ECON3131 - Week 3 - Interest Rates - Week 3 The Time Value of Money and Interest Rates Saving and Investment Saving Foregone consumption Typically

ECON3131 - Week 3 - Interest Rates - Week 3 The Time Value...

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Unformatted text preview: Week 3: The Time Value of Money and Interest Rates Saving and Investment Saving – Foregone consumption Typically, people will not let their money sit around in a box if they can do something with it that provides them with a return. Such opportunities exist. Investment – Addition to the stock of capital goods. Capital goods are goods that may be used to produce other goods or services in the future. Practice Are each of the following saving or investment? Bob deposits $10,000 in a money-market account. Bob buys a delivery truck for his furniture moving business. Bob pays his workers to go to trade school to learn about furniture delivery. Bob buys a used house as a rental property. Bob purchases 1000 shares of IBM stock. 4–3 Saving and Investment Savers are responsible for lending. Savers are paid interest in exchange for loaning their money to borrowers. Borrowers are able to pay back the principal and interest by investing in capital goods that generate even greater returns. Financial Markets Financial markets facilitate the lending of funds from saving to those who wish to undertake investments. Those who wish to borrow to finance investment projects sell IOUs to savers. The various forms of IOUs are known as financial instruments or securities. These IOUs are claims that those who lend their savings have on the future incomes of Primary and Secondary Markets - Review Primary Market – Market in which newly issued financial instruments are purchased and sold. For instance, an Initial Public Offering. Secondary Market – Market in which instruments that have not reached maturity (the time the final interest and principal payments are due) are traded. Money Markets Money Markets – Markets for financial instruments with short-tem maturities. High liquidity, lowrisk. Treasury bills (U.S. government debt) Commercial paper (short term debt by corporations) Certificates of deposit (deposits that cannot be withdrawn without penalty). Capital Markets Capital Markets – Markets for financial instruments with intermediate and long-term maturities. Equity (shares of ownership) Common stock (entitles the shareholder to say in operation of the company – common stock holders are residual claimants) Preferred stock (no say in the operation of the company, but receive priority over common stock holders) Capital Markets Long-term treasury notes and bonds. Corporate bonds Municipal Bonds Mortgage backed securities (Oh, will we talk about these more later. Yes we will.) Packaged consumer and commercial loans CRITICAL TAKEAWAY In this lecture we will discuss the basics of calculating of present and future value. Failure to have a solid understanding of PV and FV will have a strongly negative impact on your performance in the course. Besides, this is one of the things you can honestly benefit from. 4–10 Calculating Interest Yield Principal: The amount of credit extended when one makes a loan or purchases a bond. Example: Assume I bought a house for $164,000, and borrowed $161,000. That $161,000 is principal. Interest: The price paid by a borrower for the use of an asset they do not own. Interest rate: The percentage return, or percentage yield, earned by the lender. Why Pay Interest? There is a market for loanable funds. What determines a borrower’s maximum willingness-to-pay for interest on a loan? What determines a lender’s minimum willingness-to-accept for interest on a loan? 4–12 Consumption How much house will you buy at 4.25% interest? 14% interest? How much will you finance a car at 0% interest? 12.99% interest? How much will you put on your credit card at 8.99% interest? 24.99% interest? 4–13 Saving/Investment How much will you save if interest rates are 0.25%? 4.00%? Will you be more or less likely to invest in equipment upgrades if the cost of borrowing is 3% or 10%? How will you allocate your portfolio if savings accounts pay 0.50% and stocks average 11%? What if savings accounts pay 4.50% and stocks average 7%? 4–14 Supply Curve Slopes upwards because as interest goes up, more people want to lend more funds. Interest Rate The interest rate is the cost of borrowing and/or revenue from lending. R* Demand Curve Slopes down because as interest goes down, borrowing becomes cheaper. Q* Quantity of Funds 4–15 Present and Future Value We’ll start with the concept of present (discounted) value. This is based on the notion that $1.00 you receive one year from now is less valuable to you than $1.00 you receive today. This is true for three reasons, that help determine the supply of and demand for loanable funds: 1) Time preference 2) Foregone interest 3) Expected inflation. Time Preference The premium a consumer places on present consumption relative to future consumption. Neoclassical theory: The interest rate determines the price of present and future consumption, so the marginal rate of substitution between present and future consumption must equal the interest rate. In plain(er) language: The highest interest rate you’re willing to borrow at or the lowest interest rate you’re willing to lend at should make you indifferent between borrowing and not borrowing, or lending and not lending. You should be exactly compensated for changing the timing of your consumption. 4–17 A Simple Loan Lender provides borrower with principal. A gives B $100 today Borrower repays the principal at the maturity date along with some additional interest payments. B pays A $100 + $10 one year from today The simple interest rate equals interest payments divided by the amount of the loan. i = $10/$100 = 0.1 = 10% 4–18 A Simple Loan If you had $100 to lend today at i = 10%, you receive later: $100 (1 + i) = 100(1 + 0.1) = $110 If you relend $110 next year at i = 10%, you receive at the end of the second year: $100(1 + 0.1)(1 + 0.1) = $100(1 + 0.1) 2 = $121 ….$100(1 + 0.1)3 = $133 Generalizing, at the end of n years, $100 will grow into $100(1 + 0.1)n. The PV of $133 in 2013 is $100. Almost all finance formulas involve the compounding of interest. You earn interest on past interest. 4–19 Compounding “The most powerful force in the universe is compound interest.” – attrib. to Albert Einstein Imagine a checkerboard. We put one penny on the first square, two pennies on the second square, four on the third square. How much will the last square be worth? That’s a 100% interest rate (doubling each period) for 64 periods (64 squares). $0.01*(1+1.00)64 4–20 Present And Future Value FV = PV * (1 + i)n <- Memorize this! PV = FV / (1 + i)n <- This too! PV – present value – what you’d have to invest today to yield a given future return FV – future value – what you’d receive in the future given an investment today i – simple interest rate n – number of periods PV of $200 to be paid in 4 years at i = 5%? PV = $200/(1 + 0.05)4 = $164.50 4–21 The General Form of the Present-Value Formula FN F1 F2 ... P . 1 2 N (1 i ) (1 i ) (1 i ) Each F is a payment in the future. The subscripts 1,2,N etc. represent the number of time periods out from the present. Present Value of a Perpetuity Perpetuity = financial security that never matures. It pays interest forever, does not repay principal (eg. a share of stock in a corporation) Example: N = 1 year F = $10,000 Perpetuity Time (years) Payment 0 1 2 3 4 . . . F F F F . . . Perpetuities (cont’d) Would you ever want to own a perpetuity? Is the present value really infinite? How much would you pay for it? To calculate: P = F / i The present value of each successive payment is less than the last…Therefore, even the present value of a perpetuity is finite. Perpetuities (cont’d) Present value of perpetuities are also affected by the rate of discount. Compare the sensitivity of P to i where F = $1,000 i = .08 P = $12,500 i = .05 P = $20,000 i = .02 P = $50,000 Fixed Payment Securities Fixed-payment security: The dollar payments are the same every year so that the principal is amortized Amortization: The process of repaying a loan’s principal gradually over time Using Present Value Formula to Calculate Payments Sometimes we already know present value, but are concerned with size of payments to be made to the lender… Example: Mortgage loan P = $100,000 monthly payments for 30 years (N = 360) annual interest rate = 6% (therefore i = . 06/12 = 0.005) Using Present Value Formula to Calculate Payments so Real World Applications My car died over the summer after 10 years of faithful service. So I went to buy a new one that cost about $25,000. Customer incentives were available. 0.0% financing for up to 60 months $750 off the price of the car. 4–29 The Car Example Which one’s better. You can have a 0% interest rate for 60 months or $750 off the price of the car. Assume your credit qualifies you for 4% interest rate normally. I want to convince you that knowing present and future value will actually help you outside of coursework. 30 The Car Example If you take the 0.0% rate, the car costs $25,000. Since there is no interest, the final cost over 5 years is $25,000. If you take the 4.0% rate, the car costs $24,250 after the $750 cash back. Recall the formula to calculate payments. Over 60 months, the car would cost $26,793.41 (60 payments of $446.56). Thus, taking the 0.0% interest would make more sense…or does it. 31 The Car Example There are other considerations. For example, what if you had a down payment. In fact, to be extreme, what if you had $25,000 and could buy the car outright. It would cost $25,000 if you financed it at 0% and $24,250 if you took the cash back option. You’d want the cheaper car, right? Well, imagine you had a savings account that paid 1.5% interest per year, compounded annually. If you bought the car for $24,250, you’d have $750 to put into that account. At the end of 5 years, you’d have: F The actual interest earned on the account if you put the $25,000 into it and made payments at 0% interest for 60 months would be very hard to calculate as $416.67 would be pulled out each month (the payment on a $25,000 loan for 60 months at 0% interest ($25,000/60). So let’s approximate. This will come in handy later. 32 The Car Example If you’re paying off zero interest loan at a constant rate, the principal remaining looks like a right triangle. Or assume you took $100 out of the ATM each morning and spent it at a constant rate all day. The amount left in your pocket would again look like this triangle. So what’s the average amount you’d have over the course of the day? $50. The height of the midpoint of the triangle. You’d have more than $50 half the time and less than $50 half the time. 33 The Car Example So if the loan is $25,000, then on average, you owe $12,500 (half the loan). To approximate, let’s assume that on average, you’d have about $12,500 in your savings account (which would be true if you didn’t reinvest your interest earned in that account). Interest earned on $12,500 over 5 years at 1.5% = $966.05. It’s still worth financing even if you could pay cash, because you could hold onto your money and make more than $807.96 investing it into a savings account. 34 The Car Example So when might you rather take the $750 cash back? No profitable alternative option for your money. If you couldn’t earn interest on your money that would add up to more than the $750 saved plus interest on it, you’d rather take the $750 off. You’re just debt averse. Economists are not all about money. Say you owe your mother $1,000 at 0% interest and a credit card company $1,000 at 20% interest. It would clearly minimize your payments to pay down the 20% interest loan first, but if the guilt of borrowing from your mother hurts worse than paying 20% interest, paying your mother first is rational. 35 Practice Problem Say you are interested in a Nissan Altima 3.5SR that you’ve negotiated a price of $29,000 for (all taxes and fees included). You can get 0% financing for up to 60 months, or get 3% financing for up to 60 months and get $2,000 cash back. Assume there is no alternative investment option and you can not make a down payment. Which is better? 36 Present Value of a Coupon Bond Coupon Bond: Pays a regular interest payment until maturity, when face value is repaid (e.g. most corporate & government bonds) Time (years) 0 Payments Interest Face Value 1 2 3 4 . . . N F F F F . . . F V Present Value of a Coupon Bond (cont’d) To calculate present value: interest payments are fixed-payment security present value of face value Here’s an Example Three year bond. 5% interest. Face value of $10,000. Annual coupon payments of $600. To calculate the present value (price of the bond), we calculate the present value of ALL PAYMENTS and sum them. Payment 1: Coupon payment of $600 in one year. $600/1.05. Payment 2: Coupon payment of $600 in two years. $600/1.05 2 Payment 3: Face value of $10,000 plus $600 coupon in three years: $10,600/1.053 $571.43 + $544.22 + $9,156.68 = $10,272.33 4–39 Payments More Than Once Per Year Many securities require payments more frequently Semi-annually: Government & corporate bonds Quarterly: Many stock dividends Monthly: Consumer & business loans Because of compounding, this frequency must be accounted for in calculating present value Payments More Than Once Per Year Time period needs to be adjusted to account for payment frequency Assume that interest compounds each period and N = number of periods to maturity Example: 30 year mortgage at 9% N = 360 (12 months x 30 years) i = 0.0075 (0.09/12 months) Semi-annual coupons $10,000 face value. 5% interest. Three years to maturity. $300 semi-annual coupon payment. Payment Payment Payment Payment Payment Payment 1: 2: 3: 4: 5: 6: $300 in 6 months $300 in 1 year $300 in 18 months $300 in 2 years $300 in 30 months $10,300 in 3 years PV = $300/1.025 PV = $300/1.025 2 PV = $300/1.025 3 PV = $300/1.025 4 PV = $300/1.025 5 PV = $10,300/1.025 6 $292.68 + $285.54 + $278.58 + $271.79 + $265.16 + $8,881.66 = $10,275.41 Compare to $571.43 + $544.22 + $9,156.68 = $10,272.33 Why a little higher with the semi-annual coupons? 4–42 Present Value & Decision Making Comparing alternative offers A magazine subscription costs $50 for 1 year or $95 for 2 years. Which is better? Comparing coupon bonds: use one as an alternative for the other; use the interest rate on one bond as the rate of discount on other bonds in the secondary market and see if you get the same present value. Interest-Rate Risk Why does the price of a security change when the market interest rate changes? This uncertainty is interest-rate risk Reflects a change in opportunities… suppose you buy a bond paying 6% but market rates rise to 8%. Does your bond price rise or fall? A Little Emphasis Prices of existing bonds are INVERSELY related to changing market interest rates. 4–45 Think about it You purchase a three-year bond with a face value of $10,000 and a $500 annual coupon payment and the market interest rate is 5%. PV = $500/1.05 + $500/1.052 + $10500/1.053 PV = $476.19 + $453.51 + $9070.30 = $10,000.00 Now, assume market interest rate falls to 3% immediately after purchase. PV = $500/ 1.032 + $500/ 1.032 + $10500/ 1.033 PV = $485.44 + $471.30 + $9608.99 = $10,565.72 Why does the bond price increase? When the interest rate falls to 3%, an asset with a locked-in interest rate of 5% looks more attractive. It would take $10,565.72 invested now at 3% to yield the same present value as that $10,000 face value bond with $500 coupons at 5%. 4–46 Think about it You purchase a three-year bond with a face value of $10,000 and a $500 annual coupon payment and the market interest rate is 5%. PV = $500/1.05 + $500/1.052 + $10500/1.053 PV = $476.19 + $453.51 + $9070.30 = $10,000.00 Assume market interest rate rises to 7% after year 1. PV = $500/1.07 + $500/ 1.072 + $10500/ 1.073 PV = $467.29 + $436.72 + $8571.13 = $9,475.14 Why does the bond price decrease? Because after the first year, you have this 5% bond, but you could be earning 7% elsewhere. It only takes $9,475.14 invested today to earn the same as your $10,000 face value bond at 5% with $500 coupons, so people aren’t willing to pay more and you’re not willing to sell for less. That’s the price. 4–47 Interest-Rate Risk (cont’d) Bond price = present value of bond Present value of bond inversely related to i Bond price inversely related to i Interest-Rate Risk (cont’d) Another Interest Rate Change Example A $1000 bond matures in one year and pays $50 interest. Other 1-year bonds also have interest rates of 5%. P = $1050/1.05 = $1000 What if market interest rate fell (just after you bought it) to 4%? P = $1050/1.04 = $1009.62 Interest-Rate Risk (cont’d) Example (continued) What if the market interest rate rose (just after you bought it) to 10%? P = $1050/1.10 = $954.55 Why did P change? Market opportunity! Policy Insight: Annual Percentage Yield (APY) Annual Percentage Yield = The annual interest rate that would give you the same amount you would earn with more frequent compounding than with the stated annual interest rate The U.S. government requires banks to report APY on savings. APY offers a way to compare investment with different periods of compounding. Example: Which is better to invest in? A: 8.0% compounded annually B: 7.95% compounded monthly Policy Insight: Annual Percentage Yield (APY) (cont’d) Example (cont.) A: $1000 × 1.081 = $1080 B: $1000 × [1+(.0795/12)]12 = $1082.46 Option B is a better investment To compare easily, define: APY = [1 + (i/x)]x – 1 where compounding occurs x times per year APY(A) = .08; APY(B) = .08246. Different Concepts of Interest Yield Coupon return: A fixed interest return that a bond yields each year. Nominal yield: The coupon return on a bond divided by the bond’s face value; rn=C/F. Current yield: The coupon return on a bond divided by the 4–53 Example Coupon return: $600 annually on our 5%, $10,000 face value bond with a price (PV) of $10,272.33. Nominal yield: The coupon return on a bond divided by the bond’s face value; rn=C/F. $600/$10,000 = 6% Current yield: The coupon return on a bond divided by the bond’s market price; rc=C/P. $600/$10,272.33 = 5.84% 4–54 Capital Gains Capital gain: An increase in the value of a financial instrument at the time it is sold as compared with its market value at the date it was purchased. I buy a house for $161,000. I sell it for $195,000 in 4 years. Capital gain? $34,000 Professor Price buys a house in 2006 in Reno for $310,000. In 2009, it is worth $150,000. What is the capital loss on sale? 4–55 Yield to Maturity Yield to maturity: The rate of return on a bond if it is held until it matures, which reflects the market price of the bond, the bond’s coupon return, and any capital gain from holding the bond to maturity. Perpetuity: A bond with an infinite term to maturity. Perpetuity price = C/r. Simple rule: Prices of existing bonds are inversely related to changing market interest rates. Higher interest rates causes bond prices to fall. 4–56 Back to Yield to Maturity Interest rate that equates the PV of future cash flow payments received on a debt instrument with its value/price today. Used to compare the returns on alternative credit market/debt instruments. What are these instruments? In terms of the timing of their cash flow payments, there are four basic types. 4–57 Simple Loan (already mentioned) Lender provides the principal Borrower repays principal plus interest at maturity. Examples include commercial business loans Example: P = $100. P + interest = $110 next year. PV = FV / (1 + i)n $100 = $110 / (1 + i) i = 10% 4–58 Fixed-Payment Loan Lender provides borrower with principal Borrower repays some fixed amount every period Periodic payments consist of both principal and interest. Common examines: auto loans, mortgages Example: Loan = $1000 Payments = $85.81 for 25 years YTM sets PV of all future payments equal to the value of the loan today. $1000 = $85.81/(1 + i)1 + $85.81/(1+i)2 + … $85.51/(1+i)25 i = 7% 4–59 Coupon Bond Borrower pays the owner of the bond a fixed interest payment (coupon payment) every period (i...
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