Basic Financial Calculations

Discounting Future Cash Flows

Stocks, bonds, and other instruments that are discussed in the business section of newspapers and on financial web sites represent claims on future cash flows. If I buy a stock or a bond, then I receive payments in the future. The most basic concept for determining the values of those cash flows is called discounting.

If you ask me whether I would prefer $100 today or $100 a year from now, I will say that I want the $100 today. If I can earn 5 percent interest on money in a savings deposit, then with $100 today I could put the money in a savings account and have $105 a year from now.

If the interest rate is 5 percent, then I would view $105 a year from now as being equivalent to $100 today. Another way of saying this is that the discounted present value of $105 a year from now is $100. When we want to know what a future cash flow is worth today, we calculate its discounted present value.

To calculate the discounted present value of a cash flow to be received one year from now, divide by (1+i), where i is the interest rate expressed as a decimal. If the future cash flow is $105 and the interest rate is 5 percent, or .05, then the discounted present value is $105/(1.05) = $100.

What is the value of a cash flow of $105 that you will receive two years from now? Assuming the same interest rate of 5 percent, we discount twice. That is, we take $105 and divide by 1.05, and then divide by 1.05 once more, for a discounted present value of $95.24. More generally, if the interest rate is constant, we have for a cash flow C that will arrive t years from now,

discounted present value = C/(1+i)t

You may remember the way that compound interest behaves. If you have a savings balance of $100,000 and the interest compounds annually, then after six years of earning interest at a rate of 5 percent per year, your balance will equal
$100,000 (1.05)6 = $134,009.60. Discounted present value is like compounding, except that you work backwards in time. Instead of taking compound interest from today to calculate a value in the future, you start with a value in the future and discount back to the present.

Calculate the discounted present value of a payment of $100 two years from now, if the interest rate is 8 percent per year.

Calculate the present value of a payment of $100 three years from now and a payment of $200 five years from now, if the interest rate is 6 percent per year.

Does a cash flow of $100 to be received eight years from now have a discounted present value that is lower or higher than a cash flow of $100 to be received four years from now? In general, what effect does the length of time until you will receive a cash flow have on the present value of that cash flow?

For a cash flow of $100 to be received one year from now, will the discounted present value be higher if the interest rate is 5 percent or the interest rate is 10 percent? In general, what effect does a higher interest rate (sometimes called the discount rate) have on the value of a future cash flow?

Forward Interest Rates and the Yield Curve

If you check interest rates in the newspaper, you may find that the interest rate on a ten-year bond is 5 percent, while the interest rate on a one-year bond is only 3 percent. Financial pundits refer to the different interest rates for different time periods as the yield curve.

The yield curve consists of the immediate short-term interest rate as well as short-term interest rates that are expected in the future. The latter are called forward interest rates.

Here is an example of a simple two-year yield curve, consisting of the current one-year rate and next year's forward rate. Suppose that the interest rate this year is 4 percent, and the forward rate is 6 percent. What is the discounted present value of $100 to be received two years from now?

To discount $100 back to one year from now, we take $100/(1.06) = $94.34. To discount this back to the present, we take $94.34 and divide by 1.04, to obtain $90.71.

If the current one-year rate is 4 percent, the one-year forward rate is 6 percent, and the next year's forward rate is 5 percent, what is the present value of $100 to be received three years from now?

A ten-year bond pays a single interest rate for its entire term. This interest rate is something like the average of the current one-year rate and the forward rates for the following nine years. Technically, it is closer to a geometric weighted average than an arithmetic weighted average.

On July 3, 2002, the interest rate on 10-year notes issued by the U.S. Treasury was 4.75 percent. The rate on two-year notes was 2.77 percent, and the rate on the three-month bill was 1.68 percent. Thus, interest rates were much higher on long-term bonds than on short-term bonds. We say that the yield curve was steeply upward-sloping. If long-term rates are only modestly higher than short-term rates, then we say that the yield curve is mildly upward-sloping (which is normal). When long-term rates are below short-term rates, we say that they yield curve is inverted.

For this course, you will not need to know anything about doing calculations involving the forward rate and the yield curve. For teaching purposes they make things unnecessarily complicated. However, for investors on Wall Street, the forward rate and the yield curve matter a lot. The real world, unfortunately, is complex.

Interest, Rent, and Capital Gains

Suppose that there are two identical condominiums, one of which is for rent with the other one for sale. Financially, will it be to your advantage to live in the rental or to buy the other condo?

We can think in terms of borrowing the money to buy the condo and then selling it after one year. Suppose that it costs $200,000 and that after one year we can sell it for $204,000. Houses suffer from wear and tear, like lawnmowers. However, unlike lawnmowers they tend to increase in value. This is because of general inflation as well the fact that land is scarce and tends to become more valuable over time. An increase in the value of an asset is called a capital gain.

If the interest rate is 6 percent, then our cost will be
$200,000 (1.06) - $204,000 = $8000. If instead we paid $8000 in rent, that would work out to a rent of $666.67 per month. Therefore, if the rental is for $700 per month, then it would be better to buy the condo. If the rental is $600 a month, we would be better off living in the rental. (To keep things from getting too complicated, I am leaving out some factors that matter in the real world, including taxes and the closing costs involved in buying and selling a home.)

We could arrive at the same rent by looking at the condo from the perspective of an investor. Suppose that we are thinking of buying the condo and renting it out to someone else. If we can rent the condo for more than $8000 per year, then we can make a profit by buying it for $200,000. Otherwise, we cannot.

Recall the formula that we use for the profitability of owning a capital asset:

profitability = rental rate + appreciation - interest rate

The general relationship between interest, rental income, and capital gains is

Using i to stand for the interest rate, r to stand for the rental rate (the ratio of rent to price) and p to stand for the rate of capital gain (the average annual rate of appreciation), we have

profitability = r + p - i

In our example, the ratio of rent to price, r is $8000/$200,000 = .04, the interest rate, i is .06, and , the rate of capital gain, p is $204,000/$200,000 = .02. Thus, we have

profitability = .04 + .02 - .06 = 0

When profitability is zero, we are indifferent between owning and renting. There is a tendency for the price of an asset to adjust up or down so that owning and renting provide equivalent net benefits. In terms of our formula, there is a tendency for profitability to be zero. If profitability were clearly positive, people would bid up the price of the asset, which causes the ratio of rent to price (the rental rate) to go down, which brings profitability back toward zero. The opposite would happen if profitability were clearly negative.

Common Stock and the Price/Earnings Ratio

The basic relationship between the interest rate, the rental rate, and the capital gains rate that holds for a condo also holds for other capital assets. For example, for Josh's lawn mowing business, the "rent" that he derives from an additional lawnmower is equal to the value that he gets from the increase in lawns mowed. The capital gain (in this case a loss) on the lawnmower is equal to its rate of depreciation.

If you buy shares of stock, the regular income that you receive in lieu of rent consists of dividends. The ratio of a stock's dividends to its price can be used as r in the basic equation relating i, r, and p. For example, on Wednesday, July 3, 2002, the stock of Freddie Mac closed at $59.30 a share. It paid a dividend of $.88 per share, for a dividend rate or r of .0148, or 1.48 percent. If investors were using the interest rate on the 10-year Treasury note as a benchmark for pricing Freddie Mac stock, then we would set i = 4.75 percent. That means that the expected rate of capital gain on Freddie Mac stock, p, would have to equal
4.75 - 1.48 = 3.27 percent.

A stock does not have to pay dividends in order to be valuable. If the company is profitable, it brings in more in revenue than in expenses. These profits are called earnings. If a company does not distribute earnings as dividends, it can use them in other ways to enhance shareholder value. For example, a company can go into the market and buy back its own shares, increasing the demand for the stock and raising its price.

Some companies pay relatively high dividends, and others pay relatively low dividends. However, if their earnings are similar, investors would see the stocks as having similar value. Therefore, many investors prefer to use the ratio of earnings per share in place of the ratio of dividends per share. Thus, earnings per share becomes r and the overall rate of inflation becomes p in the basic equation. Economist Edward Yardeni calls this the "Fed model," because he believes that the Federal Reserve Board uses this equation to determine whether the stock market as a whole is overvalued, undervalued, or valued correctly.

[Fed model] 10-year interest rate = earnings/price ratio + inflation

On July 3, 2002, the ten-year note rate was 4.75 percent, and overall inflation appeared to be around 1 or 2 percent. Using 1.5 percent inflation, the earnings/price ratio should be:

4.75 - 1.50 = 3.25 percent

In the newspaper, what gets reported is the inverse of the ratio of earnings to price. That is, the financial press reports the ratio of price to earnings (P/E). Therefore, if the Fed model tells us that on July 3 the earnings/price ratio should have been 3.25 percent, or .0325, then the P/E ratio should have been the inverse of that, or 1/.0325, which is about 30. In fact, the market P/E ratio was slightly below 30, so that on July 3, 2002, the Fed model suggested that stocks were undervalued.

The P/E ratios for individual stocks can be all over the map. For example, on July 3, the P/E for Freddie Mac was just 9. For Coca-cola, the P/E ratio was 47. When a P/E ratio is low, that is because investors do not expect earnings to grow as fast as the overall economy. When a P/E ratio is high, investors think that earnings for the company can grow faster than the economy as a whole. For this reason, stocks with high P/E ratios are called "growth stocks" and stocks with low P/E ratios are called "value stocks."