The Time Value of Money

Why is £100 today worth more than £100 tomorrow?

• Deposit account in bank pays interest, so, overnight, £100 will have grown to be £100.03 (10% interest rate)

• Would swap £100 today for £100.03 tomorrow, but not less

• Why do banks pay interest?

Ninth Birthday Present - £20

Option1 Invest in the bank @ 5% per annum to yield £21 in one years time

Option 2

Multiplier effect – 500% return

Time Value of Money

• Money invested now must yield a positive return

• What form can that positive return take?– Riskless Return– Present Utility– Future Reward

Future Value and Compound Interest

How much interest will I earn over 1 year ? Interest = Initial sum invested x Interest rate (r)

Future Value = Initial investment + Interest

Example

If a bank pays 8% p.a. and I invest £100, then after one year

Interest = £100 x 0.08= £8 = 100 x r

Future Value = £100 + £8 = £108

= 100 + 100 x r = 100 x (1+r)

So Future Value (FV) = Initial Investment x (1+r)

What if I invest for longer than a year?

Compound interest is interest earned on interestReminder - Future Value = FVExampler=8%, Initial Investment = £100FV after 1 year = £100 x (1.08) = £108Assume that I reinvest for another yearFV after 2 years = £108 x (1.08) = £116.64OrFV after 2 years = 100 x (1.08) x (1.08) = £116.64FV after 3 years= 100 x (1.08)x(1.08)x(1.08) = £125.97

FV after t years = Initial Investment x (1+r)^t

Compound InterestInterest in 1st year = £8Interest in 2nd year = £8 + £8.64Why?

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utu

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r=8%

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Interest is earned on both the initial investment and the interest which has accrued over previous yearsEarning interest on interest is called compounding

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Class Example

Warren Buffett took control of Berkshire Hathaway in January 1965 when the share price was USD19.46.His average annually compounded return has been 21.4%.If I had invested USD100 with Warren Buffett in January 1965, how much would it now be worth?Assume that we are in January 2009.

Class Example

• Time period t = 2009 – 1965

= 44 years

• Annually compounded rate of return r = 21.4%

• Current Value = 100 x 1.214^44

= USD 507,717

Present Value (PV)

• Cash can be invested to earn interest

• £1 today is worth more than £1 tomorrow

• Future Value (FV) indicates wealth at a future point

• Present Value (PV) indicates how much is needed today to fund a specified future cashflow

How much do we need now to produce £110 at the end of the year? Future Value = Present Value x (1+r)

FV = PV x (1+r)So, rearranging

PV = FV

(1+r)

Assume that r = 8%

PV = 110/(1.08)

= 101.85

To produce £110 in 1 years time, we need to invest £101.85 today

How much would we need to produce £110 after 2 years?

We know that £100 grows to 100 x (1+r)^2 so PV = FV

(1+r)^2

Assume that r = 8%

PV = 110/(1.08)^2 = 94.30

We would need £94.30 to produce £110 after 2 years

GeneralisingFor a payment t periods away

PV = future value required(1+r)^t

Alternative Terminology

• Discounted cashflow alternative name for the present value of a future cashflow

• Discount rate interest rate (r) used to calculate the PV of future cashflows

• Discount factor1/(1+r)^t

Present Value of a future cashflow of £100

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r=0%

Present value decreases with time Present value decreases as interest rates increase.

Implied Interest Rate

Sometimes a price is quoted and the interest rate needs to be deduced

Example

I paid £95,000 for an apartment in January 1991, the valuation on January 2009 was £450,000. What is my annualised rate of return?

Class Example

• Time period t = 2009 – 1991

= 18 years

• Current Value = 450,000

• Price paid = 95,000

• Annually compounded rate of return = r

(1+r)^18 = 450,000/95,000

r = 9.03%

Future Value Multiple Cashflows

What if I expect more than one cashflow to be invested to fund my future purchase?

Example

I want to replace my car in 5 years time.

I can afford to save £2000 each year. Interest rates are 8% p.a.

What can I afford?

Class Example

• Time period t = 2009 – 1965

= 44 years

• Annually compounded rate of return r = 21.4%

• Current Value = 100 x 1.214^44

= USD 507,717

Which Car?

Renault Clio Sport £13,000

Mini Cooper £12,000

Fiat Punto

£11,000

Calculation

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£1,500

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£2,500

£3,000

£3,500

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Calculate what each cashflow will be worth at the future date then add up the values

£2000 saved in one years time will be worth £2721 four years later.

What can I buy?

I have £14,672 so I can travel in style!

Present Value Multiple Cashflows

The Car Dealer offers me the opportunity to take the car now but pay in instalments.

I can pay £13,000 now or a £5000 deposit with a payment of £4500 in 1 years time and a further £4000 in 2 years time.

Which deal is better?

