Understanding the Concept of Present Value

Interest Rates, Compounding, and Present Value In economics, an interest rate is known as the yield to maturity. Compounding is the process that gives us the value of a sum invested over time at a positive rate of interest. Present value is the process that tells us how much an expected future payment is worth today.

Compounding Assume you have $1 which you place in an account paying 10% annually. How much will you have in one year, two years, etc? –An amount of $1 at 10% interest –Year1 2 3 n $1.10 $1.21 $1.33 $1(1 + i) n Formula: FV = PV(1 + i)

Compounding over Time Extending the formula over 2 years –FV = PV(1 + i) (1 + i) or FV = PV(1 + i) 2 3 years –FV = PV(1 + i) (1 + i) (1 + i) = PV(1 + i) 3 n years –FV = PV(1 + i) n

Present Value Present value tells us how much an expected future payment is worth today. Alternatively, it tells us how much we should be willing to pay today to receive some amount in the future. –For example, if the present value of $1.10 at an interest rate of 10% is $1, we should be willing to spend $1 today to get $1.10 next year.

Present Value Formula The formula for present value can be found by rearranging the compounding formula. –FV = PV(1 + i) solve for PV FV/(1 + i) = PV

Present Value over Time Extending the formula over 2 years –FV = PV(1 + i) 2 –PV = FV/(1 + i) 2 3 years –FV = PV(1 + i) 3 –PV = FV/(1 + i) 3 n years –FV = PV(1 + i) n –PV = FV/(1 + i) n

Things to Notice An increase in the interest rate causes present value to fall. –Higher rates of interest mean smaller amounts can grow to equal some fixed amount during a specified period of time. A decrease in the interest rate causes present value to rise. –Lower rates of interest mean larger amounts are needed to reach some fixed amount during a specified period of time.

Example: How much must I invest today to get $10,000 in five years if interest rates are 10%? PV = FV/(1 + i) n PV = $10,000/(1 +.10) 5 = $10,000/ = $6,209.2 How much must I invest today to get $10,000 in five years if interest rates are 5%? PV = FV/(1 + i) n PV = $10,000/(1 +.05) 5 = $10,000/ $7,835.15

More Things to Notice Present value is always less than future value. –(1 + i) n is positive so FV/(1 + i) n < FV In addition, PV 4 < PV 3 < PV 2 < PV 1 –(1 + i) 1 < (1 + i) 2 The longer an amount has to grow to some fixed future amount, the smaller the initial amount needs to be.

Time Value of Money The longer the time to maturity, the less we need to set aside today. This is the principal lesson of present value. It is often referred to as the “time value of money.”

Example: If I want to receive $10,000 in 5 years, how much do I have to invest now if interest rates are 10%? $10,000 = PV(1 +.10) 5 $10,000/ = $ If I want to receive $10,000 in 20 years, how much do I have to invest now if interest rates are 10%? $10,000 = PV(1 +.10) 20 $10,000/6.7275= $

Understanding Interest Rates

Yield to Maturity Yield to maturity is the interest rate that equates the present value of payments received from a debt instrument with its value today. Yield to maturity can be calculated using the present value formula. –PV = FV/(1 + i) –i = (FV – PV)/PV

Simple Example: PV = FV/(1 + i) PV(1 + i) = FV PV + PVi = FV PVi = FV - PV i = (FV – PV)/PV –$1.00 = $1.10/(1 + i) –$ $1.00i = $1.10 –i = ($ $1.00)/$1.00 = 0.10 = 10%

Relationship between Yield to Maturity and Price Yields to maturity on a 10% coupon rate bond with a face value of $1000 maturing in 10 years Price of BondYield to Maturity

Relationship between Yield to Maturity and Price Three interesting facts: –Price and yield are negatively related. –When the bond is at par, yield equals coupon rate. –Yield is greater (less than) than the coupon rate when the bond price is below (above) par value.

Current Yield In more complicated cases, yield to maturity can be difficult to calculate. Tables are available that can be used. And, of course, calculators do a fine job. There are also simple formulas that can approximate yield to maturity such as current yield.

Current Yield Current yield is an approximation for yield to maturity that is used to calculate the interest rate on a bond quickly. Formula: –Current yield = Coupon/Bond Price Current yield is a better approximation to yield to maturity, the nearer price is to par and the longer is the maturity of the security. A change in current yield always signals a change in the same direction as yield to maturity.

Inverse Relationship We can use the current yield formula to see clearly the inverse relationship between interest rates and bond prices. –Current yield = Coupon/Bond Price The coupon is a fixed payment, it does not change. Therefore, if yields rise, bond prices must fall, and if yields fall, bond prices must rise.

Intuition Assume you buy a $1,000 bond today with a fixed coupon of $100. You are receiving a 10% return. Let a year pass, and you find you want to sell you bond. You call your broker and say, “Sell!” Your broker sighs and tells you that bonds just like yours now yield 12%. What price can you expect to receive?

Example Use the current yield formula: –0.12 = $100/P B –0.12P B = $100 –P B = $100/.12 = $ You must reduce your price until $100 represents a 12% rate of return.

