QM PREREQ1 – Interest Rates, Present Value and Future Value

Question 1

What are the 3 rules of money?

Answer 1

1. Money soon is worth more than money later
2. Larger cash flows are worth more than smaller
3. Less risky cash flows are worth more than more risky

Question 2

What are the 3 ways of thinking about interest rates?

Answer 2

1. Required rate of return: RoR required by an investor ot lender.
   Money today * (1 + r) = money tomorrow
2. Discount rate: rate at which some future value is discounted to arrive at a value today
   Money tomorrow / (1 + r) = Money today
3. Opportunity cost: the value an investor or lender forgoes by chhoosing a particular action.
   I.e., r is the opportunity cost of current consumption

Typically required rate of return = discount rate = opportunity cost

Question 3

What 4 premiums will be built into the rate of return if I lend someone money, on top of the risk-free rate?

Answer 3

1. Inflation premium: compensates for expected inflation ( π^e)
2. Default risk premium: compensates lender for credit risk
3. Liquidity premium: compensation for risk of loss versus fair value if an investment needs to be converted to cash quickly
4. Maturity premium: greater interest rate risk (i.e., price risk) with longer maturities. This is because as yields increase, bond price increases. So if yields increase, your bond may be devalued.
   This will also include a premium for inflation.
   It is ultimately due to uncertainty: the longer the time period, the more uncertain we are about the level of expected inflation

Ideally these would be multiplicative rather than additive, but additive is just fine

Question 4

What is the nominal risk-free rate?

Answer 4

r⌄f + π^e = nominal risk-free rate

Where r⌄f is the risk-free rate
and π^e is the inflation premium

The nominal risk-free rate might be measured by something like the return on a US Treasury 3-month T-bill
It build in an inflation premium as well as the underlying risk-free rate

Question 5

What does it mean to say that r must be in the same periodicity as N when calculating the future value of a single cash flow?

Answer 5

r represents the interest rate, N represents the number of periods

If the interest rate was 6% per year over 10 years with an annual periodicity, the final nominal value is 100(1.06)^10

If it had semi-annual periodicity it would be 100(1.03)^20

If it had quarterly periodicity it would be 100(1.015)^40

These will result in different values so we need to match the periodicity

Question 6

How do we calculate FV?

Answer 6

Future Value = Present Value x (1 + r)^N

Where r = interest rate
N = number of periods

Question 7

What is simple interest?

Answer 7

Interest calculated on the original amount
Contrasted to compounded interest, which is calculated on the amount from the last period

i.e., 5% interest on £1000 over 20 years would return (0.05 * 1000 * 20) = £2000

Question 8

How do you calculate future value of £10m you receive in 5yrs and invest at 9% RoR for 10 years?

Answer 8

It doesn’t matter when you receive it, it is still money invested for 5 years.
Method 1: FV = 10m(1.09)^10 = 23.7m
Method 2: N=10, I/Y = 9, PMT = 0
PV = -10m
CPT FV = 23.7m

To calculate value of the 10m today using this interest rate, we can discount it by (1.09)^5 and divide 10m by this amount
10m / (1.09)^5 = 6.5m

Question 9

How are interest rates stated?

Answer 9

Rates are ALWAYS quoted annually
That means if you see a 3-month T-bill yielding 3%, you do not get 3%, only 1/4 of 3% across the 3 months (which is 0.75%)

r⌄s = stated interest rate

Question 10

How do we calculate value of $1m held over 1 year with a rate of 3% that is compounded monthly?

Answer 10

FV = 1m (1 + (6% / 12)) ^ (12 x 1)

Question 11

What is continuous compounding?

Answer 11

This is really just an easier way of calculating or implementing the idea of daily compounding, which can get clunky to use (dividing rates by 365)

We have to use Euler’s constant, e, for continuous compounding. We multiply the present value by e to the power of rate x number of periouds

FV = PV x e ^(r x N)

Question 12

What do we press on the calculator to calculate using continuous compounding the future value of 50,000 at an interest rate of 7% held for 3 years?

Answer 12

On the calculator we do:
0.07 x 3 = 2nd function, LN x 50 000
We must use the equals because 0.07 x 3 should effectively be in brackets

Question 13

How to calculate stated rate if we know EAR?

Answer 13

If we know Effective Annual Rate we can work backwards to find the effective annual rate when we also know the periodicity.

Let’s say that we have an EAR of 10%

0.1 = (1 + rs /12)^12 - 1
1.1 = (1 + rs/12)^12
(1.1)^1/12 = 1 + rs/12
(1.1)^1/12 -1 = rs/12
((1.1)^1/12 - 1) x 12 = rs
0.0957 = 9.57% = rs

9.57% = stated rate

Question 14

How do we calculate stated rate if we have EAR using continuous compounding?

