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# Shadows grow taller until the sun sets

9 February 2021 (latest revision: 27 August 2021) The Invisible Hand

Stock markets are at all-time highs while future economic prospects appear temperate. That may seem weird and unnatural, perhaps even surreal, but it is the Invisible Hand doing its job. It is partly a consequence of the sinking interest rates. Asset values go up when interest rates go down because of discounting. It is like shadows growing taller when the sun is setting. The elevation of the sun represents the interest rate, while the shadows represent the asset values.

Discounting is determining the present value of future payments using the interest rate.1 If interest rates are above zero, then € 1,000 now is worth more than € 1,000 in the future. If you receive the money now, you can put it in a bank account, receive interest, and have more money in the future. For instance, if the interest rate is 3%, then € 1,000 in the present is worth € 1,030 after a year. And so the present value of € 1,030 after a year is € 1,000.

So how much is a cash flow of € 1,000 in a year worth in the present? That is the reverse calculation. The formula for the present value of a single future cash flow is:

Present Value = Future Cashflow / (1 + (Interest Rate / 100)) ^ Number of Years

If you have multiple future cash flows, then you can add up their present values. An example can illuminate this. Assume that the interest rate on government bonds is 3%, and you own a 5% government bond that still has two years to go before the principal of € 1,000 will be repaid. You will also receive € 50 in interest after one year and another € 50 in two years when the principal is due.

If you plan to sell the bond today, you want to know its present value. There are two cash flows. You will first receive € 50 after one year. The present value of that cash flow is: € 50 / (1 + (3 / 100)) ^ 1 = € 48.54. After two years, you will receive an additional € 1,050. The present value of that amount is: € 1,050 / (1 + (3 / 100)) ^ 2 = € 989.73. And so the present value of the bond is € 48.54 + € 989.73 = € 1,038.27.

At higher interest rates, the value of the bond declines. If the interest rate is 5%, its present value is (€ 50 / (1 + (5 / 100)) ^ 1) + € 1,050 / (1 + (5 / 100)) ^ 2 = € 47.62 + € 952.38 = € 1000 exactly, which is to be expected. At lower interest rates the bond will be worth more. At an interest rate of 2% the present value is (€ 50 / (1 + (2 / 100)) ^ 1) + € 1,050 / (1 + (2 / 100)) ^ 2 = € 49.02 + € 1,009.23 = € 1,058.25.

It is also possible to calculate the present value at a negative interest rate. If the interest rate is -2%, the present value of the bond is: € 50 / (1 + (-2 / 100)) ^ 1 + € 1,050 / (1 + (-2 / 100)) ^ 2 = € 51.02 + € 1,093.29 = € 1,144.31. It means that the € 1,100 you are going to receive in the future is worth € 1,144.31 in the present. In other words, investors are willing to pay € 1,144.31 now to receive € 1,100 in the future!

At lower interest rates, bonds are worth more. That also happens to other assets that provide cash flows like stock and real estate. The value of the future rises when interest rates decline. The present value of future dividends and rents rises when interest rates go lower, even though these payments are less secure than bond coupons. That may be one of the reasons why stock prices and values of real estate have risen.

When the sun sets, shadows grow taller. Imagine a € 1,000 government bond yielding only 0% that pays back the principal over 100 years while the interest rate on government bonds is -3%. The present value of that bond is: € 1,000 / (1 + (3 / 100)) ^ 100 = € 21,029.

Despite zero returns, investors are willing to pay 21 times the amount they will receive. That seems crazy and unlikely to happen, but it demonstrates why investors may be willing to bid up stock prices. Faced with such losses on long-term debts, investors flock to stocks, real estate, gold, and bitcoin. So is the economy about to go dark once interest rates go negative? Perhaps not.

Until now, banks put money into circulation with interest, so borrowers must return more than the principal. This extra money is often created out of thin air, producing inflation. With negative interest rates, the opposite may happen. Money may disappear from circulation, and inflation goes into reverse. So if you invest in gold or bitcoin, you may lose money too in the long run. The prices of real estate and stocks may go down over time, but rents and dividends may cover the losses.

And so € 1,000 in 100 years may buy a lot more than it does now. If the deflation rate is 3%, there is no loss of purchasing power at all if you pay € 21,029 to get € 1,000 in 100 years. It may seem strange, and it is a unrealistic, but it shows that the economy can do fine with negative interest rates when prices go down.

1. Discounting. James Chen (updated 5 December 2020). Investopedia. https://www.investopedia.com/terms/d/discounting.asp

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