^1/6
g= 3.36%

Question 33

Can a lump sum be an annuity?

Answer 33

Yes and an annuity can be seen as equivalent to future value. PV’s FV’s and series of cashflows can all be equivalent if they are indexed at the same point in time.

Question 34

What is the Cash Flow Addivity Principle?

Answer 34

The cash flow additivity principle—the idea that amounts of money indexed at the same point in time are additive—is one of the most important concepts in time value of money mathematics.

Question 35

What is the equation for a holding period return?

Answer 35

The following reflects return over time inclusive of capital appreciation and income.

HPR = (p1 - p0) + total income / p0

Question 36

An investor purchased 100 shares of a stock for USD34.50 per share at the beginning of the quarter. If the investor sold all of the shares for USD30.50 per share after receiving a USD51.55 dividend payment at the end of the quarter, the investor’s holding period return is closest to:

−13.0 percent.
−11.6 percent.
−10.1 percent.

Answer 36

-10.1%

Question 37

What mean should you use when including all values and outliers?

Answer 37

Arithmetic Mean

Question 38

What mean should you use when calculating compounding?

Answer 38

Geometric Mean

Question 39

What mean should you use when calculating mean where data has multiple extreme outliers?

Answer 39

Harmonised Mean, Winsorized Mean and Trimmed Mean

Question 40

What is the definition of IRR?

Answer 40

The internal rate of return is the discount rate at which the sum of present values of cash flows will equal zero.

Question 41

what do you calculate when you want the money weighted return of a fund?

Answer 41

IRR

Question 42

When use IRR (money weighted return) or geomean (time weighted return)?

Answer 42

IRR is best for calculating the return of an investment including its inflows and outflows, e.g, Real Estate

Time weighted return does not account for in/outflows and therefore better reflects the performance of a fund from a portfolio manager perspective.

Question 43

What is a gross return?

Answer 43

The return earned by an asset manager prior to deductions for management costs, custodial fees, or expenses related to administration of an investment. Trading expenses (commission) are included in gross return, they do relate to an asset managers performance.