Chapter Four Future Value Present Value and Interest

Chapter Four Future Value, Present Value, and Interest Rates Chapter 4

Learning Objectives Develop an understanding of 1. Time and the value of payments 2. Present value versus future value 3. Nominal versus real interest rates • Interest rates link the present to the future. – Tell the future reward for lending today. – Tell the cost of borrowing now and repaying later.

Valuing Monetary Payments Now and in the Future • We need a set of tools: – Future value – Present value

Future Value and Compound Interest • Future value: the value on some future date of an investment made today. – $100 invested today at 5% interest gives $105 in a year. • So the future value of $100 today at 5% interest is $105 one year from now. – The $100 investment “yields” $5, which is why interest rates are called yields. • This is an example of a simple loan of $100 for a year at 5% interest.

Future Value and Compound Interest Formula – • If the present value (PV) is $100 and the interest rate (i) is 5%, then the future value (FV) one year from now is: $105 = [$100 + $100(0. 05)] = $100(1 +. 05)= $100(1. 05) • The higher the interest rate, the higher the future value. • In general: FV = PV + PV x i = PV(1 + i)

Future Value and Compound Interest • Most financial instruments are not this simple. • We must consider compound interest to compute the value repaid more than one year from now. • Compound interest is the interest on the interest.

Future Value and Compound Interest • Suppose you deposit $100 (PV) in a bank saving account for two years at 5%(i) yearly interest rate? • The future value is: FV = $100 + $100(0. 05) + $5(0. 05) = $110. 25 FV = $100(1. 05)2 • In general: FVn = PV(1 + i)n, where n is time.

Future Value and Compound Interest Computing the future value of $100 at 5% annual interest

Future Value and Compound Interest • i and n must be in the same units. – If n is annual, i must be annual • If the annual interest rate is 5%, what is the monthly rate? • Assume im is the one-month interest rate and n is the number of months, then a deposit made for one year will have a future value of PV(1 + im)12. (NOTE: i and n are monthly)

Future Value and Compound Interest • We know that in one year the future value is $100(1. 05) so we can solve for im: (1 + im)12 = (1. 05) (1 + im) = (1. 05)1/12 = 1. 00407, which is 0. 407% • Fractions of percentage points are called basis points. – A basis point is one-hundredth of a percentage point – 1 percent is 100 basis points – 0. 407% is 40. 7 basis points

Present Value • Present value is the value today (in the present) of a payment that is promised to be made in the future. • Or, present value is the amount that must be invested today in order to realize a specific amount on a given future date.

Present Value • Solve the Future Value Formula for PV: FV = PV x (1+i), so • This is just the future value calculation inverted.

Present Value • We can generalize the process as we did for future value. Present Value of payment received n years in the future: • Present value is higher: 1. The higher future value of the payment, FVn 2. The shorter time period until payment, n. 3. The lower the interest rate, i. • Present value is the single most important relationship in our study of financial instruments.

Present Value of $100 Payment • Higher interest rates are associated with lower present values, no matter what the size or timing of the payment. • At any fixed interest rate, an increase in the time reduces its present value.

Computing Compound Annual Returns • We can turn a monthly growth rate into a compound -annual rate. – Investment grows 0. 5% per month – What is the compound annual rate? (1. 005)12 = 1. 0617 Compound annual rate = 6. 17% (Note: 6. 17% > 12 x 0. 05 = 6. 0%) Monthly Rate = Com Annual Rate PV = 1 2 3 4 5 6 7 8 9 10 11 12 0. 005 1. 061678 100. 50 101. 00 101. 51 102. 02 102. 53 103. 04 103. 55 104. 07 104. 59 105. 11 105. 64 106. 17 Simple Interest 100. 5 101. 00 101. 50 102. 00 102. 50 103. 00 103. 50 104. 00 104. 50 105. 00 105. 50 106. 00

Computing Compound Annual Rates • We can also use this to compute the percentage change per year when we know how much an investment has grown over a number of years. • Suppose an investment has increased 20 percent over five years: from $100 to $120. FVn = PV(1 + i)n 120 = 100(1 + i)5 Solve for i i = 0. 0371

Computing Compound Annual Rates i = 1. 0371 -1=> i =3. 71%

Internal Rate of Return • Imagine that you own a firm and you are considering purchasing a new machine. – Machine costs $1 million and can produce 4000 units of product per year. – You sell the product for $30 per unit, generating $120, 000 in added revenue per year. – Keeping this simple, assume the machine is the only input and you have certainty about the revenue (very simple), no maintenance (very, very simple) and a 10 year lifespan.