EAR = 5.5%

Answer 14

0.055 = e^rs - 1
1.055 = e^rs
ln(1.055) = rs
0.0535 = rs
5.35% = rs = stated return

Question 15

What is an annuity?

Answer 15

A finite set of level sequential cash flows

Something cannot be an annuity if:
- the cash flow differs
- some years are missed out
- the cash flows do not have an end date

Question 16

What is an ordinary annuity?
What is an annuity due?

Answer 16

Ordinary annuity: where the first cash flow happens at the end of the first year
Annuity due: where the first cash flow happens at the beginning of the first year

Important because the cash can earn interest over the period

Question 17

How do you calculate the future value of an ordinary annuity?

Answer 17

Enter number of years and press N
Enter payment amount and press PMT
Enter rate and press I/Y
Enter present value and press PV
Press CPT FV to calculate

Question 18

How do you calculate the future value of an annuity due?

Answer 18

An annuity due starts paying in from the beginning of the first year, rather than the end
This means that interest can accrue over the year starting from t=0
The first payment out can therefore grow for the entire duration of the annuity, rather than n-1

You can calculate the value of an annuity due by just calculating the value of an ordinary annuity and multiply it by (1+r) to account for that additional year of compounding

You could also enter Begin mode (BGN) on your financial calculator to perform the annuity due calculation. However it might be sometimes inconvenient or lead to errors if you keep flipping back and forth. Therefore MM keeps his calculator in END mode and just multiplies at the end

Question 19

How do you calculate the future value of unequal cash flows?

Answer 19

It can be calculated manually or using the calculator functions
Calculating it manually involves multiplying each annuity payment by the number of years it has to gain interest. Then adding these together.

We can use the calculator function NFV to find the future value of a series of cashflows. However not all calculators have this function. If they do, it works the same as the NPV function in terms of inputs

Question 20

How do you calculate the present value of a single cash flow?

Answer 20

Multiply the single cash flow (the future value) by:
(1 + r)^-N

Which is the same as:
FV / (1 + r)^N

Question 21

How do you calculate the present value of a series of cash flows for an ordinary annuity?

Answer 21

PV = A [(1 - 1 / {1 + r}^N) / r ]

Where PV = Present Value
A = Annuity Due
r = rate of return / interest rate
N = number of years

i.e., if A = 1000 (payment in per year)
r = 0.07
N = 6

PV = 1000 * [(1 - 1 / 1.07^6 ) / 0.07)]
PV = 4 767

Question 22

How do you calculate the present value of a series of cash flows for an annuity due?

Answer 22

It is the initial payment in plus the PV of a series of cash flows for an ordinary annuity for N - 1 years of the annuity

I.e. you would add

PV = A + A [(1 - 1 / {1 + r}^[N-1]) / r ]

Where PV = Present Value
A = Annuity Due
r = rate of return / interest rate
N = number of years

You can add the time value of money keys to calculate that

Question 23

What is a perpetuity?

Answer 23

An annuity that pays out forever.

The cash flows from a perpetuity are:
- level
- sequential
- infinite

We can find the present value of a perpetuity by dividing the amount the perpetuity pays per year by r
where r is the discount rate

I.e., if our perpetuity pays out £100 per year and the discount rate is 5%, the present value is 100/0.05 = £2000

Because of the constant discounting as time progresses, no matter what time you consider the perpetuity to start it will always have the same value, if you take the discounting into account

Question 24

How can we create a 7-year annuity from perpetuities?

Answer 24

We find 2 perpetuities that are identically matched. They pay out the same amount each period. However, one starts at t=0 and another at t=7. We go long the first one and short the second.

Until t=7, we are only exposed to cash flows from the long perpetuity. This gives us the annuity payments. When t=7 begins, we pay the perpetuity short using the cash from the perpetuity long. These balance out, and we are left with net zero cash flows.

Question 25

How do we calculate the present value of a series of unequal cash flows using the NPV function on a financial calculator?

Answer 25

First clear the calculator pressing 2nd CF 2nd CE/C in order.

Then your screen will say CF0. This is the cash flow at the beginning.
Where there is no value press 0 and then the down arrow
In years where there is a value write the amount, press enter, and down arrow twice
At the last cash flow, press the down arrow once and then hit NPV
Then I will be displayed, This is the discount rate. Write a number, then press enter and the down arrow.
Then press NPV and CPT

Your value will be displayed

Question 26

Why might you calculate the present value of a series of unequal cash flows manually rather than using the calculator’s NPV function

Answer 26

It’s almost the same number of keystrokes
You could simply divide each amount by 1+ the discount rate to the power of the number of years of discounting

i.e.,10 000 / (1.04)
+ 20 000 / (1.04)^2
+ 30 000 / (1.04)^3
= PV

Question 27

How do we determine a growth rate given FV, PV, and N?

Answer 27

r = (FV/PV)^(1/N) - 1

I.e., let’s say future value = 2 000 000
present value = 450 000
N = 20 years

(2 000 000 / 450 000)^0.05 - 1 = 0.077 = 7.7%

By contrast, if FV = 1 500 000, PV = 550 000, N = 25 years:

(1 500 000 / 550 000)^0.04 - 1 = 4.1%

We can also use this to determine growth rate per year of a financial metric of a company found on its financial statements

Question 28

How do we solve for N? I.e., how long would it take to turn £100 into £500 at 10% compounded annually?

Answer 28

Solve for N:
FV = PV (1+r)^N
(1+r)^N = FV/PV
N ln (1+r) = ln (FV / PV)
N = ln (FV / PV) / ln (1 + r)

In this case:
N = ln (500 / 100) / ln (1+0.1)
N = 16.89

Question 29

How would you determine what your monthly payment for a £500,000 mortgage would be at a 4% interest rate compounded monthly over 20 years?

Answer 29

You can simply use the annuity formula!

If PV = A [(1 - 1 / {1 + r}^N) / r ]
then A = PV / [(1 - 1 / {1 + r}^N) / r ]

Make sure to modify the periodicity by dividing the interest rate by 12 and multiplying N by 12:

A = 500 000 / [(1 - 1 / {1 + 0.04/12}^12*20) / 0.04/12 ]
A = 500 000 / 165.022
A = £3 030 per month

Question 30

How do we solve for a payment to meet a retirement goal?
I just turned 23.
At age 53, I want to retire.
I expect to live for 30 years, until 83.
I want to receive £40,000 per year during this time period.
For the next 5 years, I can save £2000 per year
From 28 onwards, how much do I need to save per year to hit my retirement goal?
Assume our return is 6.25% (1/16)

Solve by bringing the value of the retirement income back.

Answer 30

First let’s calculate the value of the initial payments on my 28th birthday:
FV5: N=5, PMT=2000, I/Y=6.25, PV=0 CPT FV
= £11,339

Then let’s calculate the value of the future retirement income when I hit 53:
PV30: N=30, PMT=40 000, I/Y=6.25, FV=0 CPT PV = 536 173

Third, let’s compare the value of the future retirement income at my 28th birthday:
PV5 = PV30 / (1.0625)^25
PV5 = £117 783

Now let’s see how far short I am:
117 783 - 11 339 = 106 444

The PV of my earnings from 28 to 53 must therefore equal £106,444

N=25, FV = 0, I/Y = 6.25, PV = 106,444 CPT PMT = 8 526

So I would need to pay in £8 526 per year from 28 to 53 to hit my retirement goal

Question 31

How do we solve for a payment to meet a retirement goal?
I just turned 23.
At age 53, I want to retire.
I expect to live for 30 years, until 83.
I want to receive a nominal £100,000 per year during this time period.
For the next 10 years, I can save a nominal £4000 per year
From 33 onwards, how much do I need to save per year to hit my retirement goal?
Assume our return is 8%

Solve by bringing the value of initial payments forward.

Answer 31

First let’s calculate the value of the initial payments on my 28th birthday:
FV10: N=10, PMT=4000, I/Y=8, PV=0 CPT FV
= £57,946

Second, let’s calculate the value of the future retirement income when I hit 53:
PV30: N=30, PMT=100 000, I/Y=8, FV=0 CPT PV = £1,125,778

Third, let’s compare the value of the initial savings when I retire:
PV30 = PV10(1.08)^20 = £270 083

Fourth, let’s find the difference:
1,125,778 - 270 083 = 855 695

Fifth, let’s calculate what annual payment I would need to make over the last 20 years of working to reach this figure:
N=20, FV=855 695, I/Y=8, PV=0 CPT PMT = £18,699

Thus I would need to save £18,699 (nominal) every year whilst working from 33 to 53 to meet my retirement goals.

Question 32

What is data?

Answer 32

A collection of numbers, characters, words or text that represents FACTS or INFORMATION

Thus,
1. Data is not knowledge
Analysis or interpretation brought to data brings knowledge
2. Data does not have to be numbers