Calculation

To compare the deals, we need to convert to Present ValueOption 1 PV = £13,000Option 2 PV immediate payment = £5,000

PV second payment = £4500/1.08= £4167

PV third payment = £4000/1.08^2

= £3429Total Option 2 PV = £12,596

Annuities

• An annuity is a series of equally spaced level cash flows with a finite maturity

• If the payment stream last forever it is called a perpetuity

• A common example of an annuity is a home mortgage where the homeowner is required to pay a fixed sum each month for a term (normally 25 years) to fund the purchase of the house

Valuing AnnuitiesAssume a constant payment of £1 over the next n years beginning in 1 years timeAssume that interest rate = rPV =1/(1+r) + 1/(1+r)^2 +…+1/(1+r)^n (1)Multiply this equation by (1+r)(1+r) x PV = 1 + 1/(1+r) + …. +1/(1+r)^n-1 (2)(2) – (1) rPV = 1-1/(1+r)^n

Divide both sides by r

PV = 1/r – 1/r(1+r)^n

This is the formula to calculate the present value of an annuity that pays £1 a year for each of the next n years. Learn it or the derivation.

Example

I have purchased a building society bond which will pay £1000 p.a. annually over the next 10 years. If interest rates are 5%, how much did I pay for it?

Class Example

• PV = 1/r – 1/r(1+r)^n

• n = 10

• r = 5%• PV = 1000 x (1/0.05 - 1/0.05 x (1.05)^10)

= 7,722

Valuing Perpetuities

PV = 1/r – 1/r(1+r)^n

For a perpetuity, n is infinite , the payments continue for ever. So the second term in the equation vanishes.

PV perpetuity = 1/r

Example

My Great Aunt has left me an annual payment of £10,000 in her will, it will continue for ever.

I would like to see if I can buy a flat with it.

How much could I spend on the flat?

Interest rates are 6%.

PV = 10000/0.06 = £166,666

Monthly Payments

• Many payments are made monthly, including mortgages and salaries.

• The easiest way to deal with monthly payments is to convert the interest rate to a monthly value, calculate the number of monthly periods then use the formula derived earlier.

Example

Suppose the monthly interest rate is 1% and a house costs £150,000.

I have put down a deposit of £20,000 but want to know how much my monthly mortgage payments over the next 25 years will be.

Example

• PV = 150k -25k = 125k

• r = 1%

• n = 300

• Mortgage payment

= 150000/(1/.01 – 1/.01(1.01)^300)

= £1369

Amortizing Loans

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monthly payment interest capital repayment outstanding loan

Ref: Example 5.10, BMM P 129

What happens if first payment is today?

• If the first payment is immediate, it is called an annuity due

• The impact is to bring all of the cashflows forward by one period

• Accordingly, each cashflow can be put on deposit for one period to earn interest r

• PV annuity due = PV annuity x (1+r)Ref BMM Example 5.12 P135

Future Value of an AnnuityAssume a constant payment of £1 over the next n years beginning in 1 years timeAssume that interest rate = rThis time, we need the future value at time nFV =1 + (1+r) + (1+r)^2 +…+(1+r)^n-1 (1)Multiply this equation by (1+r)(1+r) x FV = (1+r) + (1+r) + …. +(1+r)^n (2)(2) – (1) rFV = (1+r)^n - 1Divide both sides by r

FV = (1+r)^n - 1r

This is the formula to calculate the future value at time n of an annuity that pays £1 a year for each of the next n years. Learn it or the derivation.

Example

I am saving for a round the world trip in 5 years time. I aim to save £2000 at the end of each of the next 5 years. If interest rates are 5%, how much cash will I have to spend at the end of 5 years?

Example

• £2000 p.a.

• r = 5%

• n= 5

• FV = (1+r)^n - 1

r• FV = 2000 x (1.05)^5-1/.05

= 11,051

Comparing Interest Rates• Interest rates may be quoted for days, months,

years or any interval• How can they be compared ?Example

12% rate with monthly compounding

£100 investment

At the end of 1 year, investment = 100 x (1.01)^12

= £112.68

12% annual rate

At the end of 1 year, investment = 100 x 1.12

= £112

Effective annual interest rate

• The effective annual interest rate is defined as the rate at which money grows allowing for the effect of compounding

• This enables quoted rates to be directly compared

• APR = rate per period x # periods in year

• If APR is monthly, then

Effective annual rate = (1+ APR/12)^12

Inflation

• Prices of goods and service fluctuate

• Fluctuations are normally upwards

• An overall rise in prices is termed inflation

• Increasing prices means that purchasing power is eroded

• Increasing prices means that interest earned on bank deposits is less valuable

Nominal and Real interest rates

• Ninth birthday present is £20

• 1 year nominal interest rate is 5%

• Grows to £21 at the end of the year

• Inflation is 10%

• Guinea pigs have increased in price from £20 to £22

• No guinea pig!

Real Interest Rates

• Real interest rates reflect the rate at which the purchasing power of an investment increases

• 1 + real interest rate = 1 + nominal interest rate

1 + inflation rate

Ref: BMM Example 5.16 P141

References

• Fundamentals of Corporate finance, Brealey/Myers/Marcus Sixth edition Ch 5