Internal Rate of Return • Question: if you borrow $1 million to buy the machine, is the 10 year revenue stream enough to make the payments? • We need to compare internal rate of return (IRR) to the cost of buying the machine. • IRR is the interest rate that equates the present value of an investment with its cost.

Internal Rate of Return • Balance the cost of the machine against the PV of the future stream of revenue. – $1 million today versus $120, 000 a year for ten years. – Solve for i - the internal rate of return.

Internal Rate of Return: Example Solving for i, i = 0. 0346 or 3. 46% • So long as the interest rate at which you borrow money is less than 3. 46%, then you should buy the machine • Or, if IRR is greater than opportunity cost, you should buy the machine.

Internal Rate of Return: Example • Suppose you are let go from your job and your employer offers you two options: – An annual payment of $8000 per year for 30 years or – A lump sum of $50, 000 today. • Which do you take? • In the 1990 s, when the Defense Department downsized, they offered many personnel a similar deal. • NOTE – a fixed payment for a fixed number of years is called an annuity

Internal Rate of Return: Example • Economic theory suggests that a person must compare their discount rate with the current rate of borrowing or lending. • The military pamphlet gave the present value of the annuity using a 7% discount rate, the interest on money markets at the time. The PV= $99, 272.

Internal Rate of Return: Example • About 3/4 of military personnel took the lump sum, which was ½ the PV of the annuity. • Studying other separation packages, the study finds that people’s discount rate varies from 17 to 20%. • The discount rate that equates PV of the $8, 000 annual annuity to $50, 000 is 15. 8%.

Bond Basics • A bond is a promise to make a series of payments on specific future dates. • Bonds create obligations, and are therefore legal contracts that: – Require the borrower to make payments to the lender, and – Specify what happens if the borrower fails to do so.

Bond Basics • The most common type of bond is a coupon bond. – Issuer is required to make annual payments, called coupon payments (C). – The stated annual interest the borrower pays is called the coupon rate (ic). – The date on which the payments stop and the loan is repaid (n), is the maturity date or term to maturity. – The final payment is called the principal, face value, or par value of the bond.

Coupon Bond: the good-ole days Called a coupon bond as buyer would receive a certificate with a number of dated coupons attached. Principal Coupons

Valuing the Principal • Assume a bond has a principal (FV) payment of $1000 and its maturity date is n years in the future. • The present value of the bond principal is:

Valuing the Coupon Payments • The present value expression gives the formula for the string of yearly coupon payments made over n years. • The longer the payments go, the higher their total value. • The higher the interest rate, the lower the present value.

Valuing the Coupon Payments plus Principal • We combine the previous two equations to get the price of a coupon bond: • The value of the coupon bond, PCB, rises when – The yearly coupon payments, C, rise and – The interest rate, i, falls.

Real and Nominal Interest Rates • Nominal Interest Rates (i) – The interest rate expressed in current-dollar terms. • Real Interest Rates (r) – The inflation adjusted interest rate • Borrowers care about the resources required to repay. • Lenders care about the purchasing power of the payments they received. • Neither cares solely about the number of dollars, they care about what the dollars buy.

Real and Nominal Interest Rates • The nominal interest rate you agree on (i) must be based on expected inflation ( e) over the term of the loan plus the real interest rate you agree on (r). i = r + e • This is the Fisher Equation. • The higher expected inflation, the higher the nominal interest rate. • This equation is an approximation that works well when expected inflation and the real interest rate are low. • Exact formula: (1 + i) = (1 + r)(1 + πe) (1 + i) = 1 + r + πe +r πe • Subtract 1 from each side and ignore the cross-term.

Inflation and Nominal Interest Rates Mankiw

Inflation and Nominal Interest rates

Nominal Interest Rate, Inflation Rate and Real Interest Rate

Real and Nominal Interest Rates • Financial markets quote nominal interest rates. • When people use the term interest rate, they are referring to the nominal rate. • The real interest rate is estimated using the Fisher equation: r = i - e

Real and Nominal Interest Rates • Ex ante real interest rate is adjusted for expected changes in the price level (πe) • Ex post real interest rate is adjusted for actual changes in the price level (π) • Fisher Equation: i = r + πe • From this we get rex ante = i - πe rex post = i - π

Real and Nominal Interest Rates Real Interest Rate - interest rate that is adjusted for expected changes in the price level r = i –πe if i = 5% and πe = 3%; r = 5% - 3% = 2% if i = 8% and πe = 10%; r = 8% - 10% = -2%

A measure in inflationary expectations e π i=r+ e =i-r π http: //www. bloomberg. com/markets/rates-bonds/governmentbonds/us/ http: //research. stlouisfed. org/fred 2/

Algebra - Annuity • To compute the payment, we will use the present-value formula. If we call the size of the monthly payments C, then we need to solve the following formula: