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 £ (10% interest rate) Would swap £100 today for £ tomorrow, but not less Why do banks pay interest?

Ninth Birthday Present - £20 Option1 Invest in the 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 = x r = 100 x (1+r) SoFuture Value (FV) = Initial Investment x (1+r)

What if I invest for longer than a year? Compound interest is interest earned on interest Reminder - Future Value = FV Example r=8%, Initial Investment = £100 FV after 1 year = £100 x (1.08) = £108 Assume that I reinvest for another year FV after 2 years = £108 x (1.08) = £ Or FV after 2 years = 100 x (1.08) x (1.08) = £ FV after 3 years= 100 x (1.08)x(1.08)x(1.08) = £ FV after t years = Initial Investment x (1+r)^ t

Compound Interest Interest in 1 st year = £8 Interest in 2 nd year = £8 + £8.64 Why? Interest is earned on both the initial investment and the interest which has accrued over previous years Earning interest on interest is called compounding

Class Example Warren Buffett took control of Berkshire Hathaway in January 1965 when the share price was USD 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) = To produce £110 in 1 years time, we need to invest £ 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 = We would need £94.30 to produce £110 after 2 years Generalising For 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 factor 1/(1+r)^ t

Present Value of a future cashflow of £100 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 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 Value Option 1 PV = £13,000 Option 2 PV immediate payment = £5,000 PV second payment = £4500/1.08 = £4167 PV third payment= £4000/1.08^ 2 = £3429 Total Option 2PV= £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 Annuities Assume a constant payment of £1 over the next n years beginning in 1 years time Assume that interest rate = r PV =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 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 = /(1/.01 – 1/.01(1.01)^300) = £1369

Amortizing Loans 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 Annuity Assume a constant payment of £1 over the next n years beginning in 1 years time Assume that interest rate = r This time, we need the future value at time n FV =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 - 1 Divide both sides by r FV = (1+r)^ n - 1 r 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 = £ % 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