Yield on a Discount Basis The yield to maturity calculation for a discount bond is similar to our example of yield to maturity. A discount bond is one that is sold at a discount from its face value. The yield or interest received is determined by the difference between the price paid and its face value.

Discount Bonds Yield formula: –i = (Face - Price Paid)/Price Paid U.S. Treasury bills are sold on a discount basis. The formula used to calculate the yield is: –i = (Face - Price Paid)/Face * (360/# Days to Maturity)

Discount Bonds: Characteristics The formula used to calculate yield understates yield to maturity. The longer the maturity, the greater the understatement. A change in discount yield always signals a change in the same direction as yield to maturity.

Coupon Bond A bond is a debt instrument. A coupon bond is a bond that pays its owner a fixed coupon payment every year until maturity, at which time a specified final amount (face value) is repaid –We expect to get: Coupon payments each year Principal at maturity. How much should we be willing to pay today for a stream of income?

Coupon Bond P B = C/(1 + i) + C/(1 + i) 2 + C/(1 + i) 3 + ……..C/(1 + i) n + P/(1 + i) n –where C is a fixed coupon i is the rate of interest P B is the price or present value of the bond P is the principal n is years to maturity

Coupon Bond Example Let the coupon payment be $100, the rate of interest 10%, and the principal equal to $1000. If n is 4, how much should we pay today for this bond? –P B = 100/( ) + 100/( ) /( ) /( ) /( ) 4 –P B = 100/(1.10) + 100/(1.21) + 100/(1.331) + 100/(1.4641) /(1.4641) –P B = = –$1,000 (Note that you don’t pay $1,400).

Things to Notice When a coupon bond is priced at its face value, the yield to maturity equals the coupon rate.

More Things to Notice The yield to maturity and the coupon rate do not have to be the same. –If the bond price is less than the face value, the yield to maturity is greater than the coupon rate. In this case, the difference between the bond price and the face value adds to the total return. –If the bond price is greater than the face value, the yield to maturity is less than the coupon rate. In this case, the difference subtracts from the total return.

Even More Things to Notice The price of a coupon bond and the yield to maturity are inversely related. –An increase in the interest rate decreases the bond price. –A decrease in the interest rate increases the bond price. This is the reason the market participants are so interested in the actions of the Federal Reserve.

Interest Rates and Returns For any security, the rate of return is defined as the payments to the owner plus the change in its value, expressed as a ratio to its purchase price. The return to a bond depends on its stream of coupon payments and the price the bond receives when it is sold.

Interest Rates and Returns If the bond sells at a price in excess of its original purchase price, the owner receives a capital gain which increases his/her total return. If the bond sells at a price below its original purchase price, the owner suffers a capital loss, which decreases his/her total return.

Return on a Bond The return on a bond may be expressed by the formula: –Ret = C/P t + (P t+1 - P t )/P t where C = Coupon payment P t = Price of the bond in time t P t+1 = Price of the bond in time t + 1

Returns on Different Maturity 10% Coupon Rate Bonds TermInitial i Initial P New i New P K Gain ROR 3010% % % % % % % % % % % %

Things to Notice The only bond whose return is certain to equal the initial yield is the one whose time to maturity is the same as the holding period. A rise in interest rates is associated with a fall in bond prices, resulting in capital losses on bonds whose terms to maturity are longer than the holding period.

More Things to Notice The longer the bond’s maturity, the greater is the size of the price change associated with an interest rate change. The longer a bond’s maturity, the lower is the rate of return that occurs as a result of the increase in the interest rate. Even though the bond had a good interest rate, its return became negative when interest rates rose.

Reinvestment Risk Reinvestment risk occurs –when an investor holds a series of short bonds over a long holding period and interest rates are uncertain. If interest rates rise, the investor gains If interest rates fall, the investor loses

Reinvestment Risk: Example Assume a holding period of two years and an investor who has decided to buy two one year bonds sequentially. –Year 1 bond: Face = $1000, initial interest rate = 10% –At the end of the year, the investor has $1100. –Year 2 bond: Face = $1100, interest rate = 20% –At the end of year 2, the investor has $1320.

Reinvestment Risk: Example The investor’s two year return will be: –($ $1000)/$1000 = 0.32 = 32% over two years. –In this case the investor has benefited by buying two one year bonds. –Conversely, if interest rates had fallen to 5%, the investor would done less well.

Reinvestment Risk: Example Year 1: –($1000 x ( )) = $1100 Year 2: –($1100 x ( )) = $1155 Return = ($ $1000)/$1000 = 15.5% over two years. The investor now loses from a change in interest rates.

Real and Nominal Interest Rates Nominal interest rate is the rate of interest that makes no allowance for inflation. The real interest rate is the rate of interest that is adjusted for expected changes in the price level. –It more accurately reflects the true cost of borrowing and lending.

Fisher Equation The Fisher equation states that the nominal interest rate equals the real interest rate plus the expected rate of inflation –i n = i r +  Rearranging terms we find: –i r = i n - 

Logic behind the Inflation Premium Lenders want to be compensated for the loss in buying power due to inflation. Buyers understand that they will be repaying debt with dollars that buy less. The interest rate must reflect these